Maths+quest

SPEED - II QUESTION BANK FOR IITJEE MATHEMATICS

East Delhi : No. 1 Vigyan Vihar, New Delhi. Ph. 65270275 : North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

TARGET IIT JEE

MATHEMATICS

APPLICATION OF DERIVATIVE

Time Limit : 5 Sitting Each of 80 Minutes duration approx.

Question bank on Application of Derivative Select the correct alternative : (Only one is correct) Q.1

Suppose x1 & x2 are the point of maximum and the point of minimum respectively of the function f(x) = 2x3 − 9 ax2 + 12 a2x + 1 respectively, then for the equality x12 = x2 to be true the value of 'a' must be (A) 0 (B) 2 (C) 1 (D) 1/4

Q.2

Point 'A' lies on the curve y = e − x and has the coordinate (x, e − x ) where x > 0. Point B has the coordinates (x, 0). If 'O' is the origin then the maximum area of the triangle AOB is (A)

Q.3

1 2e

1 4e

(B)

1 e

(C)

1 8e

(D)

The angle at which the curve y = KeKx intersects the y-axis is : (A) tan−1 k2

Q.4

2

2

(B) cot−1 (k2)

(C) sec−1  1 + k 4   

(D) none

n2 {a1, a2, ....., a4, ......} is a progression where an = 3 . The largest term of this progression is : n + 200

(A) a6

(B) a7

(C) a8

(D) none

Quest x

Q.5

The angle between the tangent lines to the graph of the function f (x) = ∫ ( 2 t − 5) dt at the points where 2

the graph cuts the x-axis is (A) π/6 (B) π/4 Q.6

Q.7

The minimum value of the polynomial x(x + 1) (x + 2) (x + 3) is : (A) 0 (B) 9/16 (C) − 1

(

Q.9

)

(B) 1/2

(C) 1

(D) 3

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm3/min, when the radius is 2 cm and the height is 3 cm is (B) –

8π 5

(C) –

3π 5

(D)

2π 5

If a variable tangent to the curve x2y = c3 makes intercepts a, b on x and y axis respectively, then the value of a2b is (A) 27 c3

Q.11

(D) − 3/2

 π π The difference between the greatest and the least values of the function, f (x) = sin2x – x on − ,   2 2 3 π 3 2π + + (A) π (B) 0 (C) (D) − 2 3 2 3

(A) – 2π Q.10

(D) π/2

tan x + π6 The minimum value of is : tan x

(A) 0 Q.8

(C) π/3

(B)

4 3 c 27

(C)

27 3 c 4

Difference between the greatest and the least values of the function f (x) = x(ln x – 2) on [1, e2] is (A) 2 (B) e (C) e2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

4 3 c 9

(D) 1

[2]

Q.12

Let f (x) =

tan n x 2n

∑ tan r x

 π , n ∈ N, where x ∈ 0,   2

r =0

(A) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains exactly one integral point. (B) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains more than one integral point. (C) f (x) is bounded but minimum and maximum does not exists. (D) f (x) is not bounded as the upper bound does not exist. Q.13

Q.14

If f (x) = x3 + 7x – 1 then f (x) has a zero between x = 0 and x = 1. The theorem which best describes this, is (A) Squeeze play theorem (B) Mean value theorem (C) Maximum-Minimum value theorem (D) Intermediate value theorem π x sin for x > 0  x  Consider the function f (x) = then the number of points in (0, 1) where the  0 for x = 0 derivative f ′(x) vanishes , is (A) 0 (B) 1

Q.15

Quest (D) infinite

The sum of lengths of the hypotenuse and another side of a right angled triangle is given. The area of the triangle will be maximum if the angle between them is : (A)

Q.16

(C) 2

π 6

(B)

π 4

(C)

π 3

(D)

5π 12

In which of the following functions Rolle’s theorem is applicable?

x , 0≤ x < 1  (A) f(x) =  on [0, 1] 0 , x =1

x −x −6 on [–2,3] x −1 2

(C) f(x) =

sin x  x , − π ≤ x < 0 (B) f(x) =  on [– π, 0]  0 , x =0 x 3 − 2 x 2 − 5x + 6 if x ≠ 1, on [ −2,3]   x − 1 (D) f(x) =   − 6 if x = 1

Q.17

Suppose that f (0) = – 3 and f ' (x) ≤ 5 for all values of x. Then the largest value which f (2) can attain is (A) 7 (B) – 7 (C) 13 (D) 8

Q.18

The tangent to the graph of the function y = f(x) at the point with abscissa x = a forms with the x-axis an angle of π/3 and at the point with abscissa x = b at an angle of π/4, then the value of the integral, b

∫

f ′ (x) . f ′′ (x) dx is equal to

a

(A) 1 (B) 0 [ assume f ′′ (x) to be continuous ] Q.19

(C) − 3

(D) –1

Let C be the curve y = x3 (where x takes all real values). The tangent at A meets the curve again at B. If the gradient at B is K times the gradient at A then K is equal to (A) 4 (B) 2 (C) – 2 (D) 1/4

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[3]

Q.20

π The vertices of a triangle are (0, 0), (x, cos x) and (sin3x, 0) where 0 < x < . The maximum area for 2 such a triangle in sq. units, is (A)

Q.21 Q.22

3 3 32

(B)

3 32

(C)

(D)

The subnormal at any point on the curve xyn = an + 1 is constant for : (A) n = 0 (B) n = 1 (C) n = − 2

6 3 32

(D) no value of n

Equation of the line through the point (1/2, 2) and tangent to the parabola y = the curve y = 4 − x 2 is : (A) 2x + 2y − 5 = 0 (B) 2x + 2y − 3 = 0

Q.23

4 32

The lines y = −

(C) y − 2 = 0

− x2 + 2 and secant to 2

(D) none

3 2 x and y = − x intersect the curve 3x2 2 5

+ 4xy + 5y2 − 4 = 0 at the points P and Q respectively. The tangents drawn to the curve at P and Q (A) intersect each other at angle of 45º (B) are parallel to each other (C) are perpendicular to each other (D) none of these Q.24

Quest

The least value of 'a' for which the equation,

4 1 + = a has atleast one solution on the interval (0, π/2) is : sin x 1 − sin x

(A) 3 Q.25

(B) 5

If f(x) = 4x3 − x2 − 2x + 1 and g(x) =  1  4

 3  4

(C) 7

{f ( t ) : 0 ≤ t ≤ x} [ Min 3− x

(D) 9

; 0≤ x≤1 ; 1< x ≤ 2

then

 5  4

g   + g   + g   has the value equal to : (A)

Q.26

7 4

9 4

(B)

1  Given : f (x) = 4 −  − x  2 

2/3

(C)

13 4

5 2

ta n [ x]  , x ≠0 g (x) =  x  1 , x =0

h (x) = {x} k (x) = 5log2 ( x + 3) then in [0, 1] Lagranges Mean Value Theorem is NOT applicable to (A) f, g, h (B) h, k (C) f, g Q.27

(D)

(D) g, h, k

Two curves C1 : y = x2 – 3 and C2 : y = kx2 , k∈ R intersect each other at two different points. The tangent drawn to C2 at one of the points of intersection A ≡ (a,y1) , (a > 0) meets C1 again at B(1,y2)

(y1 ≠ y 2 ) . The value of ‘a’ is

(A) 4 Q.28

(B) 3

(C) 2

(D) 1

 1 1   2 − − f (x) = ∫   dx then f is 1+ x2 1+ x2   (A) increasing in (0, ∞) and decreasing in (– ∞, 0) (B) increasing in (– ∞, 0) and decreasing in (0, ∞) (C) increasing in (– ∞ , ∞) (D) decreasing in (– ∞ , ∞) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.29

The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. The fraction of width folded over if the area of the folded part is minimum is : (A) 5/8 (B) 2/3 (C) 3/4 (D) 4/5

Q.30

A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x, and y = 30 – 2x. The largest area of such a rectangle is (A)

Q.31

135 8

(B) 45

(C)

135 2

(D) 90

x x ≥1   3 0≤ x ≤1 Which of the following statement is true for the function f ( x ) =  x   3 (A) It is monotonic increasing ∀ x ∈ R  x − 4x x < 0 3 (B) f ′ (x) fails to exist for 3 distinct real values of x (C) f ′ (x) changes its sign twice as x varies from (–∞ ,∞ ) (D) function attains its extreme values at x1 & x2 , such that x1, x2 > 0

Q.32

A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when the depth of the liquid in it is x cm, the volume of the liquid in it is, x2 (15 − x) cu. cm. The length EF is: (A) 7.5 cm (B) 8 cm (C) 10 cm (D) 12 cm

Q.33

Coffee is draining from a conical filter, height and diameter both 15 cms into a cylinderical coffee pot diameter 15 cm. The rate at which coffee drains from the filter into the pot is 100 cu cm /min. The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 10 cm, is (A)

Q.34

Quest

9 16 π

(B)

25 9π

(C)

5 3π

(D)

16 9π

Let f (x) and g (x) be two differentiable function in R and f (2) = 8, g (2) = 0, f (4) = 10 and g (4) = 8 then (A) g ' (x) > 4 f ' (x) ∀ x ∈ (2, 4) (B) 3g ' (x) = 4 f ' (x) for at least one x ∈ (2, 4) (C) g (x) > f (x) ∀ x ∈ (2, 4) (D) g ' (x) = 4 f ' (x) for at least one x ∈ (2, 4) m n

Q.35

Let m and n be odd integers such that o < m < n. If f(x) = x for x ∈ R, then (A) f(x) is differentiable every where (B) f ′ (0) exists (C) f increases on (0, ∞) and decreases on (–∞, 0) (D) f increases on R

Q.36

A horse runs along a circle with a speed of 20 km/hr . A lantern is at the centre of the circle . A fence is along the tangent to the circle at the point at which the horse starts . The speed with which the shadow of the horse move along the fence at the moment when it covers 1/8 of the circle in km/hr is (A) 20 (B)40 (C) 30 (D) 60

Q.37

Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1: If f : R → R and c ∈ R is such that f is increasing in (c – δ, c) and f is decreasing in (c, c + δ) then f has a local maximum at c. Where δ is a sufficiently small positive quantity. Statement-2 : Let f : (a, b) → R, c ∈ (a, b). Then f can not have both a local maximum and a point of inflection at x = c. Statement-3 : The function f (x) = x2 | x | is twice differentiable at x = 0. Statement-4 : Let f : [c – 1, c + 1] → [a, b] be bijective map such that f is differentiable at c then f–1 is also differentiable at f (c). (A) FFTF (B) TTFT (C) FTTF (D) TTTF Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.38

Let f : [–1, 2] → R be differentiable such that 0 ≤ f ' (t) ≤ 1 for t ∈ [–1, 0] and – 1 ≤ f ' (t) ≤ 0 for t ∈ [0, 2]. Then (A) – 2 ≤ f (2) – f (–1) ≤ 1 (B) 1 ≤ f (2) – f (–1) ≤ 2 (C) – 3 ≤ f (2) – f (–1) ≤ 0 (D) – 2 ≤ f (2) – f (–1) ≤ 0

Q.39

A curve is represented by the equations, x = sec2 t and y = cot t where t is a parameter. If the tangent at the point P on the curve where t = π/4 meets the curve again at the point Q then PQ is equal to: (A)

5 3 2

(B)

5 5 2

(C)

2 5 3

(D)

3 5 2

Q.40

For all a, b ∈ R the function f (x) = 3x4 − 4x3 + 6x2 + ax + b has : (A) no extremum (B) exactly one extremum (C) exactly two extremum (D) three extremum .

Q.41

The set of values of p for which the equation ln x − px = 0 possess three distinct roots is  1 (A)  0,   e

(B) (0, 1)

(C) (1,e)

(D) (0,e)

Q.42

The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value of the function f (x) = x3 + 3x – 9 on the interval [– 2, 3]. If the difference between the first and the second term of the progression is equal to f ' (0) then the common ratio of the G.P. is (A) 1/3 (B) 1/2 (C) 2/3 (D) 3/4

Q.43

The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height must be equal to : (A)

Q.44

π 4

Q.47 Q.48 Q.49

(B)

2 3

(C)

1

(D) 1

3

2 3

(B) sin−1

(C) cot−1 2

(D)

π 3

In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is l. The altitude of the prism for which the volume is greatest :



(A) (B) (C) (D) 2

Q.46

Quest

The lateral edge of a regular rectangular pyramid is 'a' cm long . The lateral edge makes an angle α with the plane of the base. The value of α for which the volume of the pyramid is greatest, is : (A)

Q.45

1 3

3

3

4

x3 5 if x ≤ 1  Let f (x) =  − ( x − 2)3 if x > 1 then the number of critical points on the graph of the function is (A) 1 (B) 2 (C) 3

(D) 4

The curve y − exy + x = 0 has a vertical tangent at : (A) (1, 1) (B) (0, 1) (C) (1, 0)

(D) no point

Number of roots of the equation (A) 2 (B) 4

x2

. e2 − x = 1

is : (C) 6

(D) zero

The point(s) at each of which the tangents to the curve y = x3 − 3x2 − 7x + 6 cut off on the positive semi axis OX a line segment half that on the negative semi axis OY then the co-ordinates the point(s) is/ are given by : (A) (− 1, 9) (B) (3, − 15) (C) (1, − 3) (D) none Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[6]

Q.50

A curve with equation of the form y = ax4 + bx3 + cx + d has zero gradient at the point (0, 1) and also touches the x-axis at the point (− 1, 0) then the values of x for which the curve has a negative gradient are (A) x > − 1 (B) x < 1 (C) x < − 1 (D) − 1 ≤ x ≤ 1

Q.51

Number of solution(s) satisfying the equation, 3x2 − 2x3 = log2 (x2 + 1) − log2 x is : (A) 1 (B) 2 (C) 3 (D) none

Q.52

Consider the function f (x) = x cos x – sin x, then identify the statement which is correct . (A) f is neither odd nor even (B) f is monotonic decreasing at x = 0 (C) f has a maxima at x = π (D) f has a minima at x = – π

Q.53

Consider the two graphs y = 2x and x2 – xy + 2y2 = 28. The absolute value of the tangent of the angle between the two curves at the points where they meet, is (A) 0 (B) 1/2 (C) 2 (D) 1

Q.54

The x-intercept of the tangent at any arbitrary point of the curve

a b + = 1 is proportional to: x 2 y2

(A) square of the abscissa of the point of tangency (B) square root of the abscissa of the point of tangency (C) cube of the abscissa of the point of tangency (D) cube root of the abscissa of the point of tangency.

Quest

Q.55

For the cubic, f (x) = 2x3 + 9x2 + 12x + 1 which one of the following statement, does not hold good? (A) f (x) is non monotonic (B) increasing in (– ∞, – 2) ∪ (–1, ∞) and decreasing is (–2, –1) (C) f : R → R is bijective (D) Inflection point occurs at x = – 3/2

Q.56

The function 'f' is defined by f(x) = xp (1 − x)q for all x ∈ R, where p,q are positive integers, has a maximum value, for x equal to : (A)

Q.57

pq p+q

(B) 1

g(x) = ln (h ( x ) )

(h ' ( x ) )2

(A) g is increasing on J (C) g is concave up on J

Q.59

(D)

p p+q

Let h be a twice continuously differentiable positive function on an open interval J. Let

Suppose

Q.58

(C) 0

for each x ∈ J

> h''(x) h(x) for each x ∈ J. Then (B) g is decreasing on J (D) g is concave down on J

( x − 1)(6 x − 1) 1 if x ≠  1 2x − 1 2 then at x = Let f (x) =  2  1 0 if x = 2 (A) f has a local maxima (B) f has a local minima (C) f has an inflection point (D) f has a removable discontinuity Let f (x) and g (x) be two continuous functions defined from R → R, such that f (x1) > f (x2) and

(

)

g (x1) < g (x2), ∀ x1 > x2 , then solution set of f g (α 2 − 2α ) > f ( g(3α − 4) ) is (A) R (B) φ (C) (1, 4) (D) R – [1, 4]

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[7]

x2

Q.60

If f(x) =

∫ (t − 1) dt , 1 ≤ x ≤ 2, then global maximum value of f(x) is : x

(A) 1 Q.61

(B) 2

(C) 4

(D) 5

A right triangle is drawn in a semicircle of radius 1/2 with one of its legs along the diameter. The maximum area of the triangle is (A)

1 4

(B)

3 3 32

(C)

3 3 16

(D)

1 8

Q.62

At any two points of the curve represented parametrically by x = a (2 cos t − cos 2t) ; y = a (2 sin t − sin 2t) the tangents are parallel to the axis of x corresponding to the values of the parameter t differing from each other by : (A) 2π/3 (B) 3π/4 (C) π/2 (D) π/3

Q.63

Let x be the length of one of the equal sides of an isosceles triangle, and let θ be the angle between them. If x is increasing at the rate (1/12) m/hr, and θ is increasing at the rate of π/180 radians/hr then the rate in m2/hr at which the area of the triangle is increasing when x = 12 m and θ = π/4

Q.64

 2π   (A) 21/2 1 + 5  

(B)

If the function f (x) =

t + 3x − x 2 , where 't' is a parameter has a minimum and a maximum then the x−4

73 1/2 ·2 2

range of values of 't' is (A) (0, 4) (B) (0, ∞) Q.65

1 π (D) 21/2  +  2 5

31 2 π + 2 5

Quest (C)

(C) (– ∞, 4)

(D) (4, ∞)

The least area of a circle circumscribing any right triangle of area S is : (A) π S

(B) 2 π S

(C)

2 πS

(D) 4 π S

Q.66

A point is moving along the curve y3 = 27x. The interval in which the abscissa changes at slower rate than ordinate, is (A) (–3 , 3) (B) (– ∞ , ∞ ) (C) (–1, 1) (D) (–∞ , –3) ∪ (3,∞ )

Q.67

Let f (x) and g (x) are two function which are defined and differentiable for all x ≥ x0. If f (x0) = g (x0) and f ' (x) > g ' (x) for all x > x0 then (A) f (x) < g (x) for some x > x0 (B) f (x) = g (x) for some x > x0 (C) f (x) > g (x) only for some x > x0 (D) f (x) > g (x) for all x > x0

Q.68

P and Q are two points on a circle of centre C and radius α, the angle PCQ being 2θ then the radius of the circle inscribed in the triangle CPQ is maximum when (A) sin θ =

Q.69

3 −1 2 2

(B) sin θ =

5 −1 2

(C) sin θ =

5 +1 2

(D) sin θ =

The line which is parallel to x-axis and crosses the curve y = x at an angle of (A) y = − 1/2

(B) x = 1/2

(C) y = 1/4

5 −1 4

π is 4

(D) y = 1/2

 πt 2   dt has two critical points in the interval [1, 2.4]. One of the critical The function S(x) = ∫ sin   2   0 points is a local minimum and the other is a local maximum. The local minimum occurs at x = π (A) 1 (B) 2 (C) 2 (D) 2 x

Q.70

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[8]

Q.71

For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in km). If the speed of the current is steady at C km/hr then the most economical speed of the steamer going against the current will be (A) 1.25 C (B) 1.5 C (C) 1.75C (D) 2 C

Q.72

Let f and g be increasing and decreasing functions, respectively from [0 , ∞) to [0 , ∞). Let h (x) = f [g (x)] . If h (0) = 0, then h (x) − h (1) is : (A) always zero (B) strictly increasing (C) always negative (D) always positive

Q.73

The set of value(s) of 'a' for which the function f (x) = negative point of inflection . (A) (− ∞, − 2) ∪ (0, ∞) (C) (− 2, 0)

Q.74

a x3 + (a + 2) x2 + (a − 1) x + 2 possess a 3

(B) {− 4/5 } (D) empty set

A function y = f (x) is given by x =

1 1 & y= for all t > 0 then f is : 2 1+ t t (1 + t 2 )

(A) increasing in (0, 3/2) & decreasing in (3/2, ∞) (B) increasing in (0, 1) (C) increasing in (0, ∞) (D) decreasing in (0, 1) Q.75

Quest

The set of all values of ' a ' for which the function ,  

f (x) = (a2 − 3 a + 2)  cos2 (A) [1, ∞) Q.76

x − sin 2 4

x  + (a − 1) x + sin 1 does not possess critical points is: 4

(B) (0, 1) ∪ (1, 4)

(D) (1, 3) ∪ (3, 5)

(C) (− 2, 4)

Read the following mathematical statements carefully: I. Adifferentiable function ' f ' with maximum at x = c ⇒ f ''(c) < 0. II. Antiderivative of a periodic function is also a periodic function.

III.

If f has a period T then for any a ∈ R.

T

T

0

0

∫ f (x ) dx = ∫ f ( x + a ) dx

IV.

If f (x) has a maxima at x = c , then 'f ' is increasing in (c – h, c) and decreasing in (c, c + h) as h → 0 for h > 0. Now indicate the correct alternative. (A) exactly one statement is correct. (B) exactly two statements are correct. (C) exactly three statements are correct. (D) All the four statements are correct.

Q.77

If the point of minima of the function, f(x) = 1 + a2x – x3 satisfy the inequality

x2 + x + 2 < 0, then 'a' must lie in the interval: x 2 + 5x + 6

( (C) (2

(A) −3 3, 3 3

Q.78

3, 3 3

)

)

( (D) ( −3

) 3 ) ∪ (2

(B) −2 3, − 3 3 3, − 2

3, 3 3

)

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is 1cm the altitude is 6 cm. When the radius is 6cm, the volume is increasing at the rate of 1Cu cm/sec. When the radius is 36cm, the volume is increasing at a rate of n cu. cm/sec. The value of 'n' is equal to: (A) 12 (B) 22 (C) 30 (D) 33 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[9]

Q.79

Two sides of a triangle are to have lengths 'a' cm & 'b' cm. If the triangle is to have the maximum area, then the length of the median from the vertex containing the sides 'a' and 'b' is

1 2 a + b2 (A) 2

2a + b (B) 3

(C)

a 2 + b2 2

(D)

a + 2b 3

Q.80

Let a > 0 and f be continuous in [– a, a]. Suppose that f ' (x) exists and f ' (x) ≤ 1 for all x ∈ (– a, a). If f (a) = a and f (– a) = – a then f (0) (A) equals 0 (B) equals 1/2 (C) equals 1 (D) is not possible to determine

Q.81

The lines tangent to the curves y3 – x2y + 5y – 2x = 0 and x4 – x3y2 + 5x + 2y = 0 at the origin intersect at an angle θ equal to (A) π/6 (B) π/4 (C) π/3 (D) π/2

Q.82

The cost function at American Gadget is C(x) = x3 – 6x2 + 15x (x in thousands of units and x > 0) The production level at which average cost is minimum is (A) 2 (B) 3 (C) 5 (D) none

Q.83

A rectangle has one side on the positive y-axis and one side on the positive x - axis. The upper right hand
nx vertex on the curve y = 2 . The maximum area of the rectangle is x (A) e–1 (B) e – ½ (C) 1 (D) e½

Q.84

A particle moves along the curve y = x3/2 in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units per second. The value of dx/dt when x = 3 is (A) 4

Q.85

Q.86

Quest (B)

9 2

(C)

3 3 2

 π Number of solution of the equation 3tanx + x3 = 2 in  0,  is  4 (A) 0 (B) 1 (C) 2

(D) none

(D) 3

Let f (x) = ax2 – b | x |, where a and b are constants. Then at x = 0, f (x) has (A) a maxima whenever a > 0, b > 0 (B) a maxima whenever a > 0, b < 0 (C) minima whenever a > 0, b > 0 (D) neither a maxima nor minima whenever a > 0, b < 0 x

Q.87

ln t   Let f (x) = ∫  t ln ( t ) −  dt (x > 1) then t  1

(A) f (x) has one point of maxima and no point of minima. (B) f ' (x) has two distinct roots (C) f (x) has one point of minima and no point of maxima (D) f (x) is monotonic Q.88

f (x) = | 1 – x | 1 ≤ x ≤ 2 and π 1 0, which of the following are always true? (i) f (x) is bounded on a ≤ x ≤ b. (ii) The equation f (x) = 0 has at least one solution in a < x < b. (iii) The maximum and minimum values of f (x) on a ≤ x ≤ b occur at points where f ' (c) = 0. (iv) There is at least one point c with a < c < b where f ' (c) > 0. (v) There is at least one point d with a < d < b where f ' (c) < 0. (A) only (ii) and (iv) are true (B) all but (iii) are true (C) all but (v) are true (D) only (i), (ii) and (iv) are true Q.105 Consider the function f (x) = 8x2 – 7x + 5 on the interval [–6, 6]. The value of c that satisfies the conclusion of the mean value theorem, is (A) – 7/8 (B) – 4 (C) 7/8 (D) 0 Q.106 Consider the curve represented parametrically by the equation x = t3 – 4t2 – 3t and y = 2t2 + 3t – 5 where t ∈ R. If H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then (A) H = 2 and V = 1 (B) H = 1 and V = 2 (C) H = 2 and V = 2 (D) H = 1 and V = 1

Quest

Q.107 At the point P(a, an) on the graph of y = xn (n ∈ N) in the first quadrant a normal is drawn. The normal 1 , then n equals a →0 2 (C) 2

intersects the y-axis at the point (0, b). If Lim b = (A) 1

(B) 3

(D) 4

Q.108 Suppose that f is a polynomial of degree 3 and that f ''(x) ≠ 0 at any of the stationary point. Then (A) f has exactly one stationary point. (B) f must have no stationary point. (C) f must have exactly 2 stationary points. (D) f has either 0 or 2 stationary points. − x 2 for x < 0  Q.109 Let f (x) =  . Then x intercept of the line that is tangent to the graph of f (x) is  x 2 + 8 for x ≥ 0 (A) zero

(B) – 1

(C) –2

(D) – 4

Q.110 Suppose that f is differentiable for all x and that f '(x) ≤ 2 for all x. If f (1) = 2 and f (4) = 8 then f (2) has the value equal to (A) 3 (B) 4 (C) 6 (D) 8 Q.111 There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per additional tree drops by 10 apples. Number of trees that should be added to the existing orchard for maximising the output of the trees, is (A) 5 (B) 10 (C) 15 (D) 20 Q.112 The ordinate of all points on the curve y =

1 where the tangent is horizontal, is 2 sin x + 3 cos 2 x 2

(A) always equal to 1/2 (B) always equal to 1/3 (C) 1/2 or 1/3 according as n is an even or an odd integer. (D) 1/2 or 1/3 according as n is an odd or an even integer.

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Select the correct alternatives : (More than one are correct) n

n

 x  y Q.113 The equation of the tangent to the curve   +   = 2 (n ∈ N) at the point with abscissa equal to a  b

'a' can be :  x  a

 y  b

 x  a

 y  b

(A)   +   = 2 (B)   −   = 2

Q.114 The function y =

 y  b

 x  a

 y  b

(D)   +   = 0

2x − 1 (x ≠ 2) : x−2

(A) is its own inverse (C) has a graph entirely above x-axis Q.115 If

 x a

(C)   −   = 0

(B) decreases for all values of x (D) is bound for all x.

K x y + = 1 is a tangent to the curve x = Kt, y = , K > 0 then : a b t

(A) a > 0, b > 0

(B) a > 0, b < 0

Q.116 The extremum values of the function f(x) =

(A)

(C) a < 0, b > 0

(D) a < 0, b < 0

1 1 − , where x ∈ R is : sin x + 4 cos x − 4

Quest 4

(B)

8− 2

x

Q.117 The function f(x) =

∫

2 2

8− 2

(C)

2 2

4 2 +1

(D)

4 2

8+ 2

1 − t 4 dt is such that :

0

(A) it is defined on the interval [− 1, 1] (B) it is an increasing function (C) it is an odd function (D) the point (0, 0) is the point of inflection Q.118 Let g(x) = 2 f (x/2) + f (1 − x) and f ′′ (x) < 0 in 0 ≤ x ≤ 1 then g(x) : (A) decreases in [0, 2/3) (B) decreases in (2/3, 1] (C) increases in [0, 2/3) (D) increases in (2/3, 1] Q.119 The abscissa of the point on the curve

xy = a + x, the tangent at which cuts off equal intersects from

the co-ordinate axes is : ( a > 0) (A)

a 2

Q.120 The function

(B) −

a 2

(C) a 2

(D) − a 2

sin (x + a ) has no maxima or minima if : sin (x + b)

(A) b − a = n π , n ∈ I (C) b − a = 2n π , n ∈ I

(B) b − a = (2n + 1) π , (D) none of these .

n∈I

Q.121 The co-ordinates of the point P on the graph of the function y = e–|x| where the portion of the tangent intercepted between the co-ordinate axes has the greatest area, is

 1 (A) 1,   e

1  (B)  −1,   e

(C) (e, e–e)

(D) none

Q.122 Let f(x) = (x2 − 1)n (x2 + x + 1) then f(x) has local extremum at x = 1 when : (A) n = 2 (B) n = 3 (C) 4 (D) n = 6 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[14]

Q.123 For the function f(x) = x4 (12 ln x − 7) (A) the point (1, − 7) is the point of inflection (B) x = e1/3 is the point of minima (C) the graph is concave downwards in (0, 1) (D) the graph is concave upwards in (1, ∞) Q.124 The parabola y = x2 + px + q cuts the straight line y = 2x − 3 at a point with abscissa 1. If the distance between the vertex of the parabola and the x − axis is least then : (A) p = 0 & q = − 2 (B) p = − 2 & q = 0 (C) least distance between the parabola and x − axis is 2 (D) least distance between the parabola and x − axis is 1 Q.125 The co-ordinates of the point(s) on the graph of the function, f(x) =

x3 5x2 − + 7x - 4 where the 3 2

tangent drawn cut off intercepts from the co-ordinate axes which are equal in magnitude but opposite in sign, is (A) (2, 8/3) (B) (3, 7/2) (C) (1, 5/6) (D) none Q.126 On which of the following intervals, the function x100 + sin x − 1 is strictly increasing. (A) (− 1, 1) (B) (0, 1) (C) (π/2, π) (D) (0, π/2)

Quest

Q.127 Let f(x) = 8x3 – 6x2 – 2x + 1, then (A) f(x) = 0 has no root in (0,1) (C) f ′(c) vanishes for some c ∈(0,1)

(B) f(x) = 0 has at least one root in (0,1) (D) none

Q.128 Equation of a tangent to the curve y cot x = y3 tan x at the point where the abscissa is (A) 4x + 2y = π + 2

(B) 4x − 2y = π + 2

(C) x = 0

π is : 4

(D) y = 0

Q.129 Let h (x) = f (x) − {f (x)}2 + {f (x)}3 for every real number ' x ' , then : (A) ' h ' is increasing whenever ' f ' is increasing (B) ' h ' is increasing whenever ' f ' is decreasing (C) ' h ' is decreasing whenever ' f ' is decreasing (D) nothing can be said in general.

Q.130 If the side of a triangle vary slightly in such a way that its circum radius remains constant, then, da db dc is equal to: + + cos A cos B cos C

(A) 6 R

(B) 2 R

(C) 0

(D) 2R(dA + dB + dC)

Q.131 In which of the following graphs x = c is the point of inflection .

(A)

(B)

(C)

x

Q.132 An extremum value of y =

∫

(D)

(t − 1) (t − 2) dt is :

0

(A) 5/6

(B) 2/3

(C) 1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 2

[15]

Q.130 C,D

Q.128 A,B,D Q.129 A,C

Q.126 B,C,D Q.127 B,C

Q.125 A,B

Q.124 B,D

Q.122 A,C,D Q.123 A,B,C,D

Q.120 A,B,C Q.121 A,B

Q.119 A,B

Q.118 B,C

Q.117 A,B,C,D

Q.116 A,C

Q.115 A,D

Q.114 A,B

Q.113 A,B

Q.112 D

Q.111 C

Q.110 B

Q.109 B

Q.108 D

Q.107 C

Q.106 B

Q.105 D

Q.104 D

Q.103 C

Q.102 D

Q.101 B

Q.100 D

D

Q.99

B

Q.98

C

Q.97

C

Q.96

Q.95 C

D

Q.94

B

Q.93

A

Q.92

A

Q.91

C

Q.90

D

Q.89

Q.88 C

D

Q.87

A

Q.86

B

Q.85

A

Q.84

A

Q.83

B

Q.82

Q.81 D

A

Q.80

A

Q.79

D

Q.78

D

Q.77

A

Q.76

B

Q.75

Q.74 B

A

Q.73

A

Q.72

B

Q.71

C

Q.70

D

Q.69

B

Q.68

Q.67 D

C

Q.66

A

Q.65

C

Q.64

D

Q.63

A

Q.62

B

Q.61

Q.60 C

C

Q.59

C

Q.58

D

Q.57

D

Q.56

C

Q.55

C

Q.54

Q.53 C

B

Q.52

A

Q.51

C

Q.50

B

Q.49

B

Q.48

C

Q.47

Q.46 C

B

Q.45

C

Q.44

C

Q.43

C

Q.42

A

Q.41

B

Q.40

Q.39 D

A

Q.38

A

Q.37

B

Q.36

D

Q.35

D

Q.34

D

Q.33

Q.32 C

C

Q.31

C

Q.30

B

Q.29

C

Q.28

B

Q.27

A

Q.26

Q.25 D

D

Q.24

C

Q.23

A

Q.22

C

Q.21

A

Q.20

A

Q.19

Q.18 D

A

Q.17

D

Q.16

C

Q.15

D

Q.14

D

Q.13

A

Q.12

B

Q.11

C

Q.10

D

Q.9

A

Q.8

B

Q.4

B

Q.3

D

Q.2

B

Q.1

Q.5

D

Q.6

C

Q.7

D

Q.131 A,B,D Q.132 A,B

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

DEFINITE & INDEFINITE INTEGRATION

Question bank on Definite & Indefinite Integration There are 168 questions in this question bank. Select the correct alternative : (Only one is correct) ∞

Q.1

x +1 3− x −1 The value of the definite integral, ∫ (e + e ) dx is 1

π 4e 2

(A)

(B)

π 4e

(C) π ln   2

Q.2

The value of the definite integral,

∫ 0

(A) 1

Q.3

2 2 cos e x  · 2 x e x dx is  

(B) 1 + (sin 1) 12

Value of the definite integral

∫ ( sin

−1

π 1 1 π − tan −1  (D) 2 2  e e 2 2e

(C) 1 – (sin 1)

(D) (sin 1) – 1

(3x − 4x 3 ) − cos−1 (4x 3 − 3x ) ) dx

−12

(A) 0

Quest (B) −

x

Q.4

Let f (x) =

∫ 2

(A) 1

Q.5

π 2

1+ t4

(D)

π 2

and g be the inverse of f. Then the value of g'(0) is (B) 17

(C) 17

(D) none of these

cot −1 (e x ) ∫ e x dx is equal to :

(C)

1 cot −1 (e x ) ln (e2x + 1) − −x+c 2 ex

Lim

1 (1 + sin 2x ) x dx k ∫0

k →0

(A) 2 ln 5

∫ 0

e x e x −1 e x +3

(A) 4 − π

1 cot −1 (e x ) 2x (B) ln (e + 1) + +x+c 2 ex (D)

1 cot −1 (e x ) ln (e2x + 1) + −x+c 2 ex

1

k

Q.7

7π 2

dt

1 cot −1 (e x ) 2x (A) ln (e + 1) − +x+c 2 ex

Q.6

(C)

(B) 1

(C) e2

(D) non existent

(B) 6 − π

(C) 5 − π

(D) None

dx =

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.8

1  2  3 t 2 sin 2 t  dt  x –  If x satisfies the equation ∫ 2 dt  x – 2 = 0 (0 < α < π), then the  t + 2 t cos α + 1   ∫ t2 +1  0   −3  value x is α 2 sin α

(A) ±

(B) ± x

Q.9

If f

(x) = eg(x) and g(x) =

Q.10

α sin α

(C) ±

(D) ± 2

sin α α

t dt then f ′ (2) has the value equal to : 1 + t4

∫ 2

(A) 2/17

2 sin α α

(B) 0

(C) 1

(D) cannot be determined

(C) etan θ sec θ + c

(D) etan θ cos θ + c

(C) − 2/9

(D) − 4/9

etan θ (sec θ – sin θ) dθ equals :

∫

(A) − etan θ sin θ + c

(B) etan θ sin θ + c

π

Q.11

∫

(x · sin2x · cos x) dx =

0

(A) 0

(B) 2/9

Quest r =4 n

Q.12 The value of Lim n →∞

(A)

1 35

b−c

Q.13

n

(

r 3 r +4 n

r =1

(B)

∫ f ( x + c)dx

a −c

∑

1 14

∫ 0

sin x − cos x dx ; I2 = 1 + sin x. cos x

(A) I1 = I2 = I3 = I4 = 0 (C) I1 = I3 = I4 = 0 but I2 ≠ 0

(D)

1 5

(C)

∫ f (x )dx

a −2c

b

(D) ∫ f ( x+2c)dx a

π/ 2

2π

∫ (cos x )dx ; I3 = 6

0

1

3 1  ∫ (sin x )dx & I4 = ∫ ln x − 1 dx then −π/ 2 0

(B) I1 = I2 = I3 = 0 but I4 ≠ 0 (D) I1 = I2 = I4 = 0 but I3 ≠ 0

1− x7 dx equals : x (1 + x 7 )

(A) ln x +

2 ln (1 + x7) + c 7

(B) ln x −

2 ln (1 − x7) + c 7

(C) ln x −

2 ln (1 + x7) + c 7

(D) ln x +

2 ln (1 − x7) + c 7

π/ 2 n

Q.16

1 10

b−2c

a

π/ 2

∫

(C)

(B) ∫ f ( x+c)dx

a

Q.15

is equal to

b

(A) ∫ f ( x )dx

Let I1 =

2

=

b

Q.14

)

∫ 0

dx 1+ tan n nx

(A) 0

= ( B

)

π 4n

(C)

nπ 4

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

π (D) 2n

[3]

x

Q.17

f (x) = ∫ t( t −1)( t−2) dt takes on its minimum value when: 0

(A) x = 0 , 1

(B) x = 1 , 2

(C) x = 0 , 2

(D) x =

3+ 3 3

a

Q.18

∫ f (x ) dx =

−a

a

(A) ∫ [f ( x )+f ( −x )]dx

a

(B) ∫ [f ( x )−f (−x )]dx

(C) 2 ∫ f ( x ) dx

(D) Zero

0

0

0

Q.19

a

Let f (x) be a function satisfying f ' (x) = f (x) with f (0) = 1 and g be the function satisfying f (x) + g (x) = x2. 1

The value of the integral ∫ f ( x )g ( x ) dx is 0

(A) e – Q.20

1 2 5 e – 2 2

(B) e – e2 – 3

ln | x | 1 + ln | x | dx equals :

(A)

2 1 + ln | x | (lnx − 2) + c 3

1 1 + ln | x | (lnx − 2) + c (C) 3

Q.21

1 2

1 (e – 3) 2

(B)

2 1 + ln | x | (lnx + 2) + c 3

(D) e –

1 2 3 e – 2 2

Quest

∫x

3

(C)

1

(D) 2 1 + ln | x | (3 lnx − 2) + c



∫1  2 (| x − 3 | + | 1 − x | − 4) dx equals: 2

3 9 1 (B) (C) 2 8 4 Where {*} denotes the fractional part function.

(A) −

4/π

Q.22

∫ 0

(A) Q.23

(D)

3 2

1 1  2  3x .sin − x.cos  dx has the value : x x  8 2 π3

(B)

24 2 π3

(C)

32 2 π3

(D) None

Lim

π  2 π  π 4  π  sec   + sec 2  2 ·  + ..... + sec 2 (n − 1) + has the value equal to  6n  6n 3   6n   6n 

(A)

3 3

n →∞

(B) 3

(C) 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

2 3

[4]

3

Q.24

sin 2x sin x , x > 0 then ∫ x dx can be expressed as Suppose that F (x) is an antiderivative of f (x) = x 1 (A) F (6) – F (2)

Q.25

Q.26

x +c x 4 + x +1

4

(C)

x +1 +c x + x +1 4

(D) −

x +1 +c x + x +1 4

  Lim π  1 + cos π + cos 2 π + ..... + cos ( n − 1) π  equal to 2n 

n →∞

2n

2n

(B)

2n

1 2



(C) 2

(D) none

2  logx 2)  ( logx 2 −  dx =
n 2   

4

∫ 2

(A) 0 Q.28

x +c x + x +1

(B) −

(A) 1

Q.27

1 1 ( F (6) – F (2) ) (C) ( F (3) – F (1) ) (D) 2( F (6) – F (2) ) 2 2

3x 4 − 1 w.r.t. x is : ( x 4 + x + 1) 2

Primitive of

(A)

(B)

Quest (B) 1

(C) 2

(D) 4

If m & n are integers such that (m − n) is an odd integer then the value of the definite integral π

∫ cos mx ·sin nx dx 0

(A) 0

=

(B)

2n n −m 2

(C)

2

2m n − m2 2

(D) none 3

Q.29

Let y={x}[x] where {x}denotes the fractional part of x & [x] denotes greatest integer ≤ x, then

∫ y dx = 0

(A) 5/6

Q.30

If

∫

(B) 2/3

x4 +1

(

)

x x2 +1

2

dx = A ln x +

(A) A = 1 ; B = − 1

(C) 1

B 1+ x2

(D) 11/6

+ c , where c is the constant of integration then :

(B) A = − 1 ; B = 1

(C) A = 1 ; B = 1

(D) A = − 1 ; B = − 1

(B) ln 2

(C) 1 + ln 2

(D) none

π

Q.31

1 − sin x dx = 1 − cos x π/ 2

∫

(A) 1 − ln 2 Q.32

2t dt is : x →1 x −1 4 (D) 8 f ′ (1)

Let f : R → R be a differentiable function & f (1) = 4 , then the value of ; Lim (A) f ′ (1)

(B) 4 f ′ (1)

(C) 2 f ′ (1)

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

f (x)

∫

[5]

f (x)

Q.33

∫t

If

2

dt = x cos πx , then f ' (9)

0

(A) is equal to –

Q.34

( π / 2 )1/ 3 5

∫x

1 9

(B) is equal to –

1 3

(C) is equal to

1 3

(D) is non existent

·sin x 3 dx =

0

(A) 1 Q.35

(B) 1/2

Integral of (A) 2 ln cos (C)

(C) 2

(D) 1/3

1+2cotx(cotx +cos ecx ) w.r.t. x is : x +c 2

(B) 2 ln sin

1 x ln cos + c 2 2

x +c 2

(D) ln sin x − ln(cosec x − cot x) + c

Quest 3

Q.36

If f (x) = x + x − 1 + x − 2 , x ∈ R then

∫ f (x ) dx = 0

(A) 9/2

(B) 15/2

(C) 19/2

(D) none

3 )x + 1 (  2 28  2 Number of values of x satisfying the equation ∫  8t + t + 4  dt = , is 3 log   ( x +1) x + 1 −1 x

Q.37

(A) 0

(B) 1

(C) 2

(D) 3

tan −1 x ∫ x dx = 0 1

Q.38

π/4

(A)

∫ 0

sin x dx x

π/2

(B)

∫ 0

x dx sin x

1 (C) 2 x

Q.39

Domain of definition of the function f (x) =

∫ 0

(A) R Q.40

(B) R+

π/2

∫ 0

x dx sin x

1 (D) 2

π/ 4

∫ 0

x dx sin x

dt

is x + t2 (C) R+ ∪ {0} 2

If ∫ e3x cos 4x dx = e3x (A sin 4x + B cos 4x) + c then : (A) 4A = 3B (B) 2A = 3B (C) 3A = 4B

(D) R – {0}

(D) 4B + 3A = 1

b

Q.41

If f (a + b − x) = f (x) , then ∫ x.f (a + b − x ) dx = a

(A) 0

1 (B) 2

b

a+b f ( x ) dx (C) 2 ∫a

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

b

a−b f ( x ) dx (D) 2 ∫a

[6]

2

Q.42

 4  The set of values of 'a' which satisfy the equation ∫ ( t − log 2 a ) dt = log2  2  is a  0 ( A

)

∈R

a

(B) a ∈ R+

(C) a < 2

(D) a > 2

3

Q.43

The value of the definite integral ∫  2 x − 5(4 x − 5) + 2 x + 5(4 x − 5)  dx =   2

(A) Q.44

7 3+3 5

(B) 4 2

3 2

b

∫ x dx = 0 3

2 and ∫ x dx = a

a

(A) 0

∫

(B) 1

(C) 2

7 7−2 5 3 2

(D) 4

tan −1 x −cot −1 x dx is equal to : tan −1 x +cot −1 x

Quest

(A)

4 2 x tan−1 x + ln (1 + x2) − x + c π π

(B)

4 2 x tan−1 x − ln (1 + x2) + x + c π π

(C)

4 2 x tan−1 x + ln (1 + x2) + x + c π π

(D)

4 2 x tan−1 x − ln (1 + x2) − x + c π π

dt

d2y

Variable x and y are related by equation x =

∫

1+ t2

0

y (A)

Q.47

(D)

2 is 3

y

Q.46

4 3

Number of ordered pair(s) of (a, b) satisfying simultaneously the system of equation b

Q.45

(C) 4 3 +

1+ y

. The value of

dx 2

is equal to

2y (B) y

2

1 Let f (x) = Lim h →0 h

(A) equal to 0

x +h

∫ x

(C)

dt t + 1+ t2

1 + y2

(D) 4y

, then Lim x · f ( x ) is

(B) equal to

x → −∞

1 2

(C) equal to 1

(D) non existent

Q.48

If the primitive of f (x) = π sin πx + 2x − 4, has the value 3 for x = 1, then the set of x for which the primitive of f (x) vanishes is : (A) {1, 2, 3} (B) (2, 3) (C) {2} (D) {1, 2, 3, 4}

Q.49

If f & g are continuous functions in [0, a] satisfying f (x) = f (a − x) & g (x) + g (a − x) = 4 then a

∫ f (x ).g( x )dx = 0 a

1 (A) ∫ f (x)dx 20

a

a

(B) 2∫ f (x)dx 0

(C)

∫

a

f (x)dx

0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 4∫ f (x)dx 0

[7]

Q.50

∫

x.

(A)

ln  x + 1+x 2    1+x 2 1+ x

2

dx equals :

2  ln  x + 1+x  − x + c  

 1 − x Q.51 If f (x) =   (7 x − 6) −1 3 (A)

x

x 2  . ln2  x + 1+x  +   2

(C)

31 6

1+ x

1< x ≤ 2

x

 x + 1+x 2  −  

1+ x 2

+c

2  1+ x 2 ln  x + 1+x  + x + c

(D) 2

, then

∫ f (x ) dx is equal to 0

32 21

(C) 1

Q.52

+c

2

0 ≤ x ≤1

(B)

x . ln2 2

(B)

1 42

(D)

55 42

Quest x

e x The value of the definite integral ∫ e (1 + x · e )dx is equal to 0

(A)

ee

2

Q.53

(B)

ee

–e

(C) ee – 1

(D) e

5 4

(D) 2

1  1 sin  x −  dx has the value equal to x  x 1/ 2

∫

(A) 0

(B)

3 4

(C)

∞

Q.54

The value of the integral

∫

e −2x (sin 2x + cos 2x) dx =

0

(B) − 2

(A) 1

0

Q.55

The value of definite integral

∫

∞

(A) – Q.56

π ln 2 2

(B)

z e−z 1 − e −2z

π ln 2 2

(C) 1/2

(D) zero

(C) – π ln 2

(D) π ln 2

dz .

A differentiable function satisfies 3f 2(x) f '(x) = 2x. Given f (2) = 1 then the value of f (3) is (A)

3

(B) 3 6

24

(C) 6

(D) 2

e

Q.57

For In =

∫

(ln x)ndx, n ∈ N; which of the following holds good?

1

(A) In + (n + 1) In + 1 = e (C) In + 1 + (n +1) In = e

(B) In + 1 + n In = e (D) In + 1 + (n – 1) In = e

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

Q.58

1 for 0 < x ≤ 1  Let f be a continuous functions satisfying f ' (ln x) =  and f (0) = 0 then f (x) can be  x for x > 1 defined as 1 if x ≤ 0  (A) f (x) =   1 − e x if x > 0

1 if x ≤ 0  (B) f (x) =  x  e − 1 if x > 0

if x < 0

x if x ≤ 0  (D) f (x) =  x  e − 1 if x > 0

x  (C) f (x) =  x  e

Q.59

if x > 0

Let f : R → R be a differentiable function such that f (2) = 2. Then the value of Limit

f (x )

x→2

∫ 2

f ′ (2)

(A) 6 π/2

Q.60

∫ 0

(A)

Q.61

(B) 12

(C) 32

f ′ (2)

(D) none

Quest

dx has the value : 1+a sin 2 x 2

π

2 1+ a

Let f (x) =

(A)

π

(B)

2

1+ a

(C)

2

2π

1+ a 2

(D) none

1  x  ln  then its primitive w.r.t. x is x  ex 

1 x e – ln x + C 2 n

Q.62

f ′ (2)

4 t3 dt is x−2

(B)

1 ln x – ex + C 2

(C)

1 2 ln x – x + C 2

(D)

ex +C 2x

n

∑ n 2 + k 2 x 2 , x > 0 is equal to n →∞

Lim

k =1

(A) x tan–1(x)

Q.63

2 cos 2 x sin (2x) − sin x Let f (x) = sin 2x 2 sin 2 x cos x then sin x − cos x 0

(A) π

Q.64

π /2

(B) π/2

∫

(D)

tan −1 ( x ) x2

[f (x) + f ′ (x)] dx =

0

(C) 2 π

(D) zero

(C) 10 −7

(D) 10 −9

19 The absolute value of ∫ sin x8 is less than : 10

(A) 10 −10

1+ x

(B) 10 −11 π

Q.65

tan −1 ( x ) (C) x

(B) tan–1(x)

The value of the integral

∫

(cos px − sin qx)2 dx where p, q are integers, is equal to :

−π

(A) − π

(B) 0

(C) π

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 2 π

[9]

Q.66

Primitive of f (x) = x · 2ln ( x (A)

2ln ( x

2

2

+1)

w.r.t. x is

+1)

2( x 2 + 1)

( x 2 + 1)2ln ( x (B) ln 2 + 1

+C

( x 2 + 1)ln 2+1 (C) +C 2(ln 2 + 1) t   Lim ∫ 1 +  dt is equal to n →∞  n + 1  0

(B) e2 x+ h

∫ Limit


n 2 t dt −

a

h→0

(A) 0 Q.69

+C

( x 2 + 1)ln 2 (D) +C 2(ln 2 + 1)

(A) 0

Q.68

+1)

n

2

Q.67

2

(C) e2 – 1

(D) does not exist

x

∫
n

2

t dt

a

=

h

Quest (B) ln2 x

(C)

2
n x x

(D) does not exist

Let a, b, c be non−zero real numbers such that ; 1

∫

2

(1 + cos8x) (ax2 + bx + c) dx =

0

∫

(1 + cos8x) (ax2 + bx + c) dx , then the quadratic equation

0

ax2 + bx +

c = 0 has : (A) no root in (0, 2) (C) a double root in (0, 2) π/ 4

Q.70

Let In =

∫

tann x dx , then

0

(A) A.P. Q.71

(B) atleast one root in (0, 2) (D) none

1 1 1 ,.... are in : , , I2 + I 4 I3 + I5 I 4 + I6

(B) G.P.

(C) H.P.

(

(D) none

)

Let g (x) be an antiderivative for f (x). Then ln 1+ (g( x ) )2 is an antiderivative for

2 f (x ) g (x )

(A)

1 + ( f ( x ) )2

2 f (x) g (x )

(B)

2 f (x )

1 + ( g ( x ) )2

(C)

3π 32

(C)

1 + ( f ( x ) )2

(D) none

π/ 4

Q.72

∫

(cos 2x)3/2. cos x dx =

0

(A)

3π 16

(B)

1

Q.73

The value of the definite integral

2

∫ 0

(A)

π 4

(B)

π 1 + 4 2

3π

(D)

16 2

3π 2 16

x 2 dx 1 − x 2 (1 + 1 − x 2 ) (C)

is

π 1 − 4 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[10]

∫ ({x}

37

Q.74

2

The value of the definite integral

)

+ 3(sin 2πx ) dx where { x } denotes the fractional part function.

19

(A) 0

(B) 6

(C) 9

(D) can not be determined

π2

Q.75

The value of the definite integral 2π

(A) Q.76

(C) 2 2 π

(D)

(A)

1 [ln (6 x 2 )]3 + C 8

(B)

1 2 [ln (6 x 2 )] + C 4

(C)

1 [ln (6 x 2 )] + C 2

(D)

1 [ln (6 x 2 )]4 + C 16

1

π 2 2

Quest

∫  2 (3 sin θ)

π6

(A) π –

2

1  − (1 + sin θ) 2  dθ 2  (B) π

3

(C) π – 2 3

2x

Q.78

tan x dx , is

0

ln ( 6 x 2 ) ∫ x dx

Evaluate the integral :

5π 6

Q.77

π 2

(B)

∫

Let l = Lim x →∞

dt 1 ∫ t and m = Lim x →∞ x ln x x

(A) l m = l

(D) π +

3

x

∫ ln t dt then the correct statement is 1

(B) l m = m

(C) l = m

(D) l > m

ln 3

Q.79

If f (x) =

e–x

+2

e–2x

+3

e– 3x

+...... + ∞ , then

∫ f (x) dx

=

ln 2

(A) 1

(B)

π/ 4

π /2

Q.80

If I =

∫


n (sin x) dx then

∫

(C)

1 3

(D) ln 2


n (sin x + cos x) dx =

− π/ 4

0

(A)

1 2

I 2

(B)

I 4

(C)

I 2

 n  n 1   dx equals The value of ∫  ∏ ( x + r )   ∑  + x k    k =1 0  r =1 (A) n (B) n ! (C) (n + 1) !

(D) I

1

Q.81

Q.82

(D) n · n !

cos 3 x +cos5 x ∫ sin 2 x+sin 4 x dx (A) sin x − 6 tan−1 (sin x) + c (C) sin x − 2 (sin x)−1 − 6 tan−1 (sin x) + c

(B) sin x − 2 sin−1 x + c (D) sin x − 2 (sin x)−1 + 5 tan−1 (sin x) + c

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[11]

3

Q.83

∫ 0

 1  +  x2 + 4 x + 4 

(A) ln

 x 2 − 4 x + 4  dx =  

5 2

5 3 − 2 2

(B) ln +

3 2

(C) ln

5 5 + 2 2

(D) none

x

Q.84

The value of the function f (x) = 1 + x +

∫ (ln2t + 2 lnt) dt where f ′ (x) vanishes is : 1

(A)

Q.85

e−1

(B) 0

Limit 1  1 +  n→∞ n 

n + n+1

n + n+2

(B) 2 2 − 1

(A) 2 2

(C) 2 e−1

(D) 1 + 2 e−1  n  has the value equal to n + 3 (n − 1) 

n + ....... + n+3

(C) 2

(D) 4 ∞

Q.86

Let a function h(x) be defined as h(x) = 0, for all x ≠ 0. Also ∞

∫ h(x ) · f (x ) dx = f (0),

for every

−∞

Quest

function f (x). Then the value of the definite integral

∫ h' ( x ) · sin x dx , is

−∞

(A) equal to zero π/ 4

Q.87

∫ 0

(A)

(B) equal to 1

(C) equal to – 1

(D) non existent

(tann x + tann − 2 x)d(x − [x]) is : ( [. ] denotes greatest integer function) 1 n −1

(B)

1 n+2

(C)

2 n−1

(D) none of these

1λ

Q.88

Q.89

1  Lim  ∫ (1 + x ) λ dx   λ →0  0 

is equal to

(A) 2 ln 2

(B)

4 e

4 e

(D) 4

(B) x . ∫

dx = x ln | x | + Cx x

(C) ln

Which one of the following is TRUE. (A) x . ∫ (C)

dx = x ln | x | + C x

1 . cos x dx = tan x + C cos x ∫

(D)

1 . cos x dx = x + C cos x ∫

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]

∞

Q.90

x2n + 1· e − x dx is equal to (n ∈ N). 2

∫ 0

(A) n !

(B) 2 (n !)

(C)

n! 2

(D)

(n + 1)! 2

0

Q.91

The true set of values of 'a' for which the inequality (B) (− ∞ , − 1]

(A) [0 , 1]

Q.92

∫

(3 −2x − 2. 3−x) dx ≥ 0 is true is:

a

(C) [0, ∞)

If α ∈ (2 , 3) then number of solution of the equation

α

(D) (− ∞ , − 1] ∪ [0, ∞)

cos (x + α2) dx = sin α is :

∫ 0

(A) 1

(B) 2

(C) 3

(D) 4.

x2

Q.93

∫ f ( t ) dt where f is continuous functions then the value of f (4) is

If x · sin πx =

Quest 0

(A)

π 2

(B) 1

(2x + 1)

Q.94

∫ ( x 2 + 4x + 1)3 / 2

1 2

x3 +C (A) 2 ( x + 4x + 1)1 / 2

(B)

( x 2 + 4x + 1)1/ 2

x2 +C (C) 2 ( x + 4x + 1)1 / 2

(D)

( x + 4x + 1)1/ 2

x

1 2

(A) e4 − e − α 3

∫ 0

(A)

1 d 

 tan 2 dx 

Let A =

0

(A) Ae−a


n x dx is :

e

(B) 2 e4 − e − α

(B) −

∫

+C

(C) 2 (e4 − e) − α

(D) 2 e4 – 1 – α

−1 2 x  2  equals 1− x 

π 3 1

Q.97

+C

e4

2

x If the value of the integral ∫ e dx is α , then the value of ∫ 1

Q.96

(D) can not be determined

dx

2

Q.95

(C)

et d t then 1+ t

π 6

(C)

π 2

(D)

π 4

e − t dt ∫ t−a−1 has the value a −1 a

(B) − Ae−a

(C) − ae−a

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) Aea

[13]

π/2

Q.98

∫

sin 2θ sin θ dθ is equal to :

0

(B) π/4

(A) 0

Q.99

(C) π/2

(D) π

x2 + 2 ∫ x 4 + 4 dx is equal to (A)

1 x2 + 2 tan −1 +C 2 2x

(B)

1 tan −1 ( x 2 + 2) + C 2

(C)

1 2x tan −1 2 +C 2 x −2

(D)

1 x2 − 2 tan −1 +C 2 2x

1

1

Q.100 If β + 2 ∫ x 2 e − x dx = ∫ e − x dx then the value of β is 0

0

(A)

2

2

e−1

(B) e

(C) 1/2e

(D) can not be determined

Quest 1

Q.101 A quadratic polynomial P(x) satisfies the conditions, P(0) = P(1) = 0 &

∫ P(x) dx = 1. The leading 0

coefficient of the quadratic polynomial is : (A) 6 (B) − 6

(C) 2

(D) 3

Q.102 Which one of the following functions is not continuous on (0,π)? x

1 (B) g(x) = ∫ t sin t dt 0

(A) f(x)= cotx

1

 (C) h (x) =   2 2 sin x 9 π

Q.103 If f (x) = ∫ 0

0 a > 0. If I1 =

∫ f (tan θ + cot θ). sec2 θ d θ &

π/6

π/3

I2 =

I1 : I2

∫ f (tan θ + cot θ). cosec2 θ d θ , then the ratio

π /6

(A) is a positive integer (C) is an irrational number

(B) is a negative integer (D) cannot be determined.

sin x

∫ (1 − t + 2 t3) d t has in [ 0, 2 π ]

Q.136 f (x) =

cos x

(A) a maximum at

π 3π & a minimum at 4 4

(B) a maximum at

(C) a maximum at

5π 7π & a minimum at 4 4

(D) neither a maxima nor minima

x3

Q.137 Let S (x) =

∫ l n t d t (x > 0) and H (x) =

3π 7π & a minimum at 4 4

S′ (x) . Then H(x) is : x

Quest x2

(A) continuous but not derivable in its domain (B) derivable and continuous in its domain (C) neither derivable nor continuous in its domain (D) derivable but not continuous in its domain.

d Q.138 Number of solution of the equation dx (A) 4

Q.139 Let f (x) =

(B) 3

sin x

∫

cos x

dt = 2 2 in [0, π] is 1− t2 (C) 2

(D) 0

2 sin 2 x − 1 cos x (2 sin x + 1) then + 1 + sin x cos x

x ∫ e (f (x ) + f ' ( x ))dx

(A) ex tanx + c

(where c is the constant of integeration) (B) excotx + c

(C) ex cosec2x + c

(D) exsec2x + c

x +3

Q.140 The value of x that maximises the value of the integral

∫ t (5 − t ) dt is x

(A) 2

(B) 0

(C) 1

(D) none

Q.141 For a sufficiently large value of n the sum of the square roots of the first n positive integers i.e. 1 + 2 + 3 +......................+ n is approximately equal to (A)

1 3/ 2 n 3

(B)

2 3/ 2 n 3

(C)

1 1/ 3 n 3

2 1/ 3 (D) n 3

2

dx is (1 − x ) 2 0

Q.142 The value of ∫ (A) –2

(B) 0

(C) 15

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) indeterminate

[19]

a

Q.143 If

dx = x+a + x

∫ 0

(A)

3 4

π/8

∫ 0

2 tan θ dθ , then the value of 'a' is equal to (a > 0) sin 2θ

(B)

Q.144 The value of the integral

∫

π 4

(C)

sin (ln ( 2 + 2 x ) ) dx is x +1

3π 4

(D)

(A) – cos ln (2x + 2) + C

2    +C (B) ln  sin x +1  

 2   +C (C) cos   x +1 

 2   +C (D) sin   x +1 

πx 1 Q.145 If f(x) = A sin   + B , f′   = 2  2 

9 16

1

∫

2 and

f(x) dx = 2 A , Then the constants A and B are π

0

respectively. (A)

Quest

π π & 2 2

(B)

π2

∫e

Q.146 Let I1 =

−x 2

2 3 & π π

(C) 0 & −

π2

sin(x )dx ; I = 2

0

∫e

sin x , then x

π

2 (A) π ∫ f ( x ) dx 0

π2

π

(A) u = 4v

−x 2

4 &0 π

(1 + x) dx

0

II

III

I2 < I3

I1 = I3

(B) II only (D) Both I and II



0

π

(B) ∫ f ( x ) dx 0

ln ( x + 1) dx and v = Q.148 Let u = ∫ 2 x + 1 0

(A) π

∫e

dx ; I = 3

(D)

∫ f (x ) f  2 − x  dx =

1

Q.149 If f (x ) =

π2

0

and consider the statements I I1 < I2 Which of the following is(are) true? (A) I only (C) Neither I nor II nor III

Q.147 Let f (x) =

−x 2

4 π

π

(C) π ∫ f ( x ) dx 0

π

1 (D) π ∫ f ( x ) dx 0

π2

∫ ln (sin 2x) dx then 0

(B) 4u + v = 0

(C) u + 4v = 0

(D) 2u + v = 0

x2

π sin x ·sin θ   , is . d θ then the value of f ' 2 2 1 + cos θ 2 π /16

∫

(B) – π

(C) 2π

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 0

[20]

π2

∫

Q.150 The value of the definite integral,

0

(A) 0

sin 5x dx is sin x

π 2

(B)

(C) π

(D) 2π

Select the correct alternatives : (More than one are correct) b

Q.151

∫ sgn x dx = (where a, b∈ R) a

(A) | b | – | a |

Q.152

(B) (b–a) sgn (b–a)

(C) b sgnb – a sgna

(D) | a | – | b |

x  dx −1  m tan  + C then : = λ tan ∫ 5 + 4 cos x 2  (A) λ = 2/3 (B) m = 1/3 (C) λ = 1/3

(D) m = 2/3

Quest

Q.153 Which of the following are true ? (A)

π −a

π−a

∫

∫

a nπ

(C)

∫ 0

π x . f (sin x) dx = . 2

(

π

)

f (sin x ) dx

a

(

2x 2 + 3x + 3

0

(

)

)

(x + 1) x 2 + 2x + 2

2

a

dx = 2.

(D)

2 ∫ f (x) dx 0

b

∫ f (x + c) dx = ∫ f (x) dx 0

c

dx is :

π + 2 ln2 − tan−1 2 4

(C) 2 ln2 − cot−1 3

∫

∫ f (x)

b− c

0

1

Q.155

(B)

−a

f cos 2 x dx = n. ∫ f cos 2 x dx

Q.154 The value of ∫

(A)

a

(B)

π 1 + 2 ln2 − tan−1 4 3

(D) −

π + ln4 + cot−1 2 4

x 2 + cos 2 x cosec2 x dx is equal to : 1 + x2

(A) cot x − cot −1 x + c cos ec x

(C) − tan −1 x − sec x + c where 'c' is constant of integration . x

Q.156 Let f (x) = ∫ 0

(B) c − cot x + cot −1 x (D) − e
n tan

−1 x

− cot x + c

sin t dt (x > 0) then f (x) has : t

(A) Maxima if x = n π where n = 1, 3, 5,..... (B) Minima if x = n π where n = 2, 4, 6,...... (C) Maxima if x = n π where n = 2, 4, 6,...... (D) The function is monotonic

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[21]

1

dx

Q.157 If In = ∫

(1 + x ) 2

0

n

; n ∈ N, then which of the following statements hold good ? π 1 + 8 4 π 5 − (D) I3 = 16 48

(A) 2n In + 1 = 2 −n + (2n − 1) In π 1 − 8 4

(C) I2 = Q.158

z

(B) I2 =

1 x −1
n dx equals : x −1 x +1 2

(A)

1 2 x −1 1 2 x −1 1 2 x+1 1 2 x +1 ln + c (B) ln + c (C) ln + c (D) ln +c x +1 x +1 x −1 x −1 2 4 2 4 π /2

∫

Q.159 If An =

0

sin (2 n − 1) x d x ; Bn = sin x

π/2

∫ 0

 sin n x     sin x 

(A) An + 1 = An (C) An + 1 − An = Bn + 1 ∞

Q.160

∫ 0

π 4

(B) Bn + 1 = Bn (D) Bn + 1 − Bn = An + 1

Quest (B)

∞

(C) is same as

∫ 0

∫

d x ; for n ∈ N , then :

x dx: (1 + x) (1 + x 2 )

(A)

Q.161

2

dx (1 + x) (1 + x 2 )

π 2

(D) cannot be evaluated

1 + cscx dx equals

(A) 2 sin −1 sin x + c (C) c − 2 sin −1 (1 − 2 sin x) π/2

Q.162 If f (x) =

∫ 0

(A) f (t) = π

(B) 2 cos −1 cosx + c (D) cos −1 (1 − 2 sin x) + c


n (1 + x sin 2 θ) d θ , x ≥ 0 then : sin 2 θ

(

)

t +1−1

(B) f ′ (t) =

(C) f (x) cannot be determined

π 2 t +1

(D) none of these.

Q.163 If a, b, c ∈ R and satisfy 3 a + 5 b + 15 c = 0 , the equation ax4 + b x2 + c = 0 has : (A) atleast one root in (− 1, 0) (B) atleast one root in (0, 1) (C) atleast two roots in (− 1, 1) (D) no root in (− 1, 1) ∞

∞

dx x 2 dx Q.164 Let u = ∫ 4 &v=∫ 4 then : 2 2 0 x + 7 x +1 0 x + 7 x +1

(A) v > u

(B) 6 v = π

(C) 3u + 2v = 5π/6

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) u + v = π/3

[22]

Q.165 If ∫ eu . sin 2x dx can be found in terms of known functions of x then u can be : (A) x (B) sin x (C) cos x (D) cos 2x x

Q.166 If f(x) = ∫ 1

(A) 2


n t dt where x > 0 then the value(s) of x satisfying the equation, 1+ t

f(x) + f(1/x) = 2 is : (B) e

(C) e −2

(D) e2 1

Q.167

A polynomial function f(x) satisfying the conditions f(x) = [f ′ (x)]2 &

∫

f(x) dx =

0

(A)

x2 3 9 + x+ 4 2 4

(B)

x2 3 9 − x+ 4 2 4

(C)

x2 −x+1 4

(D)

19 can be: 12

x2 +x+1 4

Q.168 A continuous and differentiable function ' f ' satisfies the condition ,

Quest x

∫

f (t) d t = f2 (x) − 1 for all real ' x '. Then :

0

(A) ' f ' is monotonic increasing ∀ x ∈ R (B) ' f ' is monotonic decreasing ∀ x ∈ R (C) ' f ' is non monotonic (D) the graph of y = f (x) is a straight line.

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[23]

[24] Q.1 Q.6

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.165 A,B,C,D

Q.164 B,C,D

Q.163 A,B,C

Q.161 A,D

Q.160 A,C

Q.159 A,D

Q.157 A,B

Q.156 A,B

Q.155 B,C,D

Q.154 A,C,D

Q.153 A,B,C,D

Q.152 A,B

Q.151 A,C

Q.149 A

Q.148 B

Q.147 A

Q.146 D

Q.139 A Q.144 A

Q.138 C Q.143 D

Q.137 B Q.142 D

Q.136 B Q.141 B

Q.134 B

Q.133 C

Q.132 D

Q.131 D

Q.129 C

Q.128 D

Q.127 D

Q.126 B

Q.124 C

Q.123 A

Q.122 B

Q.121 D

Q.119 A

Q.118 D

Q.117 A

Q.116 C

Q.114 C

Q.113 A

Q.112 B

Q.111 C

Q.109 C

Q.108 A

Q.107 C

Q.106 B

Q.104 B

Q.103 C

Q.102 D

Q.101 B

D

Q.99

Q.98 B

B

Q.97

A

Q.96

Q.94

Q.93 A

B

Q.92

D

Q.91

Q.89

Q.88 B

A

Q.87

C

Q.86

Q.84

Q.83 C

C

Q.82

D

Q.81

Q.79

Q.78 A

B

Q.77

B

Q.76

Q.74

Q.73 C

C

Q.72

B

Q.71

Q.69

Q.68 B

C

Q.67

C

Q.66

Q.64

Q.63 A

C

Q.62

C

Q.61

Q.59

Q.58 D

C

Q.57

B

Q.56

Q.54

Q.53 A

A

Q.52

D

Q.51

Q.49

Q.48 C

D

Q.47

B

Q.46

Q.44

Q.43 D

B

Q.42

C

Q.41

Q.39

Q.38 C

B

Q.37

C

Q.36

Q.34

Q.33 A

D

Q.32

A

Q.31

Q.24 Q.29

Q.23 A Q.28 B

C A

Q.22 Q.27

C A

Q.21 Q.26

Q.19

Q.18 A

C

Q.17

B

Q.16

Q.14

Q.13 A

C

Q.12

D

Q.11

B D

C A

Q.2 Q.7

A C

Q.3 Q.8

Q.4 Q.9

B

Q.95

C

Q.90

C

Q.85

A

Q.80

B

B

Q.75

B

A

Q.70

B

D

Q.65

C

A

Q.60

C

A

Q.55

C

A

Q.50

B

D

Q.45

B

C

Q.40

D

B

Q.35

D

B C

Q.25 Q.30

A D

A

Q.20

D

C

Q.15

C

C D

Q.5 Q.10

C A

D

Quest B

B

Q.100 A Q.105 D Q.110 B Q.115 C Q.120 B Q.125 B Q.130 B Q.135 A Q.140 C Q.145 D Q.150 B

Q.158 B,D Q.162 A,B Q.166 C,D

Q.167 B,D Q.168 A,D

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

AREA UNDER THE CURVE & DIFFERENTIAL EQUATION

Select the correct alternative : (Only one is correct) Q.1

Area common to the curve y = 9 − x 2 & x² + y² = 6 x is : (A)

Q.2

π+ 3 4

(B)

π− 3 4



(C) 3  π + 

3  4 



(D) 3  π − 

3 3  4 

Spherical rain drop evaporates at a rate proportional to its surface area. The differential equation corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is K > 0, is (A)

dr +K = 0 dt

(B)

dr −K= 0 dt

(C)

dr = Kr dt

(D) none

Q.3

If y = 2 sin x + sin 2 x for 0 ≤ x ≤ 2 π , then the area enclosed by the curve and the x-axis is : (A) 9/2 (B) 8 (C) 9 (D) 4

Q.4

Number of values of m ∈ N for which y = emx is a solution of the differential equation D3y – 3D2y – 4Dy + 12y = 0, is (A) 0 (B) 1 (C) 2 (D) more than 2

Q.5

The area bounded by the curve y = x2 + 4x + 5 , the axes of co-ordinates & the minimum ordinate is: (A) 3

Quest 2 3

(B) 4

2 3

(C) 5

2 3

(D) none

Q.6

The general solution of the differential equation, y ′ + y φ ′ (x) − φ (x) . φ ′ (x) = 0 where φ (x) is a known function is : (B) y = ce+ φ (x) + φ (x) − 1 (A) y = ce− φ (x) + φ (x) − 1 (C) y = ce− φ (x) − φ (x) + 1 (D) y = ce− φ (x) + φ (x) + 1 where c is an arbitrary constant .

Q.7

The area bounded by the curve y = x2 − 1 & the straight line x + y = 3 is : (A)

Q.8

9 2

7 17 2

(D)

17 17 6

(B) x 4 3 − y 4 3 = c

(C) x 4 3 + y 4 3 = c

(D) x1 3 − y1 3 = c

The area enclosed by the curve y2 + x4 = x2 is : (A)

Q.10

(C)

Orthogonal trajectories of family of the curve x 2 3 + y 2 3 = a 2 3 , where 'a' is any arbitrary constant, is (A) x 2 3 − y 2 3 = c

Q.9

(B) 4

2 3

(B)

4 3

(C)

8 3

(D)

10 3

Equation of a curve passing through the origin if the slope of the tangent drawn at any of its point (x, y) is cos(x + y) + sin(x + y), is (A) y = 2 tan–1(ex – 1) + x (B) y = 2 tan–1(ex – 1) – x –1 x (C) y = 2 tan (e ) – x (D) y = 2 tan–1(ex) + x Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.11

The area enclosed between the curves y = sin x , y = cos x & the x-axis if 0 ≤ x ≤ (A)

(B) 2 − 2

2 −1

(C)

(

π is : 2

)

(D) 2 2 − 1

2

Q.12

The differential equation of all parabolas having their axis of symmetry coinciding with the axis of x has its order and degree respectively: (A) (2, 1) (B) (2, 2) (C) (1, 2) (D) (1, 1)

Q.13

The area bounded by the curve y = x² + 1 & the tangents to it drawn from the origin is (A)

Q.14

2 3

(B)

4 3

(C)

1 3

(D) 1

Which one of the following functions is not homogeneous? x−y (A) f (x, y) =

2

x +y

(B) f (x, y) =

2

(C) f (x, y) = x (ln x + y

2

– ln y)+yex/y

π 4

(B)

3π 2

to start with is (A) 30 min

π 4 − 2 π

(D)

π 2

(B) 45 min

1 then the time to drain the tank if the water is 4 meter deep 15 (C) 60 min

(B)

π 2 − 4 π

(D) 80 min

πx in the upper half of the circle is : 2

(C) π −

8 π

(D)

π 2 − 2 π

The solution to the differential equation y lny + xy' = 0, where y (1) = e, is (A) x (ln y) = 1

Q.19

(C) π

The area bounded by x² + y² − 2 x = 0 & y = sin (A)

Q.18

 2x 2 + y 2  x + 2y − ln ( x + y)  +y2tan (D) f(x,y)=x ln x 3x − y  

Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth y, where the constant of proportionality k > 0 depends on the acceleration due to gravity and the geometry of the hole. If t is measured in minutes and k =

Q.17

x y

The area enclosed by the curve y = x & x = – y , the circle x2 + y2 = 2 above the x-axis, is (A)

Q.16

tan −1

Quest 2

Q.15

1 2 − 3 x ·y 3

(B) xy (ln y) = 1

(C)

(ln y)2

=2

 x2  (D) ln y +   y = 1  2 

The ratio in which the x-axis divides the area of the region bounded by the curves y = x2 − 4 x & y = 2 x − x2 is : (A)

4 23

(B)

4 27

(C)

4 19

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[3]

Q.20

 

π 4

y − cos2 x

A curve passes through the point  1 ,  & its slope at any point is given by

 y   . Then the  x

curve has the equation (A) y=x tan–1(ln Q.21

9 2

(B)

(C) y =

1 –1 e tan (ln ) (D) none x x

11 3

(C)

11 4

(D)

9 4

The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point (1, 1) is x y

x y

(B) x e = e

(A) y e = e

Q.23

(B) y=x tan–1(ln + 2)

The area enclosed by the curve y = (x − 1) (x − 2) (x − 3) between the co-ordinate axes and the ordinate at x = 3 is : (A)

Q.22

e ) x

(C)

y x xe

=e

(D)

y x ye

=e

Quest

The line y = mx bisects the area enclosed by the curve y = 1 + 4x − x2 & the lines x = 0, x =

3 & 2

y = 0. Then the value of m is: (A)

13 6

(B)

6 13

(C)

3 2

(D) 4

Q.24

The differential equation of all parabolas each of which has a latus rectum '4a' & whose axes are parallel to x-axis is : (A) of order 1 & degree 2 (B) of order 2 & degree 3 (C) of order 2 and degree 1 (D) of order 2 and degree 2

Q.25

The area bounded by the curve y = f (x), the x-axis & the ordinates x =1 & x = b is (b − 1)sin(3b + 4). Then f (x) is: (A) (x − 1) cos (3x + 4) (B) sin (3x + 4) (C) sin (3x + 4) + 3 (x − 1) . cos (3x + 4) (D) none

Q.26

The foci of the curve which satisfies the differential equation (1 + y2) dx − xy dy = 0 and passes through the point (1 , 0) are :

(

(A) ± 2 , 0 Q.27

)

(

(B) 0, ± 2

)

(C) (0, ± 1)

(D) (± 2, 0)

The area of the region for which 0 < y < 3 − 2x − x2 & x > 0 is : 3

(A)

∫(

)

3 − 2 x − x 2 dx

3

(B)

1

∫ (3 − 2 x − x 0

2

) dx

0

1

(C)

∫ (3 − 2 x − x

2

) dx

3

(D)

∫ (3 − 2 x − x

2

) dx

1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.28

A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x → 0. π2

If I =

∫ f ( x ) dx then 0

π π2 (A) < I < 2 4

π π2 (B) < I < 4 2

(C) 1 < I <

π 2

(D) 0 < I < 1

Q.29

The area bounded by the curve y = f(x) , the co-ordinate axes & the line x = x1 is given by x1 . e x1 . Therefore f (x) equals: (A) ex (B) x ex (C) xex − ex (D) x ex + ex

Q.30

A curve is such that the area of the region bounded by the co-ordinate axes, the curve & the ordinate of any point on it is equal to the cube of that ordinate. The curve represents (A) a pair of straight lines (B) a circle (C) a parabola (D) an ellipse

Q.31

The limit of the area under the curve y = e−x from x = 0 to x = h as h → ∞ is : 1 (A) 2 (B) e (C) (D) 1 e Degree of the differential equation y = a 1 − e − x a , a being the parameter is (A) 1 (B) 2 (C) 3 (D) not applicables

Q.32

Q.33

)

The slope of the tangent to a curve y = f (x) at (x , f (x)) is 2x + 1 . If the curve passes through the point (1 , 2) then the area of the region bounded by the curve , the x-axis and the line x = 1 is : (A)

Q.34

Quest (

5 6

(B)

6 5

(C)

1 6

(D) 1

A curve satisfying the initial condition, y(1) = 0, satisfies the differential equation, x

dy = y – x2. The area dx

bounded by the curve and the x-axis is (A)

Q.35

Q.36

1 2

(B)

1 3

(C)

1 4

(D)

1 6

1 1  The graphs of f (x) = x2 & g(x) = cx3 (c > 0) intersect at the points (0, 0) &  , 2  . If the region which c c  lies between these graphs & over the interval [0, 1/c] has the area equal to 2/3 then the value of c is (A) 1 (B) 1/3 (C) 1/2 (D) 2 2

dy Number of straight lines which satisfy the differential equation +x dx

 dy    − y = 0 is:  dx 

(A) 1

(D) 4

(B) 2

(C) 3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.37

The area bounded by the curves y = − − x and x = − − y where x, y ≤ 0 (A) cannot be determined (B) is 1/3 (C) is 2/3 (D) is same as that of the figure bounded by the curves y = − x ; x ≤ 0 and x = − y ; y ≤ 0

Q.38

The solution of the differential equation, (x + 2y3) (A)

Q.39

x = y+ c y2

x = y2 + c y

(C)

x2 = y2 + c y

(D)

y = x2 + c x

The area bounded by the curves y = x (1 − ln x) ; x = e−1 and positive X-axis between x = e−1 and x = e is :  e 2 − 4 e −2   5  

(A) 

Q.40

(B)

dy = y is : dx

 e 2 − 5 e −2   4  

(B) 

 4 e 2 − e −2   5  

(C) 

 5 e 2 − e −2   4  

(D) 

Quest

The real value of m for which the substitution, y = um will transform the differential equation, 2x4y

dy + y4 = 4x6 into a homogeneous equation is : dx

(A) m = 0

(B) m = 1

(C) m = 3/2

(D) no value of m

Q.41

The area bounded by the curves y = x (x − 3)2 and y = x is (in sq. units) : (A) 28 (B) 32 (C) 4 (D) 8

Q.42

The solution of the differential equation, x2 (A) y = sin

1 1 – cos x x

(B) y =

x+1 x sin x1

1 1 + sin x x

(D) y =

x +1 x cos x1

(C) y = cos Q.43

dy 1 1 .cos − y sin = − 1, where y → − 1 as x → ∞ is dx x x

The positive values of the parameter 'a' for which the area of the figure bounded by the curve y = cos ax, y = 0, x =

π 5π ,x= is greater than 3 are : 6a 6a

(A) φ

(B) (0, 1/3)

(C) (3, ∞)

(D) none of these

Q.44

The equation of a curve passing through (1, 0) for which the product of the abscissa of a point P & the intercept made by a normal at P on the x-axis equals twice the square of the radius vector of the point P, is (A) x2 + y2 = x4 (B) x2 + y2 = 2 x4 (C) x2 + y2 = 4 x4 (D) none

Q.45

The curvilinear trapezoid is bounded by the curve y = x2 + 1 and the straight lines x=1 and x=2. The co-ordinates of the point ( on the given curve) with abscissa x∈ [1,2] where tangent drawn cut off from the curvilinear trapezoid an ordinary trapezium of the greatest area, is (A) (1,2)

(B) (2,5)

 3 13  (C)  ,  2 4 

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[6]

Q.46

The latus rectum of the conic passing through the origin and having the property that normal at each point (x, y) intersects the x − axis at ((x + 1), 0) is : (A) 1 (B) 2 (C) 4 (D) none

Q.47

The value of 'a' (a>0) for which the area bounded by the curves y =

x 1 + , y = 0, x = a and 6 x2

x = 2a has the least value, is (A) 2 Q.48

Q.49

(B)

Q.51

dy = tan (x2y2) − 2xy2 given y(1) = dx

(A) sinx2y2 = ex–1

(C) cosx2y2 + x = 0

(B) sin(x2y2) = x

Area of the region enclosed between the curves (B) 4/3

Q.54

(D) sin(x2y2) = e.ex

x = y2 – 1 and x = |y| 1 − y 2 is

(C) 2/3

(D) 2

dy 1 − 2 y − 4 x = is dx 1 + y + 2 x (A) 4x2 + 4xy + y2 − 2x + 2y + c = 0 (B) 4x2 – 4xy – y2 − 2x − 2y + c = 0 2 2 (C) 4x + 4xy + y + 2x + 2y + c = 0 (D) 4x2 + 4xy – y2 − 2x + 2y + c = 0

Quest

Let y = g (x) be the inverse of a bijective mapping f : R → R f (x) = 3x3 + 2x. The area bounded by the graph of g (x), the x-axis and the ordinate at x = 5 is : 5 4

(B)

7 4

The solution of the differential equation,

(C)

9 4

(D)

13 4

dy y−x = , given y (− 5) = − 5 represents dx y −x −1

(A) a pair of straight lines (C) parabola Q.53

π is 2

Solution of the differential equation,

(A) Q.52

(D) 1

The solution of the differential equation, 2 x2y

(A) 1 Q.50

(C) 21/ 3

2

(B) a circle (D) hyperbola

Area enclosed by the curves y = lnx ; y = ln | x | ; y = | ln x | and y = | ln | x | | is equal to (A) 2 (B) 4 (C) 8 (D) cannot be determined x If y = ln | c x | (where c is an arbitrary constant) is the general solution of the differential equation

x x dy y = + φ  then the function φ  is : dx x  y y x2 (A) 2 y Q.55

x2 (B) – 2 y

(C)

y2 x2

(D) –

y2 x2

If the tangent to the curve y = 1 – x2 at x = α, where 0 < α < 1, meets the axes at P and Q. Also α varies, the minimum value of the area of the triangle OPQ is k times the area bounded by the axes and the part of the curve for which 0 < x < 1 , then k is equal to (A)

2 3

(B)

75 16

(C)

25 18

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

2 3

[7]

Q.56

d3y dy − 13 3 dx = K then the value of If the function y = e4x + 2e–x is a solution of the differential equation dx y K is (A) 4

(B) 6

(C) 9

(D) 12

Q.57

If (a, 0); a > 0 is the point where the curve y = sin2x – 3 sinx cuts the x-axis first, A is the area bounded by this part of the curve , the origin and the positive x-axis, then (A) 4A + 8 cosa = 7 (B) 4A + 8 sina = 7 (C) 4A – 8 sina = 7 (D) 4A – 8 cosa = 7

Q.58

2 (x2

2

A function y = f (x) satisfies (x + 1) . f ′ (x) –

ex + x) f (x) = , ∀ x > −1 ( x + 1)

If f (0) = 5 , then f (x) is

 3x + 5  x 2 .e (A)   x +1  Q.59

 6x + 5  x 2 .e (C)  2 + ( x 1 )  

 5 − 6x  x2 .e (D)   x +1 

Quest

The curve y = ax2 + bx + c passes through the point (1, 2) and its tangent at origin is the line y = x. The area bounded by the curve, the ordinate of the curve at minima and the tangent line is (A)

Q.60

 6x + 5  x 2 .e (B)   x +1 

1 24

(B)

1 12

(C)

1 8

(D)

1 6

The differential equation whose general solution is given by,

y = (c1 cos( x + c 2 ) ) − (c3e ( − x +c 4 ) ) + (c 5 sin x ) , where c1, c2, c3, c4, c5 are arbitrary constants, is (A)

(C)

Q.61

dx

4

d5y dx

5

−

d2y dx

2

+ y=0

(B)

+ y=0

(D)

d3 y dx

3

d3 y dx

3

+

−

d2y dx

2

d2y dx

2

+

dy + y=0 dx

+

dy − y=0 dx

dy – y = cos x – sin x with initial condition that y is dx bounded when x → ∞. The area enclosed by y = f (x), y = cos x and the y-axis is

A function y = f (x) satisfies the differential equation

(A) Q.62

d4y

2 −1

(B)

2

(C) 1

(D)

1 2

The curve, with the property that the projection of the ordinate on the normal is constant and has a length equal to 'a', is

 2 2  (A) x + a ln  y − a + y  = c   2 (C) (y – a) = cx

(B) x +

a 2 − y2 = c

(D) ay = tan–1 (x + c)

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

Q.63

Q.64

Area bounded by the curve y = min {sin2x, cos2x}and x-axis between the ordinates x = 0 and x = (A)

5π square units 2

(B)

(C)

5(π − 2) square units 8

π 1 (D)  −  square units  8 2

5π is 4

5(π − 2) square units 4

The equation to the orthogonal trajectories of the system of parabolas y = ax2 is x2 + y2 = c (A) 2

y2 (B) x + =c 2 2

x2 − y2 = c (C) 2

y2 (D) x − =c 2 2

x

Q.65

If

∫ t y(t )dt = x2 + y (x) then y as a function of x is a

(A) y = 2 – (2 +

x 2 −a 2 a2) e 2 2

(C) y = 2 – (1 + Q.66

Q.67

a2)

x −a e 2

(B) y = 1 – (2 +

x 2 −a 2 a2) e 2

2

(D) none

Quest

−  1  dy  satisfies the differential equation A curve y = f (x) passing through the point 1, + xe e dx  Then which of the following does not hold good? (A) f (x) is differentiable at x = 0. (B) f (x) is symmetric w.r.t. the origin. (C) f (x) is increasing for x < 0 and decreasing for x > 0. (D) f (x) has two inflection points.

x2 2

=0.

The substitution y = zα transforms the differential equation (x2y2 – 1)dy + 2xy3dx = 0 into a homogeneous differential equation for (A) α = – 1 (B) 0 (C) α = 1 (D) no value of α. x

Q.68

∫ t y(t ) dt = x2y (x), (x >0) is

A curve passing through (2, 3) and satisfying the differential equation

0

(A) x2 + y2 = 13 Q.69

(B) y2 =

(C)

x 2 y2 + =1 8 18

(D) xy = 6

Which one of the following curves represents the solution of the initial value problem Dy = 100 – y, where y (0) = 50

(B)

(A) Q.70

9 x 2

(C)

(D)

Solution of the differential equation  e x 2 + e y2  y dy + e x 2 ( xy 2 − x ) = 0, is   dx 2

2

(B) e y (x2 – 1) + e x = C

2

2

(D) e x (y – 1) + e y = C

(A) e x (y2 – 1) + e y = C (C) e y (y2 – 1) + e x = C

2

2

2

2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[9]

Direction for Q.71 to Q.73 (3 question together) Consider the function f (x) = x3 – 8x2 + 20x – 13 Q.71

Number of positive integers x for which f (x) is a prime number, is (A) 1 (B) 2 (C) 3

(D) 4

Q.72

The function f (x) defined for R → R (A) is one one onto (B) is many one onto (C) has 3 real roots (D) is such that f (x1) · f(x2) < 0 where x1 and x2 are the roots of f ' (x) = 0

Q.73

Area enclosed by y = f (x) and the co-ordinate axes is (A)

Q.74

13 12

(C)

71 12

(D) none 3π equals 2 3π (D) 1 + 2

Quest

3π –2 2

(B)

3π 2

(C) 2 +

3π 2

The area of the region under the graph of y = xe–ax as x varies from 0 to ∞, where 'a' is a positive constant, is (A)

Q.76

(B)

The area enclosed by the curves y = cos x, y = 1 + sin 2x and x = (A)

Q.75

65 12

1 a

(B)

1 1 + a a2

(C)

1 1 − a a2

(D)

1 a2

The polynomial f (x) satisfies the condition f (x + 1) = x2 + 4x. The area enclosed by y = f (x – 1) and the curve x2 + y = 0, is (A)

16 2 3

(B)

16 3

(C)

8 2 3

(D) none

Select the correct alternatives : (More than one are correct) Q.77

Family of curves whose tangent at a point with its intersection with the curve xy = c2 form an angle of is (A) y2 − 2xy − x2 = k

(B) y2 + 2xy − x2 = k

 x

(C) y = x - 2 c tan−1   + k c where k is an arbitrary constant .

Q.78

π 4

(D) y = c ln

c+x −x+k c−x

 dy   y  = y . ln   is :  dx   x

The general solution of the differential equation, x  (A) y = xe1 − cx

(B) y = xe1 + cx

(C) y = ex . ecx

(D) y = xecx

where c is an arbitrary constant. Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[10]

Q.79

Which of the following equation(s) is/are linear. dy y (A) + = ln x dx x

Q.80

 dy  (B) y   + 4x = 0  dx 

(C) dx + dy = 0

The function f(x) satisfying the equation, f2(x) + 4 f ′ (x) . f(x) + [f ′ (x)]2 = 0 . (A) f(x) = c . e

( 2- 3 ) x

(B) f(x) = c . e

) (C) f(x) = c . e( where c is an arbitrary constant. 3−2 x

Q.81

d 2y (D) 2 = cos x dx

( 2+ 3 ) x

(D) f(x) = c . e

(

)

− 2+ 3 x

The equation of the curve passing through (3 , 4) & satisfying the differential equation, 2

dy  dy  y   + (x − y) – x = 0 can be dx  dx  (A) x − y + 1 = 0 (B) x2 + y2 = 25 Q.82

(D) x + y − 7 = 0

The area bounded by a curve, the axis of co-ordinates & the ordinate of some point of the curve is equal to the length of the corresponding arc of the curve. If the curve passes through the point P (0, 1) then the equation of this curve can be

Quest

(A) y =

1 x (e − e – x + 2) 2

(C) y = 1 Q.83

(C) x2 + y2 − 5x − 10 = 0

(B) y =

1 x −x (e + e ) 2

(D) y =

2 e + e −x x

Identify the statement(s) which is/are True. (A) f(x , y) = ey/x + tan (B) x . ln

y is homogeneous of degree zero x

y y y2 dx + sin−1 dy = 0 is homogeneous of degree one x x x

(C) f(x , y) = x2 + sin x . cos y is not homogeneous (D) (x2 + y2) dx - (xy2 − y3) dy = 0 is a homegeneous differential equation . Q.84

The graph of the function y = f (x) passing through the point (0 , 1) and satisfying the differential equation dy + y cos x = cos x is such that dx (A) it is a constant function (C) it is neither an even nor an odd function

Q.85

(B) it is periodic (D) it is continuous & differentiable for all x .

dy sin 2 x ·sin x – y cos x + = 0 is such that, A function y = f (x) satisfying the differential equation dx x2 y → 0 as x → ∞ then the statement which is correct is π/ 2

(A) Lim f(x) = 1

(B)

x →0

∫ f(x) dx is less than 0

π 2

π/ 2

(C)

∫ f(x) dx is greater than unity

(D) f(x) is an odd function

0 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[11]

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]

Select the correct alternative : (Only one is correct) Q.1

D

Q.75

Q.74 C

A

Q.73

B

Q.72

C

Q.71

D

Q.68

Q.67 A

B

Q.66

A

Q.65

A

Q.64

A

Q.61

Q.60 B

A

Q.59

B

Q.58

A

Q.57

D

Q.54

Q.53 B

C

Q.52

D

Q.51

A

Q.50

D

Q.47

Q.46 B

C

Q.45

A

Q.44

B

Q.43

C

Q.40

Q.39 B

B

Q.38

B

Q.37

B

Q.36

A

Q.33

Q.32 D

D

Q.31

C

Q.30

D

Q.29

A

Q.26

Q.25 C

C

Q.24

A

Q.23

A

Q.22

A

Q.19

Q.18 A

A

Q.17

C

Q.16

D

Q.15

A

Q.12

B

Q.11

B

Q.10

B

Q.9

B

B

Q.5

C

Q.4

B

Q.3

A

Q.2

D

Q.8

A

Q.48

D

Q.41

D

Q.34

C

Q.27

A

Q.20

Q.13

Q.6

D

Q.49

A

Q.42

C

Q.35

A

Q.28

C

Q.21

D

Q.14

A

D

Q.7

A

Quest A,B,D

Q.84

C,D

Q.80

A

Q.76

B

Q.69

A

Q.62

A

Q.55

A

Q.70

C

Q.63

D

Q.56

Select the correct alternatives : (More than one are correct)

A,B,C

Q.85

A,B,C

Q.83

B,C

Q.82

A,B

Q.81

A,C,D

Q.79

A,B,C

Q.78

A,B,C,D

Q.77

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

DETERMINANT & MATRICES

Question bank on Determinant & Matrices There are 102 questions in this question bank. Select the correct alternative : (Only one is correct)

Q.1

a2 a 1 The value of the determinant cos (nx) cos(n + 1) x cos( n + 2) x is independent of : sin ( nx) sin (n + 1) x sin (n + 2) x

(A) n Q.2

(B) a

(B)

Q.5

A −1 2

(C) −a − b − c

(D) − 1

1 cos (β − α ) cos (γ − α ) If α, β & γ are real numbers , then D = cos(α − β) 1 cos(γ − β) = cos(α − γ ) cos(β − γ ) 1

If A =

LMcos θ Nsin θ

(B) cos α cos β cos γ (D) zero

OP , A cos θ Q

− sin θ

–1

is given by

(B) AT

(C) –AT

(D) A

If the system of equations ax + y + z = 0 , x + by + z = 0 & x + y + cz = 0 (a, b, c ≠ 1) has a non-trivial solution, then the value of (A) − 1

Q.8

(D) A2

If A and B are symmetric matrices, then ABA is (A) symmetric matrix (B) skew symmetric (C) diagonal matrix (D) scalar matrix

(A) –A Q.7

A 2

Quest (B) a−1 b−1 c−1

(A) − 1 (C) cos α + cos β + cos γ

Q.6

(C)

1+ a 1 1 If a, b, c are all different from zero & 1 1 + b 1 = 0 , then the value of a−1 + b−1 + c−1 is 1 1 1+ c

(A) abc Q.4

(D) a , n and x

0 1 − 1 A A is an involutary matrix given by A = 4 − 3 4  then the inverse of will be 2 3 − 3 4  (A) 2A

Q.3

(C) x

1 1 1 + + is : 1− a 1− b 1− c

(B) 0

(C) 1

(D) none of these

 3 4 6 − 1  2 4 3 0 2  , B =  0 1 , C = 1 . Out of the given matrix products Consider the matrices A =  2 1 − 2 5  − 1 2 (i) (AB)TC (ii) CTC(AB)T (iii) CTAB and (iv) ATABBTC (A) exactly one is defined (B) exactly two are defined (C) exactly three are defined (D) all four are defined Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.9

The value of a for which the system of equations ; a3x + (a +1)3 y + (a + 2)3 z = 0 , ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0 has a non-zero solution is : (A) 1 (B) 0 (C) − 1 (D) none of these

Q.10

If A =

FG 1 aIJ , then A (where n ∈ N) equals H 0 1K F 1 n aI F 1 naIJ F 1 naIJ J (A) G (B) G (C) G H0 1 K H0 0 K H0 1 K n

2

Q.11

Q.13

(B) 4

Q.15

(D) 8

Quest

x 2 + 3x x − 1 x + 3 If px4 + qx3 + rx2 + sx + t ≡ x + 1 2 − x x − 3 then t = x − 3 x + 4 3x

(B) 0

(C) 21

(D) none

If A and B are invertible matrices, which one of the following statements is not correct (A) Adj. A = |A| A–1 (B) det (A–1) = |det (A)|–1 (C) (A + B)–1 = B–1 + A–1 (D) (AB)–1 = B–1A–1 a2 + 1 ab ac 2 If D = ba b +1 bc then D = 2 ca cb c +1

(A) 1 + a2 + b2 + c2

Q.16

(C) 6

LM3 4 OP and B = LM−2 5OP then X such that A + 2X = B equals N1 −6Q N 6 1Q L 2 3OP L 3 5OP L 5 2OP (A) M (B) M (C) M (D) none of these N − 1 0Q N − 1 0Q N −1 0 Q If A =

(A) 33 Q.14

FG n naIJ H0 n K

1 + sin 2 x cos 2 x 4 sin 2x Let f (x) = sin 2 x 1 + cos 2 x 4 sin 2x , then the maximum value of f (x) = sin 2 x cos 2 x 1 + 4 sin 2x

(A) 2

Q.12

(D)

If A =

(B) a2 + b2 + c2

(C) (a + b + c)2

(D) none

FG a bIJ satisfies the equation x – (a + d)x + k = 0, then H c dK

(A) k = bc

2

(B) k = ad

(C) k = a2 + b2 + c2 + d2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) ad–bc

[3]

(a (b (c

x

Q.17

If a, b, c > 0 & x, y, z ∈ R , then the determinant

(B) a−xb−yc−z

(A) axbycz Q.18

+ a −x

y

+ b −y

z

+ c− z

2

) (a ) (b ) (c 2

2

x

− a −x

y

− b −y

z

− c− z

(C) a2xb2yc2z

) ) )

2 2

2

1 1 = 1

(D) zero

Identify the incorrect statement in respect of two square matrices A and B conformable for sum and product. (B) tr(αA) = α tr(A), α ∈ R (A) tr(A + B) = tr(A) + tr(B) T (C) tr(A ) = tr(A) (D) tr(AB) ≠ tr(BA) cos (θ + φ) − sin (θ + φ) cos 2φ

Q.19

The determinant

sin θ − cos θ

cos θ sin θ

(B) independent of θ (D) independent of θ & φ both

(A) 0 (C) independent of φ Q.20

sin φ is : cos φ

Quest

If A and B are non singular Matrices of same order then Adj. (AB) is (A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A) (C) Adj. A + Adj. B (D) none of these a +1 a + 2 a + p

Q.21

If a + 2 a + 3 a + q = 0 , then p, q, r are in : a +3 a +4

a+r

(A) AP

Q.22

LMx + λ Let A = M x MN x

(B) GP

x x+λ x

OP PP Q

(B) λ ≠ 0 (D) x ≠ 0, λ ≠ 0

1 logx y logx z 1 logy z is For positive numbers x, y & z the numerical value of the determinant logy x log z x log z y 1

(A) 0 Q.24

(D) none

x x , then A–1 exists if x+λ

(A) x ≠ 0 (C) 3x + λ ≠ 0, λ ≠ 0

Q.23

(C) HP

(B) 1

If K ∈ R0 then det. {adj (KIn)} is equal to (A) Kn – 1 (B) Kn(n – 1)

(C) 3

(D) none

(C) Kn

(D) K

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.25

b1 + c1 The determinant b 2 + c2 b 3 + c3 a1

b1

a1 (B) 2 a 2 a3

c1

(A) a 2 b 2 c2 a3

Q.26

Q.27

b3

c1 + a 1 c2 + a 2 c3 + a 3

c3

a 1 + b1 a 2 + b2 = a 3 + b3 b1 b2

c1 c2

b3

c3

a1

b1

c1

(C) 3 a 2 b 2 c2 a3

b3

c3

a1 (D) 4 a 2 a3

b1 b2

c1 c2

b3

c3

Which of the following is an orthogonal matrix

6 / 7 2 / 7 − 3 / 7  6/7  (A) 2 / 7 3 / 7 3 / 7 − 6 / 7 2 / 7 

3/ 7  6 / 7 2 / 7 2 / 7 − 3 / 7 6 /7   (B) 3 / 7 6 / 7 − 2 / 7 

− 6 / 7 − 2 / 7 − 3 / 7  3/ 7 6/7  (C)  2 / 7  − 3 / 7 6 / 7 2 / 7 

 6 / 7 − 2/ 7 3/ 7  2 / 7 − 3 / 7 (D)  2 / 7 − 6 / 7 2 / 7 3 / 7 

Quest

1+ a + x a +y a+z The determinant b + x 1 + b + y b + z = c+ x c+ y 1+ c + z

(A) (1 + a + b + c) (1 + x + y + z) − 3 (ax + by + cz) (B) a (x + y) + b (y + z) + c (z + x) − (xy + yz + zx) (C) x (a + b) + y (b + c) + z (c + a) − (ab + bc + ca) (D) none of these Q.28

Which of the following statements is incorrect for a square matrix A. ( | A | ≠ 0) (A) If A is a diagonal matrix, A–1 will also be a diagonal matrix (B) If A is a symmetric matrix, A–1 will also be a symmetric matrix (C) If A–1 = A ⇒ A is an idempotent matrix (D) If A–1 = A ⇒ A is an involutary matrix x

Q.29

Q.30

C1 y The determinant C1 z C1

x

C2 y C2 z C2

x

C3 C3 = z C3

y

1 xyz (x + y − z) (y + z − x) 4

(A)

1 xyz (x + y) (y + z) (z + x) 3

(B)

(C)

1 xyz (x − y) (y − z) (z − x) 12

(D) none

Which of the following is a nilpotent matrix

1 0 (A)   0 1 

cos θ − sin θ (B)    sin θ cos θ 

0 0  (C)   1 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

1 1 (D)   1 1

[5]

Q.31

a a3 If a, b, c are all different and b b 3 c c3

(A) abc (ab + bc + ca) = a + b + c (C) abc (a + b + c) = ab + bc + ca Q.32

Q.33

(B) (a + b + c) (ab + bc + ca) = abc (D) none of these

Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 : If A is an invertible 3 × 3 matrix and B is a 3 × 4 matrix, then A–1B is defined Statement-2 : It is never true that A + B, A – B, and AB are all defined. Statement-3 : Every matrix none of whose entries are zero is invertible. Statement-4 : Every invertible matrix is square and has no two rows the same. (A) TFFF (B) TTFF (C) TFFT (D) TTTF 1 If ω is one of the imaginary cube roots of unity, then the value of the determinant ω 3 ω2

(A) 1 Q.34

a 4 −1 b 4 − 1 = 0 , then : c4 − 1

Quest (B) 2

(C) 3

ω3 ω2 1 ω = ω 1

(D) none

Identify the correct statement : (A) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is singular (B) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is non singular (C) If A–1 exists , (adjA)–1 may or may not exist

cos x − sin x 0   (D) F(x) =  sin x cos x 0 , then F(x) . F(y) = F(x – y)  0 0 0

Q.35

a + p 1+ x u + f If the determinant b + q m + y v + g splits into exactly K determinants of order 3, each element of c+ r n + z w + h

which contains only one term, then the value of K, is (A) 6 (B) 8 (C) 9

(D) 12

Q.36

A and B are two given matrices such that the order of A is 3×4 , if A′ B and BA′ are both defined then (A) order of B′ is 3 × 4 (B) order of B′A is 4 × 4 (C) order of B′A is 3 × 3 (D) B′A is undefined

Q.37

If the system of equations x + 2y + 3z = 4 , x + py + 2z = 3 , x + 4y + µz = 3 has an infinite number of solutions , then : (A) p = 2 , µ = 3 (B) p = 2 , µ = 4 (C) 3 p = 2 µ (D) none of these

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[6]

Q.38

Q.39

 cos 2 α  cos 2 β sin α cos α  sin β cos β      ; B= If A = sin 2 α  sin 2 β   sin α cos α  sin β cos β are such that, AB is a null matrix, then which of the following should necessarily be an odd integral π multiple of . 2 (A) α (B) β (C) α – β (D) α + β

a b a+b a c a +c D1 b+d Let D1 = c d c + d and D2 = b d then the value of where b ≠ 0 and D2 a b a−b a c a+b+c ad ≠ bc, is (A) – 2

Q.40

(B) 0

Quest

1 + a 2 x (1 + b 2 ) x (1 + c 2 ) x 2 2 2 If a2 + b2 + c2 = – 2 and f (x) = (1 + a ) x 1 + b x (1 + c ) x then f (x) is a polynomial of degree (1 + a 2 ) x (1 + b 2 ) x 1 + c 2 x (A) 0

(B) 1

 88 0 0  (B)  0 88 0   0 0 88 

(D) 3

 68 0 0  (C)  0 68 0   0 0 68 

 34 0 0  (D)  0 34 0   0 0 34 

The values of θ, λ for which the following equations sinθx – cosθy + (λ+1)z = 0; cosθx + sinθy – λz = 0; λx +(λ + 1)y + cosθ z = 0 have non trivial solution, is (A) θ = nπ, λ ∈ R – {0} (B) θ = 2nπ, λ is any rational number (C) θ = (2n + 1)π, λ ∈ R+, n ∈ I

Q.44

(C) 2

x 3 2 Matrix A =  1 y 4  , if x y z = 60 and 8x + 4y + 3z = 20 , then A (adj A) is equal to 2 2 z

 64 0 0  (A)  0 64 0   0 0 64  Q.43

π , n∈ I 2 (D) A is a skew symmetric, for θ = nπ ; n ∈ I

(B) A is symmetric, for θ = (2n + 1)

(C) A is an orthogonal matrix for θ ∈ R

Q.42

(D) 2b

 cos θ − sin θ  For a given matrix A =  which of the following statement holds good?  sin θ cos θ  (A) A = A–1 ∀ θ∈ R

Q.41

(C) – 2b

(D) θ = (2n + 1)

π , λ ∈ R, n ∈ I 2

If A is matrix such that A2 + A + 2I = O, then which of the following is INCORRECT ? (A) A is non-singular

(B) A ≠ O

(C) A is symmetric

(D) A–1 = –

1 (A + I) 2

(Where I is unit matrix of order 2 and O is null matrix of order 2 ) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[7]

Q.45

Q.46

Q.47

The system of equations : 2x cos2θ + y sin2θ – 2sinθ = 0 x sin2θ + 2y sin2θ = – 2 cosθ x sinθ – y cosθ = 0 , for all values of θ, can (A) have a unique non - trivial solution (C) have infinite solutions

(B) not have a solution (D) have a trivial solution

1 1  is The number of solution of the matrix equation X2 =   2 3 (A) more than 2 (B) 2 (C) 1

(D) 0

If x, y, z are not all simultaneously equal to zero, satisfying the system of equations (sin 3 θ) x − y + z = 0 (cos 2 θ) x + 4 y + 3 z = 0 2x + 7y+ 7z = 0 then the number of principal values of θ is (A) 2 (B) 4 (C) 5 (D) 6

Quest

2 0 1  2 − 1 5 6 − 3 3   and 2A – B = 2 − 1 6 Q.48 Let A + 2B = − 5 3 1 0 1 2 then Tr (A) – Tr (B) has the value equal to (A) 0 (B) 1 (C) 2

Q.49

For a non - zero, real a, b and c

(A) – 4

Q.50

Q.51

(B) 0

a 2 + b2 c

c

c

a

b 2 + c2 a

a

b

b

c2 + a 2 b

(D) none

= α abc, then the values of α is

(C) 2

1 3  1 0 Given A = 2 2 ; I = 0 1 . If A – λI is a singular matrix then     2 (A) λ ∈ φ (B) λ – 3λ – 4 = 0 (C) λ2 + 3λ + 4 = 0

(D) 4

(D) λ2 – 3λ – 6 = 0

If the system of equations, a2 x − ay = 1 − a & bx + (3 − 2b) y = 3 + a possess a unique solution x = 1, y = 1 then : (A) a = 1 ; b = − 1 (B) a = − 1 , b = 1 (C) a = 0 , b = 0 (D) none

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

sin θ 1   1 − sin θ 1 sin θ , where 0 ≤ θ < 2π, then Q.52 Let A =   − 1 − sin θ 1  (A) Det (A) = 0 (B) Det A ∈ (0, ∞) (C) Det (A) ∈ [2, 4] Q.53

Q.54

Number of value of 'a' for which the system of equations, a2 x + (2 − a) y = 4 + a2 a x + (2 a − 1) y = a5 − 2 possess no solution is (A) 0 (B) 1 (C) 2

(D) Det A ∈ [2, ∞)

(D) infinite

0 1 2  1 / 2 − 1 / 2 1 / 2 3 c  , then If A = 1 2 3 , A–1 =  − 4 3 a 1  5 / 2 − 3 / 2 1 / 2 (A) a = 1, c = – 1

(B) a = 2, c = –

1 2

(C) a = – 1, c = 1

(D) a =

1 1 ,c= 2 2

Quest

Q.55

Number of triplets of a, b & c for which the system of equations, ax − by = 2a − b and (c + 1) x + cy = 10 − a + 3 b has infinitely many solutions and x = 1, y = 3 is one of the solutions, is : (A) exactly one (B) exactly two (C) exactly three (D) infinitely many

Q.56

D is a 3 x 3 diagonal matrix. Which of the following statements is not true? (A) D′ = D (B) AD = DA for every matrix A of order 3 x 3 –1 (C) D if exists is a scalar matrix (D) none of these

Q.57

The following system of equations 3x – 7y + 5z = 3; 3x + y + 5z = 7 and 2x + 3y + 5z = 5 are (A) consistent with trivial solution (B) consistent with unique non trivial solution (C) consistent with infinite solution (D) inconsistent with no solution n

Q.58

If A1, A3, ..... A2n – 1 are n skew symmetric matrices of same order then B =

∑ (2r − 1)(A 2r −1 ) 2r −1 will r =1

be (A) symmetric (C) neither symmetric nor skew symmetric

(B) skew symmetric (D) data is adequate

x 3x + 2 2x − 1 2 x − 1 4x 3x + 1 = 0 is Q.59 The number of real values of x satisfying 7 x − 2 17 x + 6 12x − 1 (A) 3

(B) 0

(C) more than 3

(D) 1

λ λ + 1 λ −1 2 − 1 3  has no inverse Q.60 Number of real values of λ for which the matrix A =  λ + 3 λ − 2 λ + 7 (A) 0 (B) 1 (C) 2 (D) infinite Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[9]

1 z (y + z) − x2 Q.61 If D = y(y + z) − x2z

1 z 1 x x + 2y + z xz

(x + y) z2 1 x then, the incorrect statement is y(x + y) − xz 2 −

(A) D is independent of x (C) D is independent of z Q.62

(B) D is independent of y (D) D is dependent on x, y, z

If every element of a square non singular matrix A is multiplied by k and the new matrix is denoted by B then | A–1| and | B–1| are related as (A) | A–1| = k | B–1|

(B) | A–1| =

1 –1 |B | k

(C) | A–1| = kn | B–1|

(D) | A–1| = k–n | B–1|

where n is order of matrices.

Q.63

Quest

If f ′ (x) =

mx − p n+p

mx n

1 − 1 1  Q.64 Let A = 2 1 − 3 and 10B = 1 1 1  (A) – 2 (B) – 1 x − 1 ( x − 1) 2 If D(x) = (A) 5

then y = f(x) represents

mx + 2n mx + 2n + p mx + 2n − p

(A) a straight line parallel to x- axis (C) parabola

Q.65

mx + p n−p

(B) a straight line parallel to y- axis (D) a straight line with negative slope

2 2 4 − 5 0 α  . If B is the inverse of matrix A, then α is  1 − 2 3  (C) 2 (D) 5

x3

x −1

x2

( x + 1)3

x

( x + 1) 2

( x + 1)3

(B) – 2

then the coefficient of x in D(x) is (C) 6

(D) 0

Q.66

The set of equations λx – y + (cosθ) z = 0 3x + y + 2z =0 (cosθ)x + y + 2z = 0 0 < θ < 2π , has non- trivial solution(s) (A) for no value of λ and θ (B) for all values of λ and θ (C) for all values of λ and only two values of θ (D) for only one value of λ and all values of θ

Q.67

Matrix A satisfies A2 = 2A – I where I is the identity matrix then for n ≥ 2, An is equal to (n ∈ N) (A) nA – I (B) 2n – 1A – (n – 1)I (C) nA – (n – 1)I (D) 2n – 1A – I

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[10]

a2 +1 Q.68

If a, b, c are real then the value of determinant (A) a + b + c = 0

Q.69

(B) a + b + c = 1

ab ac

ab

ac

2

b + 1 bc = 1 if bc c2 + 1

(C) a + b + c = –1

(D) a = b = c = 0

Read the following mathematical statements carefully: I. There can exist two triangles such that the sides of one triangle are all less than 1 cm while the sides of the other triangle are all bigger than 10 metres, but the area of the first triangle is larger than the area of second triangle. II. If x, y, z are all different real numbers, then 2

1 1 1  1 1 1  + +   . + + 2 2 2 =  ( x − y) ( y − z) (z − x ) x−y y−z z−x III. log3x · log4x · log5x = (log3x · log4x) + (log4x · log5x) + (log5x · log3x) is true for exactly for one real value of x. IV. A matrix has 12 elements. Number of possible orders it can have is six. Now indicate the correct alternatively. (A) exactly one statement is INCORRECT. (B) exactly two statements are INCORRECT. (C) exactly three statements are INCORRECT. (D) All the four statements are INCORRECT. Q.70

Quest

The system of equations (sinθ)x + 2z = 0, (cosθ)x + (sinθ)y = 0 , (cosθ)y + 2z = a has (A) no unique solution (B) a unique solution which is a function of a and θ (C) a unique solution which is independent of a and θ (D) a unique solution which is independent of θ only

1 2 3  0 2 0 5   Q.71 Let A = and b = − 3 . Which of the following is true? 0 2 1  1 (A) Ax = b has a unique solution. (B) Ax = b has exactly three solutions. (C) Ax = b has infinitely many solutions. (D) Ax = b is inconsistent. Q.72

The number of positive integral solutions of the equation

x3 +1 x 2 y x 2z xy 2 y3 + 1 y 2 z = 11 is xz 2 yz 2 z 3 + 1 (A) 0 Q.73

(B) 3

(C) 6

(D) 12

If A, B and C are n × n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of the det(A2BC–1) is equal to (A)

6 5

(B)

12 5

(C)

18 5

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

24 5

[11]

(1 + x ) 2

Q.74

The equation 2 x + 1 x +1

(1 − x ) 2

− (2 + x 2 )

3x 2x

1 − 5x 2 − 3x

(A) has no real solution (C) has two real and two non-real solutions

Q.75

2 + (1 − x ) 1 − 2x

2x + 1

x +1

3x

2x

3x − 2 2 x − 3

(B) has 4 real solutions (D) has infinite number of solutions , real or non-real

(B) 9b2 (a + b)

(C) 3b2 (a + b)

(D) 7a2 (a + b)

2 1  3 4  3 − 4 Let three matrices A = 4 1 ; B = 2 3 and C = − 2 3  then        A ( BC) 2   A ( BC) 3   ABC      + ....... + ∞ =   tr(A) + tr + tr   + tr   4 8 2       (A) 6 (B) 9 (C) 12

Q.77

=0

a a + b a + 2b a + 2b a a+b The value of the determinant is a + b a + 2b a (A) 9a2 (a + b)

Q.76

(1 + x ) 2

Quest (D) none

The number of positive integral solutions

1− λ 2 1 −3 λ − 2 = 0 is 2 − 2 1+ λ (A) 0 Q.78

(B) 2

(C) 3

(D) 1

P is an orthogonal matrix and A is a periodic matrix with period 4 and Q = PAPT then X = PTQ2005P will be equal to (A) A (B) A2 (C) A3 (D) A4

a−x b b a−x b =0, then its other two roots are Q.79 If x = a + 2b satisfies the cubic (a, b∈R) f (x)= b b b a−x (A) real and different (C) imaginary Q.80

Q.81

(B) real and coincident (D) such that one is real and other imaginary

 1  − 1  1  1  A is a 2 × 2 matrix such that A − 1 =  2  and A2 − 1 = 0 . The sum of the elements of A, is         (A) –1 (B) 0 (C) 2 (D) 5

Three digit numbers x17, 3y6 and 12z where x, y, z are integers from 0 to 9, are divisible by a fixed

x 3 1 constant k. Then the determinant 7 6 z must be divisible by 1 y 2 (A) k

(B) k2

(C) k3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) None

[12]

Q.82

Q.83

In a square matrix A of order 3, ai i's are the sum of the roots of the equation x2 – (a + b)x + ab= 0; ai , i + 1's are the product of the roots, ai , i – 1's are all unity and the rest of the elements are all zero. The value of the det. (A) is equal to (A) 0 (B) (a + b) 3 (C) a3 – b3 (D) (a2 + b2)(a + b)

28 25 38 Let N = 42 38 65 , then the number of ways is which N can be resolved as a product of two 56 47 83 divisors which are relatively prime is (A) 4 (B) 8

Q.84

(D) 16

1 1 1 1 + sin A 1 + sin B 1 + sin C = 0, then If A, B, C are the angles of a triangle and 2 2 sin A + sin A sin B + sin B sin C + sin 2 C the triangle is (A) a equilateral (C) a right angled triangle

Q.85

(C) 9

(B) an isosceles (D) any triangle

Quest

x 1 ln (1 + sin x ) x 3 − 16x Lim − ; c = Lim and Let a = x →1 ln x x ln x ; b = Lim 2 x →0 x →0 4 x + x x ( x + 1) 3 a b  , then the matrix c d  is x →−1 3(sin( x + 1) − ( x + 1) )   (A) Idempotent (B) Involutary (C) Non singular

d = Lim

Q.86

Q.87

Q.88

If the system of linear equations x + 2ay + az = 0 x + 3by + bz = 0 x + 4cy + cz = 0 has a non-zero solution, then a, b, c (A) are in G..P. (C) satisfy a + 2b + 3c = 0

(D) Nilpotent

(B) are in H.P. (D) are in A.P.

Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 : If the graphs of two linear equations in two variables are neither parallel nor the same, then there is a unique solution to the system. Statement-2 : If the system of equations ax + by = 0, cx + dy = 0 has a non-zero solution, then it has infinitely many solutions. Statement-3 : The system x + y + z = 1, x = y, y = 1 + z is inconsistent. Statement-4 : If two of the equations in a system of three linear equations are inconsistent, then the whole system is inconsistent. (A) FFTT (B) TTFT (C) TTFF (D) TTTF

1 + x 2 − y 2 − z 2 2( xy + z ) 2( zx − y)    2 2 2 1+ y − z − x 2( yz + x )  then det. A is equal to Let A =  2( xy − z )  2( zx + y) 2( yz − x ) 1 + z 2 − x 2 − y 2   (A) (1 + xy + yz + zx)3 (C) (xy + yz + zx)3

(B) (1 + x2 + y2 + z2)3 (D) (1 + x3 + y3 + z3)2 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Select the correct alternatives : (More than one are correct) Q.89

The set of equations x – y + 3z = 2 , 2x – y + z = 4 , x – 2y + αz = 3 has (A) unique soluton only for α = 0 (B) unique solution for α ≠ 8 (C) infinite number of solutions for α = 8 (D) no solution for α = 8

Q.90

Suppose a1, a2, ....... real numbers, with a1 ≠ 0. If a1, a2, a3, ..........are in A.P. then

LMa (A) A = Ma MNa

1

a2

4

a5

5

a6

OP a P is singular a PQ a3 6

7

(B) the system of equations a1x + a2y + a3z = 0, a4x + a5y + a6z = 0, a7x + a8y + a9z = 0 has infinite number of solutions (C) B =

LM a Nia

1

ia 2

2

a1

OP Q

is non singular ; where i =

−1

(D) none of these Q.91

a2 The determinant b 2 c2

a 2 − ( b − c) 2 b 2 − (c − a )2 c2 − (a − b) 2

bc ca is divisible by : ab

Quest

(A) a + b + c (C) a2 + b2 + c2

(B) (a + b) (b + c) (c + a) (D) (a − b) (b − c) (c − a)

Q.92

If A and B are 3 × 3 matrices and | A | ≠ 0, then which of the following are true? (A) | AB | = 0 ⇒ | B | = 0 (B) | AB | = 0 ⇒ B = 0 –1 –1 (C) | A | = | A | (D) | A + A | = 2 | A |

Q.93

The value of θ lying between −

π π π & and 0 ≤ A ≤ and satisfying the equation 4 2 2

1 + sin 2 A cos 2 A 2 sin 4θ 2 2 sin A 1 + cos A 2 sin 4θ = 0 are : 2 2 sin A cos A 1 + 2 sin 4θ

Q.94

Q.95

(A) A =

π π , θ = − 4 8

(B) A =

3π =θ 8

(C) A =

π π , θ= − 5 8

(D) A =

π 3π , θ= 6 8

If AB = A and BA = B, then (A) A2B = A2 (B) B2A = B2

(C) ABA = A

(D) BAB = B

x a b The solution(s) of the equation a x a = 0 is/are : b b x

(A) x = − (a + b)

(B) x = a

(C) x = b

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) − b

[14]

Q.96

Q.97

If D1 and D2 are two 3 x 3 diagonal matrices, then (A) D1D2 is a diagonal matrix (B) D1D2 = D2D1 2 2 (C) D1 + D2 is a diagonal matrix (D) none of these a a2 x x 2 = 0 , then ab a 2

1 If 1 b2

(A) x = a Q.98

(B) x = b

1 a

(D) x =

a b

Which of the following determinant(s) vanish(es)? 1 ab

1 b c b c (b + c) (A) 1 ca ca (c + a ) 1 a b a b (a + b)

(B) 1 b c 1 ca

0 a−b a−c b−a 0 b−c c−a c−b 0

(C)

Q.99

(C) x =

If A =

1 1 a + b 1+1 b c 1+1 c a

logx x y z logx y logx z 1 logy z (D) logy x y z log z xy z log z y 1

Quest LMa bOP (where bc ≠ 0) satisfies the equations x + k = 0, then Nc d Q 2

(A) a + d = 0

(B) k = –|A|

(C) k = |A|

(D) none of these

Q.100 The value of θ lying between θ = 0 & θ = π/2 & satisfying the equation :

1+sin 2 θ cos 2 θ 4sin4θ 2 2 sin θ 1+cos θ 4sin4θ = 0 are : sin 2 θ cos 2 θ 1+4sin4θ (A)

7π 24

(B)

5π 24

(C)

11 π 24

(D)

π 24

p + sin x q + sin x p − r + sin x Q.101 If p, q, r, s are in A.P. and f (x) = q + sin x r + sin x − 1 + sin x such that r + sin x s + sin x s − q + sin x

2

∫ f (x)dx = – 4 then 0

the common difference of the A.P. can be : (A) − 1

LM1 Q.102 Let A = 2 MM N2

(B)

1 2

(C) 1

(D) none

OP P 1QP

2 2 1 2 , then 2

(A) A2 – 4A – 5I3 = 0 (C) A3 is not invertible

1 (A – 4I3) 5 (D) A2 is invertible (B) A–1 =

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[16]

Q.99 A,C

A,B,C,D

Q.96

Q.95 A,B,C

A,B,C,D

Q.92

Q.91 A,C,D

A,B,C

Q.90

Q.88 B

B

Q.87

B

Q.83 B

D

Q.82

A

Q.81

Q.79

Q.78 A

C

Q.77

A

Q.76

Q.74

Q.73 B

B

Q.72

A

Q.71

Q.69

Q.68 D

C

Q.67

A

Q.66

Q.64

Q.63 A

C

Q.62

D

Q.61

Q.59

Q.58 B

B

Q.57

B

Q.56

Q.54

Q.53 C

C

Q.52

A

Q.51

Q.49

Q.48 C

C

Q.47

A

Q.46

Q.44

Q.43 D

C

Q.42

C

Q.41

Q.39

Q.38 C

D

Q.37

B

Q.36

Q.34

Q.33 C

C

Q.32

A

Q.31

Q.29

Q.28 C

A

Q.27

A

Q.26

Q.24

Q.23 A

C

Q.22

A

Q.21

Q.19

Q.18 D

D

Q.17

D

Q.16

Q.14

Q.13 C

D

Q.12

C

Q.11

Q.9

C

Q.8

C

Q.7

B

D

Q.3

A

Q.2

A

Q.1 Q.6

Q.4

B

Q.70

A

Q.65

D

Q.60

B

Q.55

B

Q.50

B

Q.45

C

C

Q.40

A

B

Q.35

B

C

Q.30

C

B

Q.25

B

B

Q.20

B

A

Q.15

C

A

Q.10

C

D

Q.5

A

Quest Q.98

A,D

Q.97

Q.94

A,B,C,D

Q.93

B, D

Q.89

Q.86

Q.101 A,C

Q.84

D

A

C

D

A

D B B

D

Q.85

D

Q.80

B

Q.75

A,C A,B,C

Q.100 A,C

Q.102 A,B,D

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

DETERMINANT & MATRICES

Question bank on Determinant & Matrices There are 102 questions in this question bank. Select the correct alternative : (Only one is correct)

Q.1

a2 a 1 The value of the determinant cos (nx) cos(n + 1) x cos( n + 2) x is independent of : sin ( nx) sin (n + 1) x sin (n + 2) x

(A) n Q.2

(B) a

(B)

Q.5

A −1 2

(C) −a − b − c

(D) − 1

1 cos (β − α ) cos (γ − α ) If α, β & γ are real numbers , then D = cos(α − β) 1 cos(γ − β) = cos(α − γ ) cos(β − γ ) 1

If A =

LMcos θ Nsin θ

(B) cos α cos β cos γ (D) zero

OP , A cos θ Q

− sin θ

–1

is given by

(B) AT

(C) –AT

(D) A

If the system of equations ax + y + z = 0 , x + by + z = 0 & x + y + cz = 0 (a, b, c ≠ 1) has a non-trivial solution, then the value of (A) − 1

Q.8

(D) A2

If A and B are symmetric matrices, then ABA is (A) symmetric matrix (B) skew symmetric (C) diagonal matrix (D) scalar matrix

(A) –A Q.7

A 2

Quest (B) a−1 b−1 c−1

(A) − 1 (C) cos α + cos β + cos γ

Q.6

(C)

1+ a 1 1 If a, b, c are all different from zero & 1 1 + b 1 = 0 , then the value of a−1 + b−1 + c−1 is 1 1 1+ c

(A) abc Q.4

(D) a , n and x

0 1 − 1 A A is an involutary matrix given by A = 4 − 3 4  then the inverse of will be 2 3 − 3 4  (A) 2A

Q.3

(C) x

1 1 1 + + is : 1− a 1− b 1− c

(B) 0

(C) 1

(D) none of these

 3 4 6 − 1  2 4 3 0 2  , B =  0 1 , C = 1 . Out of the given matrix products Consider the matrices A =  2 1 − 2 5  − 1 2 (i) (AB)TC (ii) CTC(AB)T (iii) CTAB and (iv) ATABBTC (A) exactly one is defined (B) exactly two are defined (C) exactly three are defined (D) all four are defined Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.9

The value of a for which the system of equations ; a3x + (a +1)3 y + (a + 2)3 z = 0 , ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0 has a non-zero solution is : (A) 1 (B) 0 (C) − 1 (D) none of these

Q.10

If A =

FG 1 aIJ , then A (where n ∈ N) equals H 0 1K F 1 n aI F 1 naIJ F 1 naIJ J (A) G (B) G (C) G H0 1 K H0 0 K H0 1 K n

2

Q.11

Q.13

(B) 4

Q.15

(D) 8

Quest

x 2 + 3x x − 1 x + 3 If px4 + qx3 + rx2 + sx + t ≡ x + 1 2 − x x − 3 then t = x − 3 x + 4 3x

(B) 0

(C) 21

(D) none

If A and B are invertible matrices, which one of the following statements is not correct (A) Adj. A = |A| A–1 (B) det (A–1) = |det (A)|–1 (C) (A + B)–1 = B–1 + A–1 (D) (AB)–1 = B–1A–1 a2 + 1 ab ac 2 If D = ba b +1 bc then D = 2 ca cb c +1

(A) 1 + a2 + b2 + c2

Q.16

(C) 6

LM3 4 OP and B = LM−2 5OP then X such that A + 2X = B equals N1 −6Q N 6 1Q L 2 3OP L 3 5OP L 5 2OP (A) M (B) M (C) M (D) none of these N − 1 0Q N − 1 0Q N −1 0 Q If A =

(A) 33 Q.14

FG n naIJ H0 n K

1 + sin 2 x cos 2 x 4 sin 2x Let f (x) = sin 2 x 1 + cos 2 x 4 sin 2x , then the maximum value of f (x) = sin 2 x cos 2 x 1 + 4 sin 2x

(A) 2

Q.12

(D)

If A =

(B) a2 + b2 + c2

(C) (a + b + c)2

(D) none

FG a bIJ satisfies the equation x – (a + d)x + k = 0, then H c dK

(A) k = bc

2

(B) k = ad

(C) k = a2 + b2 + c2 + d2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) ad–bc

[3]

(a (b (c

x

Q.17

If a, b, c > 0 & x, y, z ∈ R , then the determinant

(B) a−xb−yc−z

(A) axbycz Q.18

+ a −x

y

+ b −y

z

+ c− z

2

) (a ) (b ) (c 2

2

x

− a −x

y

− b −y

z

− c− z

(C) a2xb2yc2z

) ) )

2 2

2

1 1 = 1

(D) zero

Identify the incorrect statement in respect of two square matrices A and B conformable for sum and product. (B) tr(αA) = α tr(A), α ∈ R (A) tr(A + B) = tr(A) + tr(B) T (C) tr(A ) = tr(A) (D) tr(AB) ≠ tr(BA) cos (θ + φ) − sin (θ + φ) cos 2φ

Q.19

The determinant

sin θ − cos θ

cos θ sin θ

(B) independent of θ (D) independent of θ & φ both

(A) 0 (C) independent of φ Q.20

sin φ is : cos φ

Quest

If A and B are non singular Matrices of same order then Adj. (AB) is (A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A) (C) Adj. A + Adj. B (D) none of these a +1 a + 2 a + p

Q.21

If a + 2 a + 3 a + q = 0 , then p, q, r are in : a +3 a +4

a+r

(A) AP

Q.22

LMx + λ Let A = M x MN x

(B) GP

x x+λ x

OP PP Q

(B) λ ≠ 0 (D) x ≠ 0, λ ≠ 0

1 logx y logx z 1 logy z is For positive numbers x, y & z the numerical value of the determinant logy x log z x log z y 1

(A) 0 Q.24

(D) none

x x , then A–1 exists if x+λ

(A) x ≠ 0 (C) 3x + λ ≠ 0, λ ≠ 0

Q.23

(C) HP

(B) 1

If K ∈ R0 then det. {adj (KIn)} is equal to (A) Kn – 1 (B) Kn(n – 1)

(C) 3

(D) none

(C) Kn

(D) K

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.25

b1 + c1 The determinant b 2 + c2 b 3 + c3 a1

b1

a1 (B) 2 a 2 a3

c1

(A) a 2 b 2 c2 a3

Q.26

Q.27

b3

c1 + a 1 c2 + a 2 c3 + a 3

c3

a 1 + b1 a 2 + b2 = a 3 + b3 b1 b2

c1 c2

b3

c3

a1

b1

c1

(C) 3 a 2 b 2 c2 a3

b3

c3

a1 (D) 4 a 2 a3

b1 b2

c1 c2

b3

c3

Which of the following is an orthogonal matrix

6 / 7 2 / 7 − 3 / 7  6/7  (A) 2 / 7 3 / 7 3 / 7 − 6 / 7 2 / 7 

3/ 7  6 / 7 2 / 7 2 / 7 − 3 / 7 6 /7   (B) 3 / 7 6 / 7 − 2 / 7 

− 6 / 7 − 2 / 7 − 3 / 7  3/ 7 6/7  (C)  2 / 7  − 3 / 7 6 / 7 2 / 7 

 6 / 7 − 2/ 7 3/ 7  2 / 7 − 3 / 7 (D)  2 / 7 − 6 / 7 2 / 7 3 / 7 

Quest

1+ a + x a +y a+z The determinant b + x 1 + b + y b + z = c+ x c+ y 1+ c + z

(A) (1 + a + b + c) (1 + x + y + z) − 3 (ax + by + cz) (B) a (x + y) + b (y + z) + c (z + x) − (xy + yz + zx) (C) x (a + b) + y (b + c) + z (c + a) − (ab + bc + ca) (D) none of these Q.28

Which of the following statements is incorrect for a square matrix A. ( | A | ≠ 0) (A) If A is a diagonal matrix, A–1 will also be a diagonal matrix (B) If A is a symmetric matrix, A–1 will also be a symmetric matrix (C) If A–1 = A ⇒ A is an idempotent matrix (D) If A–1 = A ⇒ A is an involutary matrix x

Q.29

Q.30

C1 y The determinant C1 z C1

x

C2 y C2 z C2

x

C3 C3 = z C3

y

1 xyz (x + y − z) (y + z − x) 4

(A)

1 xyz (x + y) (y + z) (z + x) 3

(B)

(C)

1 xyz (x − y) (y − z) (z − x) 12

(D) none

Which of the following is a nilpotent matrix

1 0 (A)   0 1 

cos θ − sin θ (B)    sin θ cos θ 

0 0  (C)   1 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

1 1 (D)   1 1

[5]

Q.31

a a3 If a, b, c are all different and b b 3 c c3

(A) abc (ab + bc + ca) = a + b + c (C) abc (a + b + c) = ab + bc + ca Q.32

Q.33

(B) (a + b + c) (ab + bc + ca) = abc (D) none of these

Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 : If A is an invertible 3 × 3 matrix and B is a 3 × 4 matrix, then A–1B is defined Statement-2 : It is never true that A + B, A – B, and AB are all defined. Statement-3 : Every matrix none of whose entries are zero is invertible. Statement-4 : Every invertible matrix is square and has no two rows the same. (A) TFFF (B) TTFF (C) TFFT (D) TTTF 1 If ω is one of the imaginary cube roots of unity, then the value of the determinant ω 3 ω2

(A) 1 Q.34

a 4 −1 b 4 − 1 = 0 , then : c4 − 1

Quest (B) 2

(C) 3

ω3 ω2 1 ω = ω 1

(D) none

Identify the correct statement : (A) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is singular (B) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is non singular (C) If A–1 exists , (adjA)–1 may or may not exist

cos x − sin x 0   (D) F(x) =  sin x cos x 0 , then F(x) . F(y) = F(x – y)  0 0 0

Q.35

a + p 1+ x u + f If the determinant b + q m + y v + g splits into exactly K determinants of order 3, each element of c+ r n + z w + h

which contains only one term, then the value of K, is (A) 6 (B) 8 (C) 9

(D) 12

Q.36

A and B are two given matrices such that the order of A is 3×4 , if A′ B and BA′ are both defined then (A) order of B′ is 3 × 4 (B) order of B′A is 4 × 4 (C) order of B′A is 3 × 3 (D) B′A is undefined

Q.37

If the system of equations x + 2y + 3z = 4 , x + py + 2z = 3 , x + 4y + µz = 3 has an infinite number of solutions , then : (A) p = 2 , µ = 3 (B) p = 2 , µ = 4 (C) 3 p = 2 µ (D) none of these

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[6]

Q.38

Q.39

 cos 2 α  cos 2 β sin α cos α  sin β cos β      ; B= If A = sin 2 α  sin 2 β   sin α cos α  sin β cos β are such that, AB is a null matrix, then which of the following should necessarily be an odd integral π multiple of . 2 (A) α (B) β (C) α – β (D) α + β

a b a+b a c a +c D1 b+d Let D1 = c d c + d and D2 = b d then the value of where b ≠ 0 and D2 a b a−b a c a+b+c ad ≠ bc, is (A) – 2

Q.40

(B) 0

Quest

1 + a 2 x (1 + b 2 ) x (1 + c 2 ) x 2 2 2 If a2 + b2 + c2 = – 2 and f (x) = (1 + a ) x 1 + b x (1 + c ) x then f (x) is a polynomial of degree (1 + a 2 ) x (1 + b 2 ) x 1 + c 2 x (A) 0

(B) 1

 88 0 0  (B)  0 88 0   0 0 88 

(D) 3

 68 0 0  (C)  0 68 0   0 0 68 

 34 0 0  (D)  0 34 0   0 0 34 

The values of θ, λ for which the following equations sinθx – cosθy + (λ+1)z = 0; cosθx + sinθy – λz = 0; λx +(λ + 1)y + cosθ z = 0 have non trivial solution, is (A) θ = nπ, λ ∈ R – {0} (B) θ = 2nπ, λ is any rational number (C) θ = (2n + 1)π, λ ∈ R+, n ∈ I

Q.44

(C) 2

x 3 2 Matrix A =  1 y 4  , if x y z = 60 and 8x + 4y + 3z = 20 , then A (adj A) is equal to 2 2 z

 64 0 0  (A)  0 64 0   0 0 64  Q.43

π , n∈ I 2 (D) A is a skew symmetric, for θ = nπ ; n ∈ I

(B) A is symmetric, for θ = (2n + 1)

(C) A is an orthogonal matrix for θ ∈ R

Q.42

(D) 2b

 cos θ − sin θ  For a given matrix A =  which of the following statement holds good?  sin θ cos θ  (A) A = A–1 ∀ θ∈ R

Q.41

(C) – 2b

(D) θ = (2n + 1)

π , λ ∈ R, n ∈ I 2

If A is matrix such that A2 + A + 2I = O, then which of the following is INCORRECT ? (A) A is non-singular

(B) A ≠ O

(C) A is symmetric

(D) A–1 = –

1 (A + I) 2

(Where I is unit matrix of order 2 and O is null matrix of order 2 ) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[7]

Q.45

Q.46

Q.47

The system of equations : 2x cos2θ + y sin2θ – 2sinθ = 0 x sin2θ + 2y sin2θ = – 2 cosθ x sinθ – y cosθ = 0 , for all values of θ, can (A) have a unique non - trivial solution (C) have infinite solutions

(B) not have a solution (D) have a trivial solution

1 1  is The number of solution of the matrix equation X2 =   2 3 (A) more than 2 (B) 2 (C) 1

(D) 0

If x, y, z are not all simultaneously equal to zero, satisfying the system of equations (sin 3 θ) x − y + z = 0 (cos 2 θ) x + 4 y + 3 z = 0 2x + 7y+ 7z = 0 then the number of principal values of θ is (A) 2 (B) 4 (C) 5 (D) 6

Quest

2 0 1  2 − 1 5 6 − 3 3   and 2A – B = 2 − 1 6 Q.48 Let A + 2B = − 5 3 1 0 1 2 then Tr (A) – Tr (B) has the value equal to (A) 0 (B) 1 (C) 2

Q.49

For a non - zero, real a, b and c

(A) – 4

Q.50

Q.51

(B) 0

a 2 + b2 c

c

c

a

b 2 + c2 a

a

b

b

c2 + a 2 b

(D) none

= α abc, then the values of α is

(C) 2

1 3  1 0 Given A = 2 2 ; I = 0 1 . If A – λI is a singular matrix then     2 (A) λ ∈ φ (B) λ – 3λ – 4 = 0 (C) λ2 + 3λ + 4 = 0

(D) 4

(D) λ2 – 3λ – 6 = 0

If the system of equations, a2 x − ay = 1 − a & bx + (3 − 2b) y = 3 + a possess a unique solution x = 1, y = 1 then : (A) a = 1 ; b = − 1 (B) a = − 1 , b = 1 (C) a = 0 , b = 0 (D) none

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

sin θ 1   1 − sin θ 1 sin θ , where 0 ≤ θ < 2π, then Q.52 Let A =   − 1 − sin θ 1  (A) Det (A) = 0 (B) Det A ∈ (0, ∞) (C) Det (A) ∈ [2, 4] Q.53

Q.54

Number of value of 'a' for which the system of equations, a2 x + (2 − a) y = 4 + a2 a x + (2 a − 1) y = a5 − 2 possess no solution is (A) 0 (B) 1 (C) 2

(D) Det A ∈ [2, ∞)

(D) infinite

0 1 2  1 / 2 − 1 / 2 1 / 2 3 c  , then If A = 1 2 3 , A–1 =  − 4 3 a 1  5 / 2 − 3 / 2 1 / 2 (A) a = 1, c = – 1

(B) a = 2, c = –

1 2

(C) a = – 1, c = 1

(D) a =

1 1 ,c= 2 2

Quest

Q.55

Number of triplets of a, b & c for which the system of equations, ax − by = 2a − b and (c + 1) x + cy = 10 − a + 3 b has infinitely many solutions and x = 1, y = 3 is one of the solutions, is : (A) exactly one (B) exactly two (C) exactly three (D) infinitely many

Q.56

D is a 3 x 3 diagonal matrix. Which of the following statements is not true? (A) D′ = D (B) AD = DA for every matrix A of order 3 x 3 –1 (C) D if exists is a scalar matrix (D) none of these

Q.57

The following system of equations 3x – 7y + 5z = 3; 3x + y + 5z = 7 and 2x + 3y + 5z = 5 are (A) consistent with trivial solution (B) consistent with unique non trivial solution (C) consistent with infinite solution (D) inconsistent with no solution n

Q.58

If A1, A3, ..... A2n – 1 are n skew symmetric matrices of same order then B =

∑ (2r − 1)(A 2r −1 ) 2r −1 will r =1

be (A) symmetric (C) neither symmetric nor skew symmetric

(B) skew symmetric (D) data is adequate

x 3x + 2 2x − 1 2 x − 1 4x 3x + 1 = 0 is Q.59 The number of real values of x satisfying 7 x − 2 17 x + 6 12x − 1 (A) 3

(B) 0

(C) more than 3

(D) 1

λ λ + 1 λ −1 2 − 1 3  has no inverse Q.60 Number of real values of λ for which the matrix A =  λ + 3 λ − 2 λ + 7 (A) 0 (B) 1 (C) 2 (D) infinite Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[9]

1 z (y + z) − x2 Q.61 If D = y(y + z) − x2z

1 z 1 x x + 2y + z xz

(x + y) z2 1 x then, the incorrect statement is y(x + y) − xz 2 −

(A) D is independent of x (C) D is independent of z Q.62

(B) D is independent of y (D) D is dependent on x, y, z

If every element of a square non singular matrix A is multiplied by k and the new matrix is denoted by B then | A–1| and | B–1| are related as (A) | A–1| = k | B–1|

(B) | A–1| =

1 –1 |B | k

(C) | A–1| = kn | B–1|

(D) | A–1| = k–n | B–1|

where n is order of matrices.

Q.63

Quest

If f ′ (x) =

mx − p n+p

mx n

1 − 1 1  Q.64 Let A = 2 1 − 3 and 10B = 1 1 1  (A) – 2 (B) – 1 x − 1 ( x − 1) 2 If D(x) = (A) 5

then y = f(x) represents

mx + 2n mx + 2n + p mx + 2n − p

(A) a straight line parallel to x- axis (C) parabola

Q.65

mx + p n−p

(B) a straight line parallel to y- axis (D) a straight line with negative slope

2 2 4 − 5 0 α  . If B is the inverse of matrix A, then α is  1 − 2 3  (C) 2 (D) 5

x3

x −1

x2

( x + 1)3

x

( x + 1) 2

( x + 1)3

(B) – 2

then the coefficient of x in D(x) is (C) 6

(D) 0

Q.66

The set of equations λx – y + (cosθ) z = 0 3x + y + 2z =0 (cosθ)x + y + 2z = 0 0 < θ < 2π , has non- trivial solution(s) (A) for no value of λ and θ (B) for all values of λ and θ (C) for all values of λ and only two values of θ (D) for only one value of λ and all values of θ

Q.67

Matrix A satisfies A2 = 2A – I where I is the identity matrix then for n ≥ 2, An is equal to (n ∈ N) (A) nA – I (B) 2n – 1A – (n – 1)I (C) nA – (n – 1)I (D) 2n – 1A – I

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[10]

a2 +1 Q.68

If a, b, c are real then the value of determinant (A) a + b + c = 0

Q.69

(B) a + b + c = 1

ab ac

ab

ac

2

b + 1 bc = 1 if bc c2 + 1

(C) a + b + c = –1

(D) a = b = c = 0

Read the following mathematical statements carefully: I. There can exist two triangles such that the sides of one triangle are all less than 1 cm while the sides of the other triangle are all bigger than 10 metres, but the area of the first triangle is larger than the area of second triangle. II. If x, y, z are all different real numbers, then 2

1 1 1  1 1 1  + +   . + + 2 2 2 =  ( x − y) ( y − z) (z − x ) x−y y−z z−x III. log3x · log4x · log5x = (log3x · log4x) + (log4x · log5x) + (log5x · log3x) is true for exactly for one real value of x. IV. A matrix has 12 elements. Number of possible orders it can have is six. Now indicate the correct alternatively. (A) exactly one statement is INCORRECT. (B) exactly two statements are INCORRECT. (C) exactly three statements are INCORRECT. (D) All the four statements are INCORRECT. Q.70

Quest

The system of equations (sinθ)x + 2z = 0, (cosθ)x + (sinθ)y = 0 , (cosθ)y + 2z = a has (A) no unique solution (B) a unique solution which is a function of a and θ (C) a unique solution which is independent of a and θ (D) a unique solution which is independent of θ only

1 2 3  0 2 0 5   Q.71 Let A = and b = − 3 . Which of the following is true? 0 2 1  1 (A) Ax = b has a unique solution. (B) Ax = b has exactly three solutions. (C) Ax = b has infinitely many solutions. (D) Ax = b is inconsistent. Q.72

The number of positive integral solutions of the equation

x3 +1 x 2 y x 2z xy 2 y3 + 1 y 2 z = 11 is xz 2 yz 2 z 3 + 1 (A) 0 Q.73

(B) 3

(C) 6

(D) 12

If A, B and C are n × n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of the det(A2BC–1) is equal to (A)

6 5

(B)

12 5

(C)

18 5

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

24 5

[11]

(1 + x ) 2

Q.74

The equation 2 x + 1 x +1

(1 − x ) 2

− (2 + x 2 )

3x 2x

1 − 5x 2 − 3x

(A) has no real solution (C) has two real and two non-real solutions

Q.75

2 + (1 − x ) 1 − 2x

2x + 1

x +1

3x

2x

3x − 2 2 x − 3

(B) has 4 real solutions (D) has infinite number of solutions , real or non-real

(B) 9b2 (a + b)

(C) 3b2 (a + b)

(D) 7a2 (a + b)

2 1  3 4  3 − 4 Let three matrices A = 4 1 ; B = 2 3 and C = − 2 3  then        A ( BC) 2   A ( BC) 3   ABC      + ....... + ∞ =   tr(A) + tr + tr   + tr   4 8 2       (A) 6 (B) 9 (C) 12

Q.77

=0

a a + b a + 2b a + 2b a a+b The value of the determinant is a + b a + 2b a (A) 9a2 (a + b)

Q.76

(1 + x ) 2

Quest (D) none

The number of positive integral solutions

1− λ 2 1 −3 λ − 2 = 0 is 2 − 2 1+ λ (A) 0 Q.78

(B) 2

(C) 3

(D) 1

P is an orthogonal matrix and A is a periodic matrix with period 4 and Q = PAPT then X = PTQ2005P will be equal to (A) A (B) A2 (C) A3 (D) A4

a−x b b a−x b =0, then its other two roots are Q.79 If x = a + 2b satisfies the cubic (a, b∈R) f (x)= b b b a−x (A) real and different (C) imaginary Q.80

Q.81

(B) real and coincident (D) such that one is real and other imaginary

 1  − 1  1  1  A is a 2 × 2 matrix such that A − 1 =  2  and A2 − 1 = 0 . The sum of the elements of A, is         (A) –1 (B) 0 (C) 2 (D) 5

Three digit numbers x17, 3y6 and 12z where x, y, z are integers from 0 to 9, are divisible by a fixed

x 3 1 constant k. Then the determinant 7 6 z must be divisible by 1 y 2 (A) k

(B) k2

(C) k3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) None

[12]

Q.82

Q.83

In a square matrix A of order 3, ai i's are the sum of the roots of the equation x2 – (a + b)x + ab= 0; ai , i + 1's are the product of the roots, ai , i – 1's are all unity and the rest of the elements are all zero. The value of the det. (A) is equal to (A) 0 (B) (a + b) 3 (C) a3 – b3 (D) (a2 + b2)(a + b)

28 25 38 Let N = 42 38 65 , then the number of ways is which N can be resolved as a product of two 56 47 83 divisors which are relatively prime is (A) 4 (B) 8

Q.84

(D) 16

1 1 1 1 + sin A 1 + sin B 1 + sin C = 0, then If A, B, C are the angles of a triangle and 2 2 sin A + sin A sin B + sin B sin C + sin 2 C the triangle is (A) a equilateral (C) a right angled triangle

Q.85

(C) 9

(B) an isosceles (D) any triangle

Quest

x 1 ln (1 + sin x ) x 3 − 16x Lim − ; c = Lim and Let a = x →1 ln x x ln x ; b = Lim 2 x →0 x →0 4 x + x x ( x + 1) 3 a b  , then the matrix c d  is x →−1 3(sin( x + 1) − ( x + 1) )   (A) Idempotent (B) Involutary (C) Non singular

d = Lim

Q.86

Q.87

Q.88

If the system of linear equations x + 2ay + az = 0 x + 3by + bz = 0 x + 4cy + cz = 0 has a non-zero solution, then a, b, c (A) are in G..P. (C) satisfy a + 2b + 3c = 0

(D) Nilpotent

(B) are in H.P. (D) are in A.P.

Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1 : If the graphs of two linear equations in two variables are neither parallel nor the same, then there is a unique solution to the system. Statement-2 : If the system of equations ax + by = 0, cx + dy = 0 has a non-zero solution, then it has infinitely many solutions. Statement-3 : The system x + y + z = 1, x = y, y = 1 + z is inconsistent. Statement-4 : If two of the equations in a system of three linear equations are inconsistent, then the whole system is inconsistent. (A) FFTT (B) TTFT (C) TTFF (D) TTTF

1 + x 2 − y 2 − z 2 2( xy + z ) 2( zx − y)    2 2 2 1+ y − z − x 2( yz + x )  then det. A is equal to Let A =  2( xy − z )  2( zx + y) 2( yz − x ) 1 + z 2 − x 2 − y 2   (A) (1 + xy + yz + zx)3 (C) (xy + yz + zx)3

(B) (1 + x2 + y2 + z2)3 (D) (1 + x3 + y3 + z3)2 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Select the correct alternatives : (More than one are correct) Q.89

The set of equations x – y + 3z = 2 , 2x – y + z = 4 , x – 2y + αz = 3 has (A) unique soluton only for α = 0 (B) unique solution for α ≠ 8 (C) infinite number of solutions for α = 8 (D) no solution for α = 8

Q.90

Suppose a1, a2, ....... real numbers, with a1 ≠ 0. If a1, a2, a3, ..........are in A.P. then

LMa (A) A = Ma MNa

1

a2

4

a5

5

a6

OP a P is singular a PQ a3 6

7

(B) the system of equations a1x + a2y + a3z = 0, a4x + a5y + a6z = 0, a7x + a8y + a9z = 0 has infinite number of solutions (C) B =

LM a Nia

1

ia 2

2

a1

OP Q

is non singular ; where i =

−1

(D) none of these Q.91

a2 The determinant b 2 c2

a 2 − ( b − c) 2 b 2 − (c − a )2 c2 − (a − b) 2

bc ca is divisible by : ab

Quest

(A) a + b + c (C) a2 + b2 + c2

(B) (a + b) (b + c) (c + a) (D) (a − b) (b − c) (c − a)

Q.92

If A and B are 3 × 3 matrices and | A | ≠ 0, then which of the following are true? (A) | AB | = 0 ⇒ | B | = 0 (B) | AB | = 0 ⇒ B = 0 –1 –1 (C) | A | = | A | (D) | A + A | = 2 | A |

Q.93

The value of θ lying between −

π π π & and 0 ≤ A ≤ and satisfying the equation 4 2 2

1 + sin 2 A cos 2 A 2 sin 4θ 2 2 sin A 1 + cos A 2 sin 4θ = 0 are : 2 2 sin A cos A 1 + 2 sin 4θ

Q.94

Q.95

(A) A =

π π , θ = − 4 8

(B) A =

3π =θ 8

(C) A =

π π , θ= − 5 8

(D) A =

π 3π , θ= 6 8

If AB = A and BA = B, then (A) A2B = A2 (B) B2A = B2

(C) ABA = A

(D) BAB = B

x a b The solution(s) of the equation a x a = 0 is/are : b b x

(A) x = − (a + b)

(B) x = a

(C) x = b

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) − b

[14]

Q.96

Q.97

If D1 and D2 are two 3 x 3 diagonal matrices, then (A) D1D2 is a diagonal matrix (B) D1D2 = D2D1 2 2 (C) D1 + D2 is a diagonal matrix (D) none of these a a2 x x 2 = 0 , then ab a 2

1 If 1 b2

(A) x = a Q.98

(B) x = b

1 a

(D) x =

a b

Which of the following determinant(s) vanish(es)? 1 ab

1 b c b c (b + c) (A) 1 ca ca (c + a ) 1 a b a b (a + b)

(B) 1 b c 1 ca

0 a−b a−c b−a 0 b−c c−a c−b 0

(C)

Q.99

(C) x =

If A =

1 1 a + b 1+1 b c 1+1 c a

logx x y z logx y logx z 1 logy z (D) logy x y z log z xy z log z y 1

Quest LMa bOP (where bc ≠ 0) satisfies the equations x + k = 0, then Nc d Q 2

(A) a + d = 0

(B) k = –|A|

(C) k = |A|

(D) none of these

Q.100 The value of θ lying between θ = 0 & θ = π/2 & satisfying the equation :

1+sin 2 θ cos 2 θ 4sin4θ 2 2 sin θ 1+cos θ 4sin4θ = 0 are : sin 2 θ cos 2 θ 1+4sin4θ (A)

7π 24

(B)

5π 24

(C)

11 π 24

(D)

π 24

p + sin x q + sin x p − r + sin x Q.101 If p, q, r, s are in A.P. and f (x) = q + sin x r + sin x − 1 + sin x such that r + sin x s + sin x s − q + sin x

2

∫ f (x)dx = – 4 then 0

the common difference of the A.P. can be : (A) − 1

LM1 Q.102 Let A = 2 MM N2

(B)

1 2

(C) 1

(D) none

OP P 1QP

2 2 1 2 , then 2

(A) A2 – 4A – 5I3 = 0 (C) A3 is not invertible

1 (A – 4I3) 5 (D) A2 is invertible (B) A–1 =

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[16]

Q.99 A,C

A,B,C,D

Q.96

Q.95 A,B,C

A,B,C,D

Q.92

Q.91 A,C,D

A,B,C

Q.90

Q.88 B

B

Q.87

B

Q.83 B

D

Q.82

A

Q.81

Q.79

Q.78 A

C

Q.77

A

Q.76

Q.74

Q.73 B

B

Q.72

A

Q.71

Q.69

Q.68 D

C

Q.67

A

Q.66

Q.64

Q.63 A

C

Q.62

D

Q.61

Q.59

Q.58 B

B

Q.57

B

Q.56

Q.54

Q.53 C

C

Q.52

A

Q.51

Q.49

Q.48 C

C

Q.47

A

Q.46

Q.44

Q.43 D

C

Q.42

C

Q.41

Q.39

Q.38 C

D

Q.37

B

Q.36

Q.34

Q.33 C

C

Q.32

A

Q.31

Q.29

Q.28 C

A

Q.27

A

Q.26

Q.24

Q.23 A

C

Q.22

A

Q.21

Q.19

Q.18 D

D

Q.17

D

Q.16

Q.14

Q.13 C

D

Q.12

C

Q.11

Q.9

C

Q.8

C

Q.7

B

D

Q.3

A

Q.2

A

Q.1 Q.6

Q.4

B

Q.70

A

Q.65

D

Q.60

B

Q.55

B

Q.50

B

Q.45

C

C

Q.40

A

B

Q.35

B

C

Q.30

C

B

Q.25

B

B

Q.20

B

A

Q.15

C

A

Q.10

C

D

Q.5

A

Quest Q.98

A,D

Q.97

Q.94

A,B,C,D

Q.93

B, D

Q.89

Q.86

Q.101 A,C

Q.84

D

A

C

D

A

D B B

D

Q.85

D

Q.80

B

Q.75

A,C A,B,C

Q.100 A,C

Q.102 A,B,D

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

COMPOUND ANGLES TRIGONOMETRIC EQUATIONS & INEQUATIONS SOLUTION OF TRIANGLES SEQUENCES & PROGRESSION

Question bank on Compound angles, Trigonometric eqn and ineqn, Solutions of Triangle, Sequence & Progression There are 132 questions in this question bank. Select the correct alternative : (Only one is correct) Q.1

Q.2

If x + y = 3 – cos4θ and x – y = 4 sin2θ then (A) x4 + y4 = 9

(B)

x + y =16

(C) x3 + y3 = 2(x2 + y2)

(D)

x + y =2

If in a triangle ABC, b cos2 (A) in A.P.

Q.3

Q.4

(B) in G.P.

sin A (1 − n ) cos A

(B)

(n − 1) cos A sin A

s2 3 3

The exact value of cos (A) – 1/2

sin A (n − 1) cos A

(D)

sin A (n + 1) cos A

x ∈I 2 (D) a , x are finite but not possible to find (B) a = –1 ;

(B) A =

s2 2

(C) A >

s2 3

(D) None

2π 3π 6π 9π 18π 27 π cos ec + cos cos ec + cos cos ec is equal to 28 28 28 28 28 28 (B) 1/2 (C) 1 (D) 0

In any triangle ABC, (a + b)2 sin2 (A) c (a + b)

Q.8

(C)

If A is the area and 2s the sum of the 3 sides of a triangle, then : (A) A ≤

Q.7

None

FG π (a + x)IJ = 0 then, which of the following holds good? H2 K

x ∈I 2 (C) a ∈ R ; x ∈φ

Q.6

(D)

Quest

Given a2 + 2a + cosec2

(A) a = 1 ;

Q.5

(C) in H.P.

n sin A cos A then tan(A + B) equals 1 − n cos2 A

If tanB = (A)

A B 3 + a cos2 = c then a, b, c are : 2 2 2

C C + (a − b)2 cos2 = 2 2

(B) b (c + a)

(C) a (b + c)

(D) c2

) . cos ( 32π + x) − sin3 ( 72π − x) when simplified reduces to : cos ( x − π2 ) . tan ( 32π + x)

tan ( x −

π 2

(A) sin x cos x

(B) − sin2 x

(C) − sin x cos x

(D) sin2x

Q.9

If in a ∆ ABC, sin3A + sin3B + sin3C = 3 sinA · sinB · sinC then (A) ∆ ABC may be a scalene triangle (B) ∆ ABC is a right triangle (C) ∆ ABC is an obtuse angled triangle (D) ∆ ABC is an equilateral triangle

Q.10

In a triangle ABC, CH and CM are the lengths of the altitude and median to the base AB. If a = 10, b = 26, c = 32 then length (HM) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

(A) 5 Q.11

(B) 7

(C) 9

(D) none

sin 2 θ sin θ + cos θ − for all permissible vlaues of θ sin θ − cos θ tan 2 θ − 1 (A) is less than – 1 (B) is greater than 1 (C) lies between – 1 and 1 including both (D) lies between – 2 and The value of

Q.12

sin 3θ = 4 sin θ sin 2θ sin 4θ in 0 ≤ θ ≤ π has : (A) 2 real solutions (B) 4 real solutions (C) 6 real solutions (D) 8 real solutions.

Q.13

In a triangle ABC, CD is the bisector of the angle C. If cos

2

C 1 has the value and l (CD) = 6, then 2 3

 1 1  +  has the value equal to  a b

(A) Q.14

1 9

(B)

1 12

(C)

Quest

RS π , 5π , 19π , 23π UV T12 12 12 12 W R 5π , 13π , 19π UV (C) S T 12 12 12 W

π 7π 17π 23π  (B)  , , , 

Q.17

RS T

1 2

(B) 2

If cos (α + β) = 0 then sin (α + 2β) = (A) sin α (B) − sin α

UV W

tan A has the value equal to tan B

1 2

(C) − 2

(D) −

(C) cos β

(D) − cos β

With usual notations, in a triangle ABC, a cos(B – C) + b cos(C – A) + c cos(A – B) is equal to (A)

Q.18

12 12 12 12  π 7 π 19 π 23π , , , (D) 12 12 12 12

If the median of a triangle ABC through A is perpendicular to AB then (A)

Q.16

(D) none

The set of angles btween 0 & 2π satisfying the equation 4 cos2 θ − 2 2 cos θ − 1 = 0 is (A)

Q.15

1 6

abc R2

(B)

sin 3 θ − cos 3 θ − sin θ − cos θ

 

π 2

(A) θ ∈  0 , 

abc 4R 2

cos θ 1 + cot 2 θ

(C)

4abc R2

(D)

abc 2R 2

− 2 tan θ cot θ = − 1 if : π 2

 

(B) θ ∈  , π

 

(C) θ ∈  π ,

3π   2

 3π  , 2 π  2 

(D) θ ∈ 

Q.19

With usual notations in a triangle ABC, ( I I1 ) · ( I I2 ) · ( I I3 ) has the value equal to (A) R2r (B) 2R2r (C) 4R2r (D) 16R2r

Q.20

In a triangle ABC, angle B < angle C and the values of B & C satisfy the equation 2 tan x - k (1 + tan2 x) = 0 where (0 < k < 1) . Then the measure of angle A is :

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[3]

(A) π/3 Q.21

If cos α =

2π π π , , 3 4 12

(B)

k −1 k +1

(B)

The equation, sin2 θ −

3

(C)

π 3π π , , 2 8 8

(D)

π 3π π , , 2 10 5

(D)

k+1 k

C A B C  = k sin , then tan tan = 2 2 2 2

k +1 k −1

(C)

k k+1

4 4 =1− has : 3 sin θ − 1 sin θ − 1 3

Quest (B) one root

(C) two roots

(D) infinite roots

3  1 1  1 1  1 1 +   +   +  = KR where K has the value  r1 r2   r2 r3   r3 r1  a 2 b2 c2

(B) 16

(C) 64

(D) 128

1 − sin x + 1 + sin x 5π < x < 3π , then the value of the expression is 2 1 − sin x − 1 + sin x x x x x (A) –cot (B) cot (C) tan (D) –tan 2 2 2 2

Q.26

If

Q.27

If x sin θ = y sin  θ +

 

(A) x + y + z = 0

2π 4 π   = z sin  θ +  then :  3 3

(B) xy + yz + zx = 0

In a ∆ ABC, the value of (A)

Q.29

(D)

With usual notation in a ∆ ABC  equal to : (A) 1

Q.28

π π π , , 2 3 6

 

(A) no root Q.25

(C) 3

2

If A + B + C = π & sin  A + (A)

Q.24

(B)

In a ∆ ABC, if the median, bisector and altitude drawn from the vertex A divide the angle at the vertex into four equal parts then the angles of the ∆ ABC are : (A)

Q.23

(D) 3π/4

α β 2 cos β − 1 then tan cot has the value equal to, where(0 < α < π and 0 < β < π) 2 2 2 − cos β

(A) 2 Q.22

(C) π/2

(B) 2π/3

r R

The value of cos (A)

1 32

(C) xyz + x + y + z = 1 (D) none

a cos A + b cos B + c cos C is equal to : a+b+c

(B)

R 2r

(C)

R r

(D)

2r R

π 2π 4π 8π 16 π cos cos cos cos is : 10 10 10 10 10

(B)

1 16

(C)

cos ( π / 10 ) 16

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) −

10 + 2 5 64

[4]

Q.30

With usual notation in a ∆ ABC, if R = k (A) 1

(B) 2

(r1 + r2 ) ( r2 + r3 ) ( r3 + r1 ) r1 r2 + r2 r3 + r3 r1

where k has the value equal to:

(C) 1/4

(D) 4

Q.31

If a cos3 α + 3a cos α sin2 α = m and a sin3 α + 3a cos2 α sin α = n . Then (m + n)2/3 + (m − n)2/3 is equal to : (A) 2 a2 (B) 2 a1/3 (C) 2 a2/3 (D) 2 a3

Q.32

In a triangle ABC , AD is the altitude from A . Given b > c , angle C = 23° & AD = then angle B = (A) 157°

Q.33

(B) 113°

(C) 147°

[JEE ’94, 2] (D) none

The value of cot x + cot (60º + x) + cot (120º + x) is equal to : (A) cot 3x

(B) tan 3x

(C) 3 tan 3x

3 − 9 tan 2 x (D) 3 tan x − tan 3 x

Quest

Q.34

In a ∆ ABC, cos 3A + cos 3B + cos 3C = 1 then : (A) ∆ ABC is right angled (B) ∆ ABC is acute angled (C) ∆ ABC is obtuse angled (D) nothing definite can be said about the nature of the ∆.

Q.35

The value of

3 + cot 76° cot 16° is : cot 76° + cot 16°

(A) cot 44º Q.36

(B) tan 44º

(C) tan 2º

(D) cot 46º

If the incircle of the ∆ ABC touches its sides respectively at L, M and N and if x, y, z be the circumradii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz is equal to : (A) R r2

(B) r R2

(C)

1 R r2 2

(D)

1 r R2 2

Q.37

The number of solutions of tan (5π cos θ) = cot (5 π sin θ) for θ in (0, 2π) is : (A) 28 (B) 14 (C) 4 (D) 2

Q.38

If A = 3400 then 2 sin

Q.39

a bc b 2 − c2

A is identical to 2

(A)

1 + sin A + 1 − sin A

(B) − 1 + sin A − 1 − sin A

(C)

1 + sin A − 1 − sin A

(D) − 1 + sin A + 1 − sin A

AD, BE and CF are the perpendiculars from the angular points of a ∆ ABC upon the opposite sides. The perimeters of the ∆ DEF and ∆ ABC are in the ratio : (A)

2r R

(B)

r 2R

(C)

r R

(D)

r 3R

where r is the in radius and R is the circum radius of the ∆ ABC

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.40

The value of cosec

π – 18

π 3 sec 18 is a

(A) surd (C) negative natural number Q.41

(B) rational which is not integral (D) natural number

In a ∆ ABC if b + c = 3a then cot (A) 4

B C · cot has the value equal to : 2 2

(B) 3

(C) 2

(D) 1

Q.42

The set of values of ‘a’ for which the equation, cos 2x + a sin x = 2a − 7 possess a solution is : (A) (− ∞, 2) (B) [2, 6] (C) (6, ∞) (D) (− ∞, ∞)

Q.43

In a right angled triangle the hypotenuse is 2 2 times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are (A)

Q.44

π π & 3 6

(B)

π 3π & 8 8

(C)

(A) 1/4

Quest (B) 1/2

(C) 1

In ∆ ABC, the minimum value of

(A) 1

is

(C) 3

(D) non existent

(B)

1 3

(C) – 3

(D) –

The general solution of sin x + sin 5x = sin 2x + sin 4x is : (A) 2nπ (B) nπ (C) nπ/3 where n ∈ I

1 3

(D) 2 nπ/3

The product of the distances of the incentre from the angular points of a ∆ ABC is : (A) 4 R2 r

Q.49

A 2

If the orthocentre and circumcentre of a triangle ABC be at equal distances from the side BC and lie on the same side of BC then tanB tanC has the value equal to : (A) 3

Q.48

(B) 2

(D) 2

B

∑ cot 2 2 . cot 2 2 ∏ cot 2

Q.47

π 3π & 5 10

abc a b c then the value of λ is : + + =λ fgh f g h

A

Q.46

(D)

Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ∆ ABC on the sides a, b and c respectively . If

Q.45

π π & 4 4

(B) 4 Rr2

2 Number of roots of the equation cos x +

[−π, π] is (A) 2

(B) 4

(C)

(a b c) R s

(D)

(a bc )s R

3 +1 3 sin x − − 1 = 0 which lie in the interval 2 4 (C) 6

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 8

[6]

Q.50

sec 8θ − 1 sec 4θ − 1 is equal to (A) tan 2θ cot 8θ

(B) tan 8θ tan 2θ

(

(C) cot 8θ cot 2θ

(D) tan 8θ cot 2θ

)

Q.51

In a ∆ABC if b = a 3 − 1 and ∠C = 300 then the measure of the angle A is (A) 150 (B) 450 (C) 750 (D) 1050

Q.52

Number of values of θ ∈ [ 0 , 2 π] satisfying the equation cotx – cosx = 1 – cotx. cosx (A) 1 (B) 2 (C) 3 (D) 4

Q.53

The exact value of cos273º + cos247º + (cos73º . cos47º) is (A) 1/4 (B) 1/2 (C)3/4

Q.54

In a ∆ABC, a = a1 = 2 , b = a2 , c = a3 such that ap+1 = where p = 1,2 then (A) r1 = r2

Q.55

(

) (

3π

3π

(C) r2 = 2r1

)



(D) none

 3π   5π  − α + cos  − α  when simplified  2   2 



(B) − sin 2α

(C) 1 − sin 2α

(D) 1 + sin 2α

If ‘O’ is the circumcentre of the ∆ ABC and R1, R2 and R3 are the radii of the circumcircles of triangles

(A)

a bc 2 R3

(B)

a b c + + has the value equal to: R1 R2 R 3

R3 a bc

(C)

4∆ R2

(D)

∆ 4R 2

The maximum value of ( 7 cosθ + 24 sinθ ) × ( 7 sinθ – 24 cosθ ) for every θ ∈ R . (A) 25

Q.59

(C) − 1

The expression [1 − sin (3π − α) + cos (3π + α)] 1 − sin 

OBC, OCA and OAB respectively then

Q.58

(D) r2 = 3r1

tan 2 − α cos 2 − α π π   The expression, + cos  α −  sin (π − α) + cos (π + α) sin  α −  when 2 2 cos(2 π − α )

reduces to : (A) sin 2α Q.57

5p 4p − 2   a 22 − p − ap  2−p p  3 5p  

Quest (B) r3 = 2r1

simplified reduces to : (A) zero (B) 1 Q.56

(D) 1

(B) 625

(C)

625 2

(D)

625 4

4 sin50 sin550 sin650 has the values equal to (A)

3 +1 2 2

(B)

3 −1 2 2

(C)

3 −1 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

d

i

3 3−1 2 2

[7]

Q.60

If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then a bc x y z is equal to (A)

A

∏ tan 2

(B)

A

∑ cot 2

(C)

A

∑ tan 2

(D)

A

∑ sin 2

Q.61

The medians of a ∆ ABC are 9 cm, 12 cm and 15 cm respectively . Then the area of the triangle is (A) 96 sq cm (B) 84 sq cm (C) 72 sq cm (D) 60 sq cm

Q.62

If x =

nπ x x x π 3π , satisfies the equation sin − cos = 1 − sin x & the inequality − ≤ , then: 2 2 2 2 2 4

(A) n = −1, 0, 3, 5 (C) n = 0, 2, 4 Q.63

FG H

The value of 1 + cos (A)

9 16

(B) n = 1, 2, 4, 5 (D) n = −1, 1, 3, 5

π 9

IJ FG1 + cos 3π IJ FG1+ cos 5π IJ FG1+ cos 7π IJ is KH 9K H 9KH 9K

Quest (B)

10 16

(C)

12 16

(D)

5 16

Q.64

The number of all possible triplets (a1 , a2 , a3) such that a1+ a2 cos 2x + a3 sin² x = 0 for all x is (A) 0 (B) 1 (C) 3 (D) infinite

Q.65

In a ∆ABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius of the semicircle is 2∆ 2∆ (B) a+b a +b−c Where ∆ is the area of the triangle ABC. (A)

(C)

2∆ s

(D)

c 2

Q.66

For each natural number k , let Ck denotes the circle with radius k centimeters and centre at the origin. On the circle Ck , a particle moves k centimeters in the counter- clockwise direction. After completing its motion on Ck , the particle moves to Ck+1 in the radial direction. The motion of the particle continues in this manner .The particle starts at (1, 0).If the particle crosses the positive direction of the x- axis for the first time on the circle Cn then n equal to (A) 6 (B) 7 (C) 8 (D) 9

Q.67

If in a ∆ ABC,

cos A cos B cos C = = a b c

(A) right angled Q.68

Q.69

then the triangle is

(B) isosceles

(C) equilateral

If cos A + cosB + 2cosC = 2 then the sides of the ∆ ABC are in (A) A.P. (B) G.P (C) H.P.

(D) obtuse

(D) none

If A and B are complimentary angles, then :  

A  2 

B 2

(A) 1 + tan   1 + tan  = 2

 

A  2 

B 2

(B) 1 + cot  1 + cot  = 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

 

A B (C) 1 + sec  1 + cos ec  = 2



Q.70

2 

2

B 2

The value of , 3 cosec 20° − sec 20° is : 2 sin 20°

(A) 2 Q.71

A  2 

(D)  1 − tan   1 − tan  = 2

(B) sin 40°

4 sin 20°

(C) 4

(D) sin 40°

If in a ∆ ABC, cosA·cosB + sinA sinB sin2C = 1 then, the statement which is incorrect, is (A) ∆ ABC is isosceles but not right angled (B) ∆ ABC is acute angled (C) ∆ ABC is right angled

(D) least angle of the triangle is

π 4

2 π ( ) − 2 (0.25)sincos(x2−x 4 ) + 1 = 0, is

tan x − π 4

Q.72

The set of values of x satisfying the equation, 2 (A) an empty set (B) a singleton (C) a set containing two values (D) an infinite set

:

Q.73

The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to : (A) ∆ (B) 2 ∆ (C) 3 ∆ (D) 4 ∆ [ where ∆ is the area of the triangle ABC ]

Q.74

If in a triangle sin A : sin C = sin (A − B) : sin (B − C) then a2 : b2 : c2 (A) are in A.P. (B) are in G.P. (C) are in H.P. (D) none of these

Quest

[ Y G ‘99 Tier - I ]

5

Q.75

The number of solution of the equation,

∑ cos(r x) = 0

lying in (0, p) is :

r =1

(A) 2

Q.76

If θ = 3 α and sin θ = 1

(A)

Q.78

(B) 3

2

a +b

a 2

a + b2

If in a triangle ABC (A)

π 8

(C) a + b

(D) none

10 10 10 10 + tan 67 – cot 67 – tan7 is : 2 2 2 2

(A) a rational number (B) irrational number Q.79

(D) more than 5

. The value of the expression , a cosec α − b sec α is

(B) 2 a 2 + b 2

2

The value of cot 7

(C) 5

(C) 2(3 + 2 3 )

(D) 2 (3 – 3 )

2 cos A cos B 2 cos C a b + + = + then the value of the angle A is : a b c b c ca

(B)

π 4

(C)

π 3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

π 2

[9]

Q.80

The value of the expression (sinx + cosecx)2 + (cosx + secx)2 – ( tanx + cotx)2 wherever defined is equal to (A) 0 (B) 5 (C) 7 (D) 9

Q.81

If A = 5800 then which one of the following is true

Q.82

A (A) 2 sin   = 1 + sin A − 1 − sin A 2

A (B) 2 sin   = − 1 + sin A + 1 − sin A 2

A (C) 2 sin   = − 1 + sin A − 1 − sin A 2

A (D) 2 sin   = 2

1 + sin A + 1 − sin A

With usual notations in a triangle ABC, if r1 = 2r2 = 2r3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a

(D) 2a = 3b

1 π x2 − x Q.83 If tan α = 2 and tan β = (x ≠ 0, 1), where 0 < α, β < , then tan 2 2 2x − 2x + 1 x − x +1 (α + β) has the value equal to :

Quest

(A) 1

Q.84

Q.86

(C) 2

(D)

If r1, r2, r3 be the radii of excircles of the triangle ABC, then (A)

Q.85

(B) – 1

A

∑ cot 2

(B)

A

B

∑ cot 2 cot 2

(C)

∑ r1 ∑ r1r2

A

∑ tan 2

3 4

is equal to :

(D)

A

∏ tan 2

Minimum value of 8cos2x + 18sec2x ∀ x ∈ R wherever it is defined, is : (A) 24 (B) 25 (C) 26 (D) 18

In a ∆ABC

 a2 b2 c 2  A B C  + +  sin A sin B sin C  . sin 2 sin 2 sin 2 simplifies to  

(B) ∆

(A) 2∆

(C)

∆ 2

(D)

∆ 4

where ∆ is the area of the triangle Q.87

If θ is eliminated from the equations x = a cos(θ – α) and y = b cos (θ – β) then

x2

y2

2xy cos(α − β) is equal to ab a b (A) cos2 ( α – β) (B) sin2 (α – β) 2

Q.88

+

2

−

(C) sec2 ( α – β)

The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is π (A) x = nπ (B) nπ ± (C) x = 2nπ 3 where n ∈ I Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) cosec2 (α – β)

(D) x =

nπ 3

[10]

Q.89

If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (logbc)3 + (logca)3 is (A) an odd prime (B) an even prime (C) an odd composite (D) an irrational number

Q.90

If the arcs of the same length in two circles S1 and S2 subtend angles 75° and 120° respectively at the S centre. The ratio 1 is equal to S2 (A)

Q.91

Q.92

1 5

81 16

(B)

(C)

Number of principal solution of the equation tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5

(D)

(C) 7

25 64

(D) more than 7

tan 2 20° − sin 2 20° simplifies to tan 2 20° · sin 2 20°

The expression

Quest

(A) a rational which is not integral (C) a natural which is prime Q.93

64 25

(B) a surd (D) a natural which is not composite

The value of x that satisfies the relation x = 1 – x + x2 – x3 + x4 – x5 + ......... ∞ (A) 2 cos36° (B) 2 cos144° (C) 2 sin18°

(D) none

Select the correct alternatives : (More than one are correct) Q.94

If sin θ = sin α then sin (A) sin

Q.95

θ = 3

π 3

α 3

π 3

α 3

(C) sin  + 

π 3

α 3

(D) − sin  + 

Choose the INCORRECT statement(s). °

Q.96

α 3

(B) sin  − 

°

°

°

(A

1 1 1 1 sin 82 . cos 37 and sin 127 . sin 97 have the same value. 2 2 2 2

(B)

If tan A =

(C) (D)

The sign of the product sin 2 . sin 3 . sin 5 is positive. There exists a value of θ between 0 & 2 π which satisfies the equation ; sin4 θ – sin2 θ – 1 = 0.

3 4− 3

& tan B =

3 4+ 3

then tan (A − B) must be irrational.

Which of the following functions have the maximum value unity ? (A) sin2 x − cos2 x (C) −

sin 2x − cos 2x 2

(B)

(D)

sin 2x − cos 2x 2

6  1 sin x + 1 cos x    3 5  2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[11]

Q.97

If the sides of a right angled triangle are {cos2α + cos2β + 2cos(α + β)} and {sin2α + sin2β + 2sin(α + β)}, then the length of the hypotenuse is : (A) 2[1+cos(α − β)] (B) 2[1 − cos(α + β)] (C) 4 cos2

Q.98

Q.99

α−β 2

(D) 4sin2

α+β 2

An extreme value of 1 + 4 sin θ + 3 cos θ is : (A) − 3 (B) − 4 (C) 5

The sines of two angles of a triangle are equal to (A) 245/1313

(B) 255/1313

(D) 6

5 99 & . The cosine of the third angle is : 13 101

(C) 735/1313

4 Q.100 It is known that sin β = & 0 < β < π then the value of 5

(A) independent of α for all β in (0, π/2) (C)

5

(B)

(D) 765/1313

3 sin (α + β) − cos2 π cos(α + β) 6 is: sin α

for tan β > 0

Quest

3 (7 + 24 cot α ) for tan β < 0 15

3

(D) none

Q.101 If x = sec φ − tan φ & y = cosec φ + cot φ then : (A) x =

y +1 y −1

(B) y =

1+ x 1− x

(C) x =

y−1 y+1

(D) xy + x − y + 1 = 0

Q.102 If 2 cosθ + sinθ = 1, then the value of 4 cosθ + 3sinθ is equal to (A) 3 Q.103 If sin t + cos t = (A) −1

(B) –5

(C)

7 5

(D) –4

1 t then tan is equal to : 5 2

(*B) –

1 3

(C) 2

(D) −

1 6

SEQUENCE & PROGRESSION Select the correct alternative : (Only one is correct) Q.104 If a, b, c be in A.P., b, c, d in G.P. & c, d, e in H.P., then a, c, e will be in : (A) A.P. (B) G.P. (C) H.P. (D) none of these Q.105 If a, b, c are in H.P., then a, a − c, a − b are in : (A) A.P. (B) G.P. (C) H.P.

(D) none of these

Q.106 If three positive numbers a , b, c are in H.P. then e
n ( a + c ) +
n ( a − 2 b + c ) simplifies to (A) (a – c)2 (B) zero (C) ( a – c) (D) 1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]

∞

Q.107 The sum

∑ r=2

1 is equal to : r −1 2

(A) 1

(B) 3/4

(C) 4/3

(D) none

Q.108 In a potato race , 8 potatoes are placed 6 metres apart on a straight line, the first being 6 metres from the basket which is also placed in the same line. A contestant starts from the basket and puts one potato at a time into the basket. Find the total distance he must run in order to finish the race. (A) 420 (B) 210 (C) 432 (D) none Q.109 If the roots of the cubic x3 – px2 + qx – r = 0 are in G.P. then (A) q3 = p3r (B) p3 = q3r (C) pq = r

(D) pr = q

Q.110 Along a road lies an odd number of stones placed at intervals of 10 m. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Then the number of stones is (A) 15 (B) 29 (C) 31 (D) 35

Quest

Q.111 If log (5. 2x +1) 2 ; log ( 21− x +1) 4 and 1 are in Harmonical Progression then (A) x is a positive real (C) x is rational which is not integral

(B) x is a negative real (D) x is an integer

Q.112 If a, b, c are in G.P., then the equations, ax2 + 2bx + c = 0 & dx2 + 2ex + f = 0 have a common root, if d e f , , are in : a b c

(A) A.P.

(B) G.P.

(C) H.P.

(D) none

Q.113 If the sum of the roots of the quadratic equation, ax2 + bx + c = 0 is equal to sum of the squares of their a b c , , are in : c a b

reciprocals, then (A) A.P.

(B) G.P.

(C) H.P.

Q.114 If for an A.P. a1 , a2 , a3 ,.... , an ,.... a1 + a3 + a5 = – 12 and a1 a2 a3 = 8 then the value of a2 + a4 + a6 equals (A) – 12 (B) – 16

Q

. 1

1

5

G

a

i v

G

e n

. P .

( A

)

t h

( B

)

c o

( C

)

c o

( D

)

c o

f o

,

u

r

p

t h e n

e

o

s i t i v

w

h

i c h

m

o

n

e

n

u

o

f

t h e

r a t i o

m

e r

i n

f o

l l o

G

. P .

. P .

i n g

,

s ?

m

o

n

r a t i o

m

m

o

n

d

i f f e r e n c e

o

f

t h

e

A

. P .

i s

3

/ 2

m

m

o

n

d

i f f e r e n

o

f

t h

e

A

. P .

i s

2

/ 3

. P .

c e

i s

2

3

5

l d

m

G

i s

I f

h o

m

f

f

A

w

(C) – 18 (D) – 21 [ Apex : Q.62 of Test - 1 Scr. 2004 ]

c o

o

o

b

(D) none

6

,

9

a n

d

1

5

a r e

a d

d

e d

r e s p

e c t i v

e l y

t o

t h e s e

n u

m

b e r s

,

w

e

g

e t

/ 2

/ 3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Q.116 Consider an A.P. with first term 'a' and the common difference d. Let Sk denote the sum of the first K Skx terms. Let S is independent of x, then x (A) a = d/2 (B) a = d

(C) a = 2d

(D) none

Q.117 Concentric circles of radii 1, 2, 3......100 cms are drawn. The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green regions in sq. cm is equal to (A) 1000π (B) 5050π (C) 4950π (D) 5151π Q.118 Consider

the A.P. a1 , a2 ,..... , an ,.... the G.P. b1 , b2 ,....., bn ,..... 9

such that a1 = b1 = 1 ; a9 = b9 and ∑ a r = 369 then r =1

(A) b6 = 27

(B) b7 = 27

(C) b8 = 81 (D) b9 = 18 [ Apex : Q.68 of Test - 1 Scr. 2004 ]

Quest

Q.119 For an increasing A.P. a1, a2, ...... an if a1 + a3 + a5 = – 12 : a1a3a5 = 80 then which of the following does not hold? (B) a2 = – 1 (C) a3 = – 4 (D) a5 = 2 (A) a1 = – 10 Q.120 Consider a decreasing G.P. : g1, g2, g3, ...... gn ....... such that g1 + g2 + g3 = 13 and g12 + g 22 + g 32 =91 then which of the following does not hold? (A) The greatest term of the G.P. is 9. (C) g1 = 1

(B) 3g4 = g3 (D) g2 = 3

Q.121 If p , q, r in H.P. and p & r be different having same sign then the roots of the equation px2 + qx + r = 0 are (A) real & equal (B) real & distinct (C) irrational (D) imaginary Q.122 The point A(x1, y1) ; B(x2, y2) and C(x3, y3) lie on the parabola y = 3x2. If x1, x2, x3 are in A.P. and y1, y2, y3 are in G.P. then the common ratio of the G.P. is (A) 3 + 2 2

(B) 3 +

2

(C) 3 –

2

(D) 3 – 2 2

Q.123 If a, b, c are in A.P., then a2 (b + c) + b2 (c + a) + c2 (a + b) is equal to : (A)

(a + b + c ) 3 8

Q.124 If Sn =

(B)

2 (a + b + c)3 9

(C)

3 (a + b + c)3 10

(D)

1 (a + b + c)3 9

1 1+ 2 1 + 2 + 3 + ...... + n + 3 3 +...... + 3 3 3 , n = 1, 2, 3,...... Then Sn is not greater than: 3 1 1 +2 1 + 2 + 3 + ...... + n 3

(A) 1/2

(B) 1

(C) 2

(D) 4

Q.125 If Sn denotes the sum of the first n terms of a G.P. , with the first term and the common ratio both positive, then (A) Sn , S2n , S3n form a G.P. (B) Sn , S2n , – Sn , S3n , –S2n form a G.P. (C) S2n – Sn , S3n – S2n , S3n – Sn form a G.P. (D) S2n–Sn , S3n–S2n , S3n–Sn form a G.P. Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[14]

Q.126

1 1 .3 1 .3 .5 1 .3 .5 .7 + + + + ................∞ is equal to 2.4 2.4.6 2.4.6.8 2.4.6.8.10 (A)

1 4

(B)

1 3

(C)

1 2

(D) 1

Q.127 Consider an A.P. a1 , a2 , a3 ,......... such that a3 + a5 + a8 = 11 and a4 + a2 = –2, then the value of a1 + a6 + a7 is (A) –8 (B) 5 (C) 7 (D) 9 Q.128 A circle of radius r is inscribed in a square. The mid points of sides of the square have been connected by line segment and a new square resulted. The sides of the resulting square were also connected by segments so that a new square was obtained and so on, then the radius of the circle inscribed in the nth square is

 1− n  2 (A) 2  r  

 3−3n  2 r (B) 2  

 −n  2 (C) 2  r  

 − 5 −3 n  2 r (D) 2  

Quest 2k + 2 ∑ 3k equal to k =1 ∞

Q.129 The sum (A) 12

(B) 8

2n+2 is equal to n −2 n =1 4 (A) 1372 (B) 440

(C) 6

(D) 4

(C) 320

(D) 388

∞

Q.130 The sum 5 ∑

Q.131 Given am+n = A ; am–n = B as the terms of the G.P. a1 , a2 , a3 ,............. then for A ≠ 0 which of the following holds? (B) a n = 2 n A n Bn

(A) a m = AB m 2 − m − n − mn  A  m+n

m 2 −m −n − n 2  m+n

A (D) a n = a1   B

(C) a m = a1   B

Q.132 The sum of the infinite series, 12 − (A)

1 2

(B)

25 24

22 32 4 2 52 6 2 + 2 − 3 + 4 − 5 +........ is : 5 5 5 5 5

(C)

25 54

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

125 252

[15]

Answers Select the correct alternative : (Only one is correct) Q.1 D Q.2 D Q.3 A Q.4 B Q.5 Q.8 D Q.9 D Q.10 C Q.11 D Q.12 Q.15 C Q.16 A Q.17 A Q.18 B Q.19 Q.22 C Q.23 A Q.24 D Q.25 C Q.26 Q.29 D Q.30 C Q.31 C Q.32 B Q.33 Q.36 C Q.37 A Q.38 D Q.39 C Q.40 Q.43 B Q.44 A Q.45 A Q.46 A Q.47 Q.50 D Q.51 D Q.52 B Q.53 C Q.54 Q.57 C Q.58 C Q.59 B Q.60 B Q.61 Q.64 D Q.65 A Q.66 B Q.67 C Q.68 Q.71 C Q.72 A Q.73 B Q.74 A Q.75 Q.79 D Q.80 B Q.81 C Q.82 C Q.83 Q.86 B Q.87 B Q.88 D Q.89 A Q.90 Q.93 C Select the correct alternatives : (More than one are correct) Q.94 ABD Q.95 BCD Q.96 ABCD Q.97 AC Q.100 ABC Q.101 BCD Q.102 AC Q.103 BC

A D D D D D C D C A C A C

Q.6 Q.13 Q.20 Q.27 Q.34 Q.41 Q.48 Q.55 Q.62 Q.69 Q.76 Q.84 Q.91

D A C B C C B A B A B C C

Q.7 Q.14 Q.21 Q.28 Q.35 Q.42 Q.49 Q.56 Q.63 Q.70 Q.78 Q.85 Q.92

Q.98

BD

Q.99

BC

Quest

SEQUENCE & PROGRESSION Select the correct alternative : (Only one is correct) Q.104 B Q.105 C Q.106 A Q.107 Q.109 A Q.110 C Q.111 B Q.112 Q.114 D Q.115 A Q.116 A Q.117 Q.119 B Q.120 C Q.121 D Q.122 Q.124 C Q.125 B Q.126 C Q.127 Q.129 B Q.130 c Q.131 A Q.132

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

B A B A C C

Q.108 Q.113 Q.118 Q.123 Q.128

D B D A A B B B A C B C D

C C B B A

[16]

TARGET IIT JEE

MATHEMATICS

STRAIGHT LINES & CIRCLES - I

Select the correct alternative : (Only one is correct) Q.1

If the lines x + y + 1 = 0 ; 4x + 3y + 4 = 0 and x + αy + β = 0, where α2 + β2 = 2, are concurrent then (A) α = 1, β = – 1 (B) α = 1, β = ± 1 (C) α = – 1, β = ± 1 (D) α = ± 1, β = 1

Q.2

The axes are translated so that the new equation of the circle x² + y² − 5x + 2y – 5 = 0 has no first degree terms. Then the new equation is : (A) x2 + y2 = 9

(B) x2 + y2 =

49 4

(C) x2 + y2 =

81 16

(D) none of these

Q.3

Given the family of lines, a (3x + 4y + 6) + b (x + y + 2) = 0 . The line of the family situated at the greatest distance from the point P (2, 3) has equation : (A) 4x + 3y + 8 = 0 (B) 5x + 3y + 10 = 0 (C) 15x + 8y + 30 = 0 (D) none

Q.4

The ends of a quadrant of a circle have the coordinates (1, 3) and (3, 1) then the centre of the such a circle is (A) (1, 1) (B) (2, 2) (C) (2, 6) (D) (4, 4)

Q.5

The straight line, ax + by = 1 makes with the curve px2 + 2a xy + qy2 = r a chord which subtends a right angle at the origin . Then : (A) r (a2 + b2) = p + q (B) r (a2 + p2) = q + b 2 2 (C) r (b + q ) = p + a (D) none

Q.6

The circle described on the line joining the points (0 , 1) , (a , b) as diameter cuts the x−axis in points whose abscissae are roots of the equation : (A) x² + ax + b = 0 (B) x² − ax + b = 0 (C) x² + ax − b = 0 (D) x² − ax − b = 0

Q.7

Centroid of the triangle, the equations of whose sides are 12x2 – 20xy + 7y2 = 0 and 2x – 3y + 4=0 is

Quest 8 8 (B)  ,  3 3

(A) (3, 3) Q.8

 8 (C)  3,   3

8  (D)  , 3  3 

The line 2x – y + 1 = 0 is tangent to the circle at the point (2, 5) and the centre of the circles lies on x – 2y = 4. The radius of the circle is (B) 5 3

(A) 3 5

(C) 2 5

(D) 5 2

Q.9

The line x + 3y − 2 = 0 bisects the angle between a pair of straight lines of which one has equation x − 7y + 5 = 0 . The equation of the other line is : (A) 3x + 3y − 1 = 0 (B) x − 3y + 2 = 0 (C) 5x + 5y − 3 = 0 (D) none

Q.10

Given two circles x² + y² − 6x − 2y + 5 = 0 & x² + y² + 6x + 22y + 5 = 0. The tangent at (2, −1) to the first circle : (A) passes outside the second circle (B) touches the second circle (C) intersects the second circle in 2 real points (D) passes through the centre of the second circle.

Q.11

A variable rectangle PQRS has its sides parallel to fixed directions. Q & S lie respectively on the lines x = a, x = − a & P lies on the x − axis . Then the locus of R is : (A) a straight line (B) a circle (C) a parabola (D) pair of straight lines

Q.12

  To which of the following circles, the line y − x + 3 = 0 is normal at the point  3 + 3 , 3  ? 

 

(A)  x − 3 −

2

2

 3  3   + y −  =9  2 2

(C) x² + (y − 3)² = 9

 

(B)  x −

2

2

2

2

 3  3   + y −  =9  2 2

(D) (x − 3)² + y² = 9 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.13

On the portion of the straight line, x + 2y = 4 intercepted between the axes, a square is constructed on the side of the line away from the origin. Then the point of intersection of its diagonals has co-ordinates (A) (2, 3) (B) (3, 2) (C) (3, 3) (D) (2, 2)

Q.14

The locus of the mid point of a chord of the circle x² + y² = 4 which subtends a right angle at the origin is (A) x + y = 2 (B) x² + y² = 1 (C) x² + y² = 2 (D) x + y = 1

Q.15

Given the family of lines, a (2x + y + 4) + b (x − 2y − 3) = 0 . Among the lines of the family, the number of lines situated at a distance of 10 from the point M (2, − 3) is : (A) 0 (B) 1 (C) 2

(D) ∞

Q.16

The equation of the line passing through the points of intersection of the circles ; 3x² + 3y² − 2x + 12y − 9 = 0 & x² + y² + 6x + 2y − 15 = 0 is : (A) 10x − 3y − 18 = 0 (B) 5x + 3y − 18 = 0 (C) 5x − 3y − 18 = 0 (D) 10x + 3y + 1 = 0

Q.17

Through a point A on the x-axis a straight line is drawn parallel to y-axis so as to meet the pair of straight lines ax2 + 2hxy + by2 = 0 in B and C. If AB = BC then (A) h2 = 4ab (B) 8h2 = 9ab (C) 9h2 = 8ab (D) 4h2 = ab

Q.18

The number of common tangent(s) to the circles x2 + y2 + 2x + 8y – 23 = 0 and x2 + y2 – 4x – 10y + 19 = 0 is (A) 1 (B) 2 (C) 3 (D) 4

Q.19

A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then (A) ABC is a unique triangle (B) There can be only two such triangles. (C) No such triangle is possible (D) There can be infinite number of such triangles.

Q.20

From the point A (0 , 3) on the circle x² + 4x + (y − 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is : (A) x² + 8x + y² = 0 (B) x² + 8x + (y − 3)² = 0 (C) (x − 3)² + 8x + y² = 0 (D) x² + 8x + 8y² = 0

Q.21

If A (1, p2) ; B (0, 1) and C (p, 0) are the coordinates of three points then the value of p for which the area of the triangle ABC is minimum, is

Quest

(A)

1 3

(B) –

1 3

(C)

1 1 or – 3 3

(D) none

Q.22

The area of the quadrilateral formed by the tangents from the point (4 , 5) to the circle x² + y² − 4x − 2y − 11 = 0 with the pair of radii through the points of contact of the tangents is : (A) 4 sq.units (B) 8 sq.units (C) 6 sq.units (D) none

Q.23

The area of triangle formed by the lines x + y – 3 = 0 , x – 3y + 9 = 0 and 3x – 2y + 1= 0 16 10 sq. units (B) sq. units (C) 4 sq. units (D) 9 sq. units 7 7 Two circles of radii 4 cms & 1 cm touch each other externally and θ is the angle contained by their direct common tangents. Then sin θ =

(A)

Q.24

(A) Q.25

24 25

(B)

12 25

(C)

3 4

(D) none

The set of lines ax + by + c = 0, where 3a + 2b + 4c = 0, is concurrent at the point :  3 3  4 4

(A)  , 

 1 1  2 2

(B)  , 

 3 1  4 2

(C)  , 

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) (1, 1)

[3]

Q.26

The locus of poles whose polar with respect to x² + y² = a² always passes through (K , 0) is (A) Kx − a² = 0 (B) Kx + a² = 0 (C) Ky + a² = 0 (D) Ky − a² = 0

Q.27

The co−ordinates of the point of reflection of the origin (0, 0) in the line 4x − 2y − 5 = 0 is : (A) (1, − 2)

Q.28

(B) (2, − 1)

4 5

2 5

(C)  , − 

(D) (2, 5)

The locus of the mid points of the chords of the circle x2 + y2 − ax − by = 0 which subtend a right angle  a b  2 2

at  ,  is (B) ax + by = a2 + b2

(A) ax + by = 0 (C)

x2 + y2 − ax − by +

a 2 +b 2 =0 8

(D)

x2 +

y2 − ax − by −

a 2 +b 2 =0 8

Q.29

A ray of light passing through the point A (1, 2) is reflected at a point B on the x − axis and then passes through (5, 3) . Then the equation of AB is : (A) 5x + 4y = 13 (B) 5x − 4y = − 3 (C) 4x + 5y = 14 (D) 4x − 5y = − 6

Q.30

From (3, 4) chords are drawn to the circle x² + y² − 4x = 0. The locus of the mid points of the chords is (A) x² + y² − 5x − 4y + 6 = 0 (B) x² + y² + 5x − 4y + 6 = 0 (C) x² + y² − 5x + 4y + 6 = 0 (D) x² + y² − 5x − 4y − 6 = 0

Q.31

m, n are integer with 0 < n < m. A is the point (m, n) on the cartesian plane. B is the reflection of A in the line y = x. C is the reflection of B in the y-axis, D is the reflection of C in the x-axis and E is the reflection of D in the y-axis. The area of the pentagon ABCDE is (A) 2m(m + n) (B) m(m + 3n) (C) m(2m + 3n) (D) 2m(m + 3n)

Q.32

Which one of the following is false ? The circles x² + y² − 6x − 6y + 9 = 0 & x² + y² + 6x + 6y + 9 = 0 are such that : (A) they do not intersect (B) they touch each other (C) their exterior common tangents are parallel (D) their interior common tangents are perpendicular.

Q.33

The lines y − y1 = m (x − x1) ± a 1 + m 2 are tangents to the same circle . The radius of the circle is (A) a/2 (B) a (C) 2a (D) none

Q.34

The centre of the smallest circle touching the circles x² + y² − 2y − 3 = 0 and x² + y² − 8x − 18y + 93 = 0 is : (A) (3 , 2) (B) (4 , 4) (C) (2 , 7) (D) (2 , 5)

Q.35

The ends of the base of an isosceles triangle are at (2, 0) and (0, 1) and the equation of one side is x = 2 then the orthocentre of the triangle is 3 3  5  3  4 7  (A)  ,  (B)  ,1  (C)  ,1  (D)  ,  4 2  4  4   3 12 

Q.36

A rhombus is inscribed in the region common to the two circles x2 + y2 − 4x − 12 = 0 and x2 + y2 + 4x − 12 = 0 with two of its vertices on the line joining the centres of the circles. The area of the rhombous is

Quest

(A) 8 3 sq.units

(B) 4 3 sq.units

(C) 16 3 sq.units

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[4]

Q.37

A variable straight line passes through a fixed point (a, b) intersecting the co−ordinates axes at A & B. If 'O' is the origin then the locus of the centroid of the triangle OAB is : (A) bx + ay − 3xy = 0 (B) bx + ay − 2xy = 0 (C) ax + by − 3xy = 0 (D) none

Q.38

The angle between the two tangents from the origin to the circle (x − 7)2 + (y + 1)2 = 25 equals (A)

π 4

(B)

π 3

(C)

π 2

(D) none

Q.39

If P = (1, 0); Q = (−1, 0) & R = (2, 0) are three given points, then the locus of the points S satisfying the relation, SQ2 + SR2 = 2 SP2 is : (A) a straight line parallel to x−axis (B) a circle passing through the origin (C) a circle with the centre at the origin (D) a straight line parallel to y−axis .

Q.40

The equation of the circle having normal at (3 , 3) as the straight line y = x and passing through the point (2 , 2) is : (A) x² + y² − 5x + 5y + 12 = 0 (B) x² + y² + 5x − 5y + 12 = 0 (C) x² + y² − 5x − 5y − 12 = 0 (D) x² + y² − 5x − 5y + 12 = 0

Q.41

The equation of the base of an equilateral triangle ABC is x + y = 2 and the vertex is (2, − 1) . The area of the triangle ABC is : (A)

Q.42

Quest

2 6

(B)

3 6

(C)

3 8

(D) none

In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to AB ⋅ AD

(A)

AB + AD 2

2

(B)

AB ⋅ AD AB + AD

(C)

AB ⋅ AD

AB ⋅ AD

(D)

AB2 − AD2

Q.43

The equation of the pair of bisectors of the angles between two straight lines is, 12x2 − 7xy − 12y2 = 0. If the equation of one line is 2y − x = 0 then the equation of the other line is : (A) 41x − 38y = 0 (B) 38x − 41y = 0 (C) 38x + 41y = 0 (D) 41x + 38y = 0

Q.44

If the circle C1 : x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/4, then the co-ordinates of the centre of C2 are :  9  5

(A)  ± , ± Q.45

12   5

 9  5

(B)  ± , ∓

12   5

9  12 ,±   5 5

(C)  ±

9  12 ,∓   5 5

(D)  ±

Area of the rhombus bounded by the four lines, ax ± by ± c = 0 is : (A)

c2 2 ab

(B)

2 c2 ab

(C)

4 c2 ab

(D)

ab 4 c2

Q.46

Two lines p1x + q1y + r1 = 0 & p2x + q2y + r2 = 0 are conjugate lines w.r.t. the circle x² + y² = a² if (A) p1p2 + q1q2 = r1r2 (B) p1p2 + q1q2 + r1r2 = 0 (C) a²(p1p2 + q1q2) = r1r2 (D) p1p2 + q1q2 = a² r1r2

Q.47

Area of the quadrilateral formed by the lines x + y = 2 is : (A) 8 (B) 6 (C) 4

Q.48

(D) none

If the two circles (x − 1)² + (y − 3)² = r² & x² + y² − 8x + 2y + 8 = 0 intersect in two distinct points then (A) 2 < r < 8 (B) r < 2 (C) r = 2 (4) r > 2 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.49

Let the algebraic sum of the perpendicular distances from the points (3, 0), (0, 3) & (2, 2) to a variable straight line be zero, then the line passes through a fixed point whose co-ordinates are : (A) (3, 2)

(B) (2, 3)

 3 3 (C)  ,   5 5

 5 5 (D)  ,   3 3

Q.50

If a circle passes through the point (a , b) & cuts the circle x² + y² = K² orthogonally, then the equation of the locus of its centre is : (A) 2ax + 2by − (a² + b² + K²) = 0 (B) 2ax + 2by − (a² − b² + K²) = 0 (C) x² + y² − 3ax − 4by + (a² + b² − K²) = 0 (D) x² + y² − 2ax − 3by + (a² − b² − K²) = 0

Q.51

Consider a quadratic equation in Z with parameters x and y as Z2 – xZ + (x – y)2 = 0 The parameters x and y are the co-ordinates of a variable point P w.r.t. an orthonormal co-ordinate system in a plane. If the quadratic equation has equal roots then the locus of P is (A) a circle (B) a line pair through the origin of co-ordinates with slope 1/2 and 2/3 (C) a line pair through the origin of co-ordinates with slope 3/2 and 2 (D) a line pair through the origin of co-ordinates with slope 3/2 and 1/2

Q.52

Consider the circle S ≡ x2 + y2 – 4x – 4y + 4 = 0. If another circle of radius 'r' less than the radius of the circle S is drawn, touching the circle S, and the coordinate axes, then the value of 'r' is

Quest

(A) 3 – 2 2 Q.53

(C) 7 – 4 2

(D) 6 – 4 2

Vertices of a parallelogram ABCD are A(3, 1), B(13, 6), C(13, 21) and D(3, 16). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is (A)

Q.54

(B) 4 – 2 2

11 12

(B)

11 8

(C)

25 8

(D)

13 8

The distance between the chords of contact of tangents to the circle ; x2+ y2 + 2gx + 2fy + c = 0 from the origin & the point (g , f) is : (A) g2 + f 2

(B)

g2 + f 2 − c 2

(C)

g2 + f 2 − c 2 g2 + f 2

(D)

g2 + f 2 + c 2 g2 + f 2

Q.55

Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line 2x + y = 5. Then the area of the triangle is (A) 5 (B) 3 (C) 5/2 (D) 1

Q.56

The locus of the centers of the circles which cut the circles x2 + y2 + 4x − 6y + 9 = 0 and x2 + y2 − 5x + 4y − 2 = 0 orthogonally is (C) 9x − 10y + 11=0 (D) 9x + 10y + 7 = 0 (A) 9x + 10y − 7 = 0 (B) x − y + 2 = 0

Q.57

Distance between the two lines represented by the line pair, x2 − 4xy + 4y2 + x − 2y − 6 = 0 is : (A)

Q.58

1 5

(B)

5

(C) 2 5

(D) none

The locus of the center of the circles such that the point (2 , 3) is the mid point of the chord 5x + 2y = 16 is : (A) 2x − 5y + 11 = 0 (B) 2x + 5y − 11 = 0 (C) 2x + 5y + 11 = 0 (D) none

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[6]

Q.59

The distance between the two parallel lines is 1 unit. A point 'A' is chosen to lie between the lines at a distance 'd' from one of them. Triangle ABC is equilateral with B on one line and C on the other parallel line . The length of the side of the equilateral triangle is (A)

Q.60

2 2 d + d +1 3

(B) 2

d2 − d +1 3

(C) 2 d 2 − d + 1

(D)

d2 − d +1

The locus of the mid points of the chords of the circle x² + y² + 4x − 6y − 12 = 0 which subtend an angle of

π radians at its circumference is : 3

(A) (x − 2)² + (y + 3)² = 6.25 (C) (x + 2)² + (y − 3)² = 18.75

(B) (x + 2)² + (y − 3)² = 6.25 (D) (x + 2)² + (y + 3)² = 18.75

Q.61

Given A(0, 0) and B(x, y) with x ∈ (0, 1) and y > 0. Let the slope of the line AB equals m1. Point C lies on the line x = 1 such that the slope of BC equals m2 where 0 < m2 < m1. If the area of the triangle ABC can be expressed as (m1 – m2) f (x), then the largest possible value of f (x) is (A) 1 (B) 1/2 (C) 1/4 (D) 1/8

Q.62

If two chords of the circle x2 + y2 − ax − by = 0, drawn from the point (a, b) is divided by the x − axis in the ratio 2 : 1 then: (A) a2 > 3 b2 (B) a2 < 3 b2 (C) a2 > 4 b2 (D) a2 < 4 b2

Q.63

P lies on the line y = x and Q lies on y = 2x. The equation for the locus of the mid point of PQ, if | PQ | = 4, is (A) 25x2 + 36xy + 13y2 = 4 (B) 25x2 – 36xy + 13y2 = 4 (C) 25x2 – 36xy – 13y2 = 4 (D) 25x2 + 36xy – 13y2 = 4

Q.64

The points (x1, y1) , (x2, y2) , (x1, y2) & (x2, y1) are always : (A) collinear (B) concyclic (C) vertices of a square (D) vertices of a rhombus

Q.65

If the vertices P and Q of a triangle PQR are given by (2, 5) and (4, –11) respectively, and the point R moves along the line N: 9x + 7y + 4 = 0, then the locus of the centroid of the triangle PQR is a straight line parallel to (A) PQ (B) QR (C) RP (D) N

Q.66

The angle at which the circles (x – 1)2 + y2 = 10 and x2 + (y – 2)2 = 5 intersect is π π π π (A) (B) (C) (D) 6 4 3 2 The co−ordinates of the points A, B, C are (− 4, 0) , (0, 2) & (− 3, 2) respectively. The point of intersection of the line which bisects the angle CAB internally and the line joining C to the middle point of AB is 5 13 7 10  5 3  7 4 (A)  − ,  (B)  − ,  (C)  , −  (D)  − ,   2 2  3 3 3   2 2 3

Q.67

Q.68

Quest

Two congruent circles with centres at (2, 3) and (5, 6) which intersect at right angles has radius equal to (A) 2 2

Q.69

(B) 3

(C) 4

(D) none

Three lines x + 2y + 3 = 0 ; x + 2y – 7 = 0 and 2x – y – 4 = 0 form the three sides of two squares. The equation to the fourth side of each square is (A) 2x – y + 14 = 0 & 2x – y + 6 = 0 (B) 2x – y + 14 = 0 & 2x – y – 6 = 0 (C) 2x – y – 14 = 0 & 2x – y – 6 = 0 (D) 2x – y – 14 = 0 & 2x – y + 6 = 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[7]

Q.70

A circle of radius unity is centred at origin. Two particles start moving at the same time from the point (1, 0) and move around the circle in opposite direction. One of the particle moves counterclockwise with constant speed v and the other moves clockwise with constant speed 3v. After leaving (1, 0), the two particles meet first at a point P, and continue until they meet next at point Q. The coordinates of the point Q are (A) (1, 0) (B) (0, 1) (C) (0, –1) (D) (–1, 0)

Q.71

The points A(a, 0), B(0, b), C(c, 0) & D(0, d) are such that ac = bd & a, b, c, d are all non−zero. The the points (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic

Q.72

The value of 'c' for which the set, {(x, y)x2 + y2 + 2x ≤ 1} ∩ {(x, y)x − y + c ≥ 0} contains only one point in common is : (A) (− ∞, − 1] ∪ [3, ∞) (B) {− 1, 3} (C) {− 3} (D) {− 1 }

Q.73

Given A ≡ (1, 1) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid- point P of BC is (A) x + y = 1 (B) x + y = 2 (C) x + y = 2xy (D) 2x + 2y = 1

Q.74

A circle is inscribed into a rhombous ABCD with one angle 60º. The distance from the centre of the circle to the nearest vertex is equal to 1 . If P is any point of the circle, then

Quest

2

2

2

PA + PB + PC + PD

(A) 12

2

is equal to :

(B) 11

(C) 9

(D) none

Q.75

The number of possible straight lines , passing through (2, 3) and forming a triangle with coordinate axes, whose area is 12 sq. units , is (A) one (B) two (C) three (D) four

Q.76

P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the co-ordinate axes cut at right angles, then : (A) a2 − 6ab + b2 = 0 (B) a2 + 2ab − b2 = 0 2 2 (C) a − 4ab + b = 0 (D) a2 − 8ab + b2 = 0

Q.77

In a triangle ABC , if A (2, – 1) and 7x – 10y + 1 = 0 and 3x – 2y + 5 = 0 are equations of an altitude and an angle bisector respectively drawn from B, then equation of BC is (A) x + y + 1 = 0 (B) 5x + y + 17 = 0 (C) 4x + 9y + 30 = 0 (D) x – 5y – 7 = 0

Q.78

The range of values of 'a' such that the angle θ between the pair of tangents drawn from the point (a, 0) to the circle x2 + y2 = 1 satisfies (A) (1, 2)

(

(B) 1 , 2

)

π < θ < π is : 2

(

)

(C) − 2 , − 1

(

) (

(D) − 2 , − 1 ∪ 1 , 2

)

Q.79

Distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x − 4y + 8 = 0 is (A) 15/2 (B) 9/2 (C) 5 (D) None

Q.80

Three concentric circles of which the biggest is x2 + y2 = 1, have their radii in A.P. If the line y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is  

1 4

(A)  0 , 



(B)  0 , 

1   2 2



(C)  0 , 

2 − 2  4 

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[8]

Q.81

The co-ordinates of the vertices P, Q, R & S of square PQRS inscribed in the triangle ABC with vertices A ≡ (0, 0) , B ≡ (3, 0) & C ≡ (2, 1) given that two of its vertices P, Q are on the side AB are respectively 1 4

 3  8

  3 1   8 8

 1 1  4 8

3 3 1 1 1 1 (B)  , 0 ,  , 0 ,  ,  &  , 

1 2

3 9 9 3 3 3 (D)  , 0 ,  , 0 ,  ,  &  , 

(A)  , 0 ,  , 0 ,  ,  &  ,  3 2

  3 1   3 2

 

(C) (1, 0) ,  , 0 ,  ,  & 1 , 

2

2

 4

 4

  4 4

  4 4

 2 4

 2 4

Q.82

A tangent at a point on the circle x2 + y2 = a2 intersects a concentric circle C at two points P and Q. The tangents to the circle X at P and Q meet at a point on the circle x2 + y2 = b2 then the equation of circle is (A) x2 + y2 = ab (B) x2 + y2 = (a – b)2 2 2 2 (C) x + y = (a + b) (D) x2 + y2 = a2 + b2

Q.83

AB is the diameter of a semicircle k, C is an arbitrary point on the semicircle (other than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of (A) angle ASB changes as C moves on k. (B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius. (C) angle ASB = 135° for all C. (D) angle ASB = 150° for all C.

Q.84

Tangents are drawn to the circle x2 + y2 = 1 at the points where it is met by the circles, x2 + y2 − (λ + 6) x + (8 − 2 λ) y − 3 = 0 . λ being the variable . The locus of the point of intersection of these tangents is : (A) 2x − y + 10 = 0 (B) x + 2y − 10 = 0 (C) x − 2y + 10 = 0 (D) 2x + y − 10 = 0

Q.85

Given

Quest

x y + = 1 and ax + by = 1 are two variable lines, 'a' and 'b' being the parameters connected by a b

the relation a2 + b2 = ab. The locus of the point of intersection has the equation (A) x2 + y2 + xy − 1 = 0 (B) x2 + y2 – xy + 1 = 0 (C) x2 + y2 + xy + 1 = 0 (D) x2 + y2 – xy – 1 = 0 Q.86

B & C are fixed points having co−ordinates (3, 0) and (− 3, 0) respectively. If the vertical angle BAC is 90º, then the locus of the centroid of the ∆ ABC has the equation : (A) x2 + y2 = 1 (B) x2 + y2 = 2 (C) 9 (x2 + y2) = 1 (D) 9 (x2 + y2) = 4

Q.87

The set of values of 'b' for which the origin and the point (1, 1) lie on the same side of the straight line, a2x + a by + 1 = 0 ∀ a ∈ R, b > 0 are : (A) b ∈ (2, 4) (B) b ∈ (0, 2) (C) b ∈ [0, 2] (D) (2, ∞)

Q.88

If  a ,  ,  b ,  ,  c ,  &  d ,  are four distinct points on a circle of radius 4 units then,

 

1 a

 

abcd is equal to (A) 4

1  b 

1 c

(B) 1/4

 

1 d

(C) 1

(D) 16

Q.89

Triangle formed by the lines x + y = 0 , x – y = 0 and lx + my = 1. If l and m vary subject to the condition l 2 + m2 = 1 then the locus of its circumcentre is (A) (x2 – y2)2 = x2 + y2 (B) (x2 + y2)2 = (x2 – y2) 2 2 2 2 (C) (x + y ) = 4x y (D) (x2 – y2)2 = (x2 + y2)2

Q.90

Tangents are drawn to a unit circle with centre at the origin from each point on the line 2x + y = 4. Then the equation to the locus of the middle point of the chord of contact is (A) 2 (x2 + y2) = x + y (B) 2 (x2 + y2) = x + 2 y (C) 4 (x2 + y2) = 2x + y (D) none Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[9]

Q.91

The co−ordinates of three points A(−4, 0) ; B(2, 1) and C(3, 1) determine the vertices of an equilateral trapezium ABCD. The co−ordinates of the vertex D are : (A) (6, 0) (B) (− 3, 0) (C) (− 5, 0) (D) (9, 0)

Q.92

ABCD is a square of unit area. A circle is tangent to two sides of ABCD and passes through exactly one of its vertices. The radius of the circle is (A) 2 − 2

2 −1

(B)

(C)

1 2

(D)

1 2

Q.93

A parallelogram has 3 of its vertices as (1, 2), (3, 8) and (4, 1). The sum of all possible x-coordinates for the 4th vertex is (A) 11 (B) 8 (C) 7 (D) 6

Q.94

A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is

2 π – 3 6

(A)

π 3 – 3

(B)

(C)

π 3 – 3 6

π  (D) 3 1 −  6 

Quest

Q.95

The image of the pair of lines represented by ax2 + 2h xy + by2 = 0 by the line mirror y = 0 is (A) ax2 − 2h xy − by2 = 0 (B) bx2 − 2h xy + ay2 = 0 (C) bx2 + 2h xy + ay2 = 0 (D) ax2 − 2h xy + by2 = 0

Q.96

A straight line with slope 2 and y-intercept 5 touches the circle, x2 + y2 + 16x + 12y + c = 0 at a point Q. Then the coordinates of Q are (A) (–6, 11) (B) (–9, –13) (C) (–10, – 15) (D) (–6, –7)

Q.97

The acute angle between two straight lines passing through the point M(− 6, − 8) and the points in which the line segment 2x + y + 10 = 0 enclosed between the co-ordinate axes is divided in the ratio 1 : 2 : 2 in the direction from the point of its intersection with the x-axis to the point of intersection with the y-axis is (A) π/3 (B) π/4 (C) π/6 (D) π/12

Q.98

A variable circle cuts each of the circles x2 + y2 − 2x = 0 & x2 + y2 − 4x − 5 = 0 orthogonally. The variable circle passes through two fixed points whose co−ordinates are :  −5 ± 3  , 0 2  

(A)  Q.99

 −5 ± 3 5  , 0 2  

(B) 

 −5 ± 5 3  , 0 2  

(C) 

(

If in triangle ABC , A ≡ (1, 10) , circumcentre ≡ − 13 , 23 co-ordinates of mid-point of side opposite to A is : (A) (1, − 11/3) (B) (1, 5) (C) (1, − 3)

)

 −5 ± 5  , 0 2  

(D) 

(

)

,4 and orthocentre ≡ 11 3 3 then the

(D) (1, 6)

Q.100 The radical centre of three circles taken in pairs described on the sides of a triangle ABC as diametres is the (A) centroid of the ∆ ABC (B) incentre of the ∆ ABC (C) circumcentre o the ∆ ABC (D) orthocentre of the ∆ ABC Q.101 The line x + y = p meets the axis of x & y at A & B respectively . A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q . P and Q lie respectively on OB and AB. If the area of the triangle APQ is 3/8th of the area of the triangle OAB, then

AQ is equal to : BQ

(A) 2

(D) 3

(B) 2/3

(C) 1/3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[10]

Q.102 Two circles are drawn through the points (1, 0) and (2, − 1) to touch the axis of y. They intersect at an angle (A) cot–1

3 4

(B) cos −1

4 5

(C)

π 2

(D) tan−1 1

Q.103 In a triangle ABC, side AB has the equation 2 x + 3 y = 29 and the side AC has the equation, x + 2 y = 16 . If the mid − point of BC is (5, 6) then the equation of BC is : (A) x − y = − 1 (B) 5 x − 2 y = 13 (C) x + y = 11 (D) 3 x − 4 y = − 9 Q.104 If the line x cos θ + y sin θ = 2 is the equation of a transverse common tangent to the circles x2 + y2 = 4 and x2 + y2 − 6 3 x − 6y + 20 = 0, then the value of θ is : (A) 5π/6 (B) 2π/3 (C) π/3 (D) π/6 Q.105 ABC is an isosceles triangle. If the co-ordinates of the base are (1, 3) and (− 2, 7) , then co-ordinates of vertex A can be :

(

)

(A) − 1 , 5 2

(

)

(B) − 1 , 5 8

(C)

( 65 , − 5)

(

(D) − 7 , 1 8

)

Q.106 A circle is drawn with y-axis as a tangent and its centre at the point which is the reflection of (3, 4) in the line y = x. The equation of the circle is (B) x2 + y2 – 8x – 6y + 16 = 0 (A) x2 + y2 – 6x – 8y + 16 = 0 (C) x2 + y2 – 8x – 6y + 9 = 0 (D) x2 + y2 – 6x – 8y + 9 = 0

Quest 4

Q.107 A is a point on either of two lines y + 3 x = 2 at a distance of units from their point of intersection. 3 The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are  

(A)  −

 , 2 3 

2

(B) (0, 0)

 2  , 2  3 

(C) 

(D) (0, 4)

Q.108 A circle of constant radius ' a ' passes through origin ' O ' and cuts the axes of co−ordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is : (A) (x2 + y2)  12 + 12  = 4 a2 x y 

(B) (x2 + y2)2  12 + 12  = a2 x y 

(C) (x2 + y2)2  12 + 12  = 4 a2 x y 

(D) (x2 + y2)  12 + 12  = a2 x y 

Q.109 Three straight lines are drawn through a point P lying in the interior of the ∆ ABC and parallel to its sides. The areas of the three resulting triangles with P as the vertex are s1, s2 and s3. The area of the triangle in terms of s1, s2 and s3 is : s1 s 2 + s 2 s 3 + s 3 s1

(A) (C)

(

s1 + s 2 + s 3

)

2

(B)

3

s1 s 2 s 3

(D) none

Q.110 The circle passing through the distinct points (1, t) , (t, 1) & (t, t) for all values of ' t ' , passes through the point : (A) (− 1, − 1) (B) (− 1, 1) (C) (1, − 1) (D) (1, 1) Q.111 The sides of a ∆ ABC are 2x − y + 5 = 0 ; x + y − 5 = 0 and x − 2y − 5 = 0 . Sum of the tangents of its interior angles is : (A) 6 (B) 27/4 (C) 9 (D) none

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[11]

Q.112 If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and B then the locus of the centroid of the triangle OAB is (A) x2 + y2 = (2k)2 (B) x2 + y2 = (3k)2 (C) x2 + y2 = (4k)2 (D) x2 + y2 = (6k)2 Q.113 Chords of the curve 4x2 + y2 − x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose co-ordinates are 1 5

4 5

(A)  , − 

 1 4  5 5

(B)  − , 

 1 4  5 5

(C)  , 

 1  5

4 5

(D)  − , − 

Q.114 Let x & y be the real numbers satisfying the equation x2 − 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are M & m respectively, then the numerical value of M − m is : (A) 2 (B) 8 (C) 15 (D) none of these Q.115 If the straight lines joining the origin and the points of intersection of the curve 5x2 + 12xy − 6y2 + 4x − 2y + 3 = 0 and x + ky − 1 = 0 are equally inclined to the co-ordinate axes then the value of k : (A) is equal to 1 (B) is equal to − 1 (C) is equal to 2 (D) does not exist in the set of real numbers . Q.116 A line meets the co-ordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d1 & d2 are the distances of the tangent to the circle at the origin O from the points A and B respectively, the diameter of the circle is : (A)

Quest

2d1 + d 2

(B)

d1 + 2d 2

(C) d1 + d2

d1d 2 (D) d + d 1 2

2 2 Q.117 A pair of perpendicular straight lines is drawn through the origin forming with the line 2x + 3y = 6 an isosceles triangle right angled at the origin. The equation to the line pair is : (B) 5x2 − 26xy − 5y2 = 0 (A) 5x2 − 24xy − 5y2 = 0 2 2 (C) 5x + 24xy − 5y = 0 (D) 5x2 + 26xy − 5y2 = 0 π to the axis X, such that the two circles 4 x2 + y2 = 4, x2 + y2 – 10x – 14y + 65 = 0 intercept equal lengths on it, is (A) 2x – 2y – 3 = 0 (B) 2x – 2y + 3 = 0 (C) x – y + 6 = 0 (D) x – y – 6 = 0

Q.118 The equation of a line inclined at an angle

Q.119 If the line y = mx bisects the angle between the lines ax2 + 2h xy + by2 = 0 then m is a root of the quadratic equation : (A) hx2 + (a − b) x − h = 0 (B) x2 + h (a − b) x − 1 = 0 (C) (a − b) x2 + hx − (a − b) = 0 (D) (a − b) x2 − hx − (a − b) = 0 Q.120 Tangents are drawn from any point on the circle x2 + y2 = R2 to the circle x2 + y2 = r2. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals (A)

2r

(B) 2r

(C)

2r 2− 3

(D)

4r 3− 5

Q.121 An equilateral triangle has each of its sides of length 6 cm . If (x1, y1); (x2, y2) & (x3, y3) are its vertices then the value of the determinant,

x1 x2 x3

y1 1 y2 1 y3 1

(A) 192

2

is equal to : (B) 243

(C) 486

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 972

[12]

Q.122 A variable circle C has the equation x2 + y2 – 2(t2 – 3t + 1)x – 2(t2 + 2t)y + t = 0, where t is a parameter. If the power of point P(a,b) w.r.t. the circle C is constant then the ordered pair (a, b) is

 1 1 (B)  − ,   10 10 

1 1 (A)  , −  10   10

1 1 (C)  ,   10 10 

1  1 (D)  − , −   10 10 

Q.123 Points A & B are in the first quadrant; point 'O' is the origin . If the slope of OA is 1, slope of OB is 7 and OA = OB, then the slope of AB is: (A) − 1/5 (B) − 1/4 (C) − 1/3 (D) − 1/2 Q.124 Let C be a circle with two diameters intersecting at an angle of 30 degrees. A circle S is tangent to both the diameters and to C, and has radius unity. The largest radius of C is (A) 1 +

6+ 2

(B) 1 +

6− 2

(C)

6+ 2 –1

(D) none of these

Q.125 The co-ordinates of a point P on the line 2x − y + 5 = 0 such that PA − PB is maximum where A is (4, − 2) and B is (2, − 4) will be : (A) (11, 27) (B) (− 11, − 17) (C) (− 11, 17) (D) (0, 5) Q.126 A straight line l1 with equation x – 2y + 10 = 0 meets the circle with equation x2 + y2 = 100 at B in the first quadrant. A line through B, perpendicular to l1 cuts the y-axis at P (0, t). The value of 't' is (A) 12 (B) 15 (C) 20 (D) 25

Quest

Q.127 A variable circle C has the equation x2 + y2 – 2(t2 – 3t + 1)x – 2(t2 + 2t)y + t = 0, where t is a parameter. The locus of the centre of the circle is (A) a parabola (B) an ellipse (C) a hyperbola (D) pair of straight lines Q.128 Let a and b represent the length of a right triangle's legs. If d is the diameter of a circle inscribed into the triangle, and D is the diameter of a circle superscribed on the triangle, then d + D equals (A) a + b (B) 2(a + b) 1 (C) (a + b) (D) a 2 + b 2 2 Select the correct alternatives : (More than one are correct)

Q.129 The area of triangle ABC is 20 cm2. The co−ordinates of vertex A are (− 5, 0) and B are (3, 0). The vertex C lies on the line, x − y = 2 . The co−ordinates of C are (A) (5, 3) (B) (− 3, − 5) (C) (− 5, − 7) (D) (7, 5) Q.130 A point (x1, y1) is outside the circle, x2 + y2 + 2gx + 2fg + c = 0 with centre at the origin and AP, AQ are tangents to the circle. Then : (A) area of the quadriletral formed by the pair of tangents and the corresponding radii through the points of contact is

(g

2

)(

+ f 2 − c x12 + y12 + 2gx1 + 2fy1 + c

)

(B) equation of the circle circumscribing the ∆APQ is, x2 + y2 + x(g – x1) + y(f − y1) – (gx1 + fy1) = 0 (C) least radius of a circle passing through the points 'A' & the origin is,

(

)(

(x1 + g)2 + (y1 + f )2

2 2 2 2  1 −1  2 g + f − c x1 + y1 + 2gx1 + 2fy1 + c (D) the ∠ between the two tangent is, sin  2 (x1 + g) 2 + (y1 + f )2 

)   

Q.131 Let u ≡ ax + by + a 3 b = 0 v ≡ bx − ay + b 3 a = 0 a, b ∈ R be two straight lines. The equation of the bisectors of the angle formed by k1u − k2v = 0 & k1u + k2v = 0 for non zero real k1 & k2 are: (A) u = 0 (B) k2u + k1v = 0 (C) k2u − k1v = 0 (D) v = 0 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Q.132

y − y1 x − x1 = = r , represents : sinθ cosθ

(A) equation of a straight line, if θ is constant & r is variable (B) equation of a circle, if r is constant & θ is a variable (C) a straight line passing through a fixed point & having a known slope (D) a circle with a known centre & a given radius. Q.133 All the points lying inside the triangle formed by the points (1, 3), (5, 6) & (− 1, 2) satisfy (A) 3x + 2y ≥ 0 (B) 2x + y + 1 ≥ 0 (C) 2x + 3y − 12 ≥ 0 (D) − 2x + 11 ≥ 0 Q.134 The equations of the tangents drawn from the origin to the circle, x² + y² − 2rx − 2hy + h² = 0 are (A) x = 0 (B) y = 0 (C) (h² − r²) x − 2rhy = 0 (D) (h² − r²)x + 2rhy = 0 Q.135 The co-ordinates of the fourth vertex of the parallelogram where three of its vertices are (− 3, 4); (0, − 4) & (5, 2) can be : (A) (8, − 6) (B) (2, 10) (C) (− 8, − 2) (D) none Q.136 The equation of a circle with centre (4, 3) and touching the circle x2 + y2 = 1 is : (A) x2 + y2 − 8x − 6y − 9 = 0 (B) x2 + y2 − 8x − 6y + 11 = 0 2 2 (C) x + y − 8x − 6y − 11 = 0 (D) x2 + y2 − 8x − 6y + 9 = 0

Quest

Q.137 Two vertices of the ∆ ABC are at the points A(− 1, − 1) and B(4, 5) and the third vertex lines on the straight line y = 5(x − 3) . If the area of the ∆ is 19/2 then the possible co−ordinates of the vertex C are: (A) (5, 10) (B) (3, 0) (C) (2, − 5) (D) (5, 4) Q.138 A circle passes through the points (− 1, 1) , (0, 6) and (5, 5) . The point(s) on this circle, the tangent(s) at which is/are parallel to the straight line joining the origin to its centre is/are : (A) (1, − 5) (B) (5, 1) (C) (− 5, − 1) (D) (− 1, 5) Q.139 Line

x y x y + = 1 cuts the co−ordinate axes at A(a, 0) & B (0, b) & the line + = − 1 at a b a ′ b′

A′ (−a′, 0) & B′ (0, −b′). If the points A, B, A′, B′ are concyclic then the orthocentre of the triangle ABA′ is: (A) (0, 0)

aa ′  (C)  0 , 

(B) (0, b')



b 

 

(D)  0 ,

b b'   a 

Q.140 Point M moved along the circle (x − 4)2 + (y − 8)2 = 20 . Then it broke away from it and moving along a tangent to the circle, cuts the x−axis at the point (− 2, 0) . The co−ordinates of the point on the circle at which the moving point broke away can be : (A)  − 3 , 46   5

5

2 44 (B)  − , 

 5

5

(C) (6, 4)

(D) (3, 5)

Q.141 If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x − 3 y = 0 then the co-ordinates of the third vertex are :  3a



a

3a a

(B)  2 , − 2  (C) (0, − a) (D)  − 2 , 2      2 2 2 2 Q.142 The circles x + y + 2x + 4y − 20 = 0 & x + y + 6x − 8y + 10 = 0 (A) are such that the number of common tangents on them is 2 (B) are not orthogonal (C) are such that the length of their common tangent is 5 (12/5)1/4 (A) (0, a)

(D) are such that the length of their common chord is 5

3 . 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[14]

Q.143 Two straight lines u = 0 and v = 0 passes through the origing forming an angle of tan −1 (7/9) with each other . If the ratio of the slopes of u = 0 and v = 0 is 9/2 then their equations are: (A) y = 3x & 3y = 2x (B) 2y = 3x & 3y = x (C) y + 3x = 0 & 3y + 2x = 0 (D) 2y + 3x = 0 & 3y + x = 0 Q.144 The centre(s) of the circle(s) passing through the points (0, 0) , (1, 0) and touching the circle x2 + y2=9 is/are :  2 2

 2 2

1 2

1 (C)  , 21/ 2 

1 3 (B)  , 

(A)  3 , 1 

2

 

(D)  , − 21/ 2 



Q.145 Given two straight lines x − y − 7 = 0 and x − y + 3 = 0. Equation of a line which divides the distance between them in the ratio 3 : 2 can be : (A) x − y − 1 = 0 (B) x − y − 3 = 0 (C) y = x (D) x − y + 1 = 0 Q.146 The circles x2 + y2 − 2x − 4y + 1 = 0 and x2 + y2 + 4x + 4y − 1 = 0 (A) touch internally (B) touch externally (C) have 3x + 4y − 1 = 0 as the common tangent at the point of contact. (D) have 3x + 4y + 1 = 0 as the common tangent at the point of contact.

Quest

Q.147 Three vertices of a triangle are A(4, 3) ; B(1, − 1) and C(7, k) . Value(s) of k for which centroid, orthocentre, incentre and circumcentre of the ∆ ABC lie on the same straight line is/are : (A) 7 (B) − 1 (C) − 19/8 (D) none Q.148 A and B are two fixed points whose co-ordinates are (3, 2) and (5, 4) respectively. The co-ordinates of a point P if ABP is an equilateral triangle, is/are :

(

(A) 4 − 3 , 3 + 3

)

(

(B) 4 + 3 , 3 − 3

)

(

(C) 3 − 3 , 4 + 3

)

(

(D) 3 + 3 , 4 − 3

)

Q.149 Which of the following lines have the intercepts of equal lengths on the circle, x2 + y2 − 2x + 4y = 0? (A) 3x − y = 0 (B) x + 3y = 0 (C) x + 3y + 10 = 0 (D) 3x − y − 10 = 0 Q.150 Straight lines 2x + y = 5 and x − 2y = 3 intersect at the point A . Points B and C are chosen on these two lines such that AB = AC . Then the equation of a line BC passing through the point (2, 3) is (A) 3x − y − 3 = 0 (B) x + 3y − 11 = 0 (C) 3x + y − 9 = 0 (D) x − 3y + 7 = 0 Q.151 Equation of a straight line passing through the point (2, 3) and inclined at an angle of arc tan

1 with the line y + 2x = 5 is: 2

(A) y = 3

(B) x = 2

(C) 3x + 4y − 18 = 0 (D) 4x + 3y − 17 = 0

Q.152 The x − co-ordinates of the vertices of a square of unit area are the roots of the equation x2 − 3x + 2 = 0 and the y − co-ordinates of the vertices are the roots of the equation y2 − 3y + 2 = 0 then the possible vertices of the square is/are : (A) (1, 1), (2, 1), (2, 2), (1, 2) (B) (− 1, 1), (− 2, 1), (− 2, 2), (− 1, 2) (C) (2, 1), (1, − 1), (1, 2), (2, 2) (D) (− 2, 1), (− 1, − 1), (− 1, 2), (− 2, 2) Q.153 Consider the equation y − y1 = m (x − x1). If m & x1 are fixed and different lines are drawn for different values of y1, then (A) the lines will pass through a fixed point (B) there will be a set of parallel lines (C) all the lines intersect the line x = x1 (D) all the lines will be parallel to the line y = x1.

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

ANSWER KEY Select the correct alternative : (Only one is correct) Q.1

D

Q.2

B

Q.3

A

Q.4

A

Q.5

A

Q.6

B

Q.7

B

Q.8

A

Q.9

C

Q.10

B

Q.11

A

Q.12

D

Q.13

C

Q.14

C

Q.15

B

Q.16

A

Q.17

B

Q.18

C

Q.19

D

Q.20

B

Q.21

D

Q.22

B

Q.23

B

Q.24

A

Q.25

C

Q.26

A

Q.27

B

Q.28

C

Q.29

A

Q.30

A

Q.31

B

Q.32

B

Q.33

B

Q.34

D

Q.35

B

Q.36

A

Q.37

A

Q.38

C

Q.39

D

Q.40

D

Q.41

B

Q.42

D

Q.43

A

Q.44

B

Q.45

B

Q.46

C

Q.47

A

Q.48

A

Q.49

D

Q.50

A

Q.51

D

Q.52

D

Q.53

B

Q.54

C

Q.55

A

Q.56

C

Q.57

B

Q.58

A

Q.59

B

Q.60

B

Q.61

D

Q.62

A

Q.63

B

Q.64

B

Q.65

D

Q.66

B

Q.67

D

Q.68

B

Q.69

D

Q.70

D

Q.71

D

Q.72

D

Q.73

A

Q.74

B

Q.75

C

Q.76

C

Q.77

B

Q.78

D

Q.79

C

Q.80

C

Q.81

D

Q.82

A

Q.83

C

Q.84

A

Q.85

A

Q.86

A

Q.87

B

Q.88

C

Q.89

A

Q.90

C

Q.91

D

Q.92

A

Q.93

D

Q.94

B

Q.95

D

Q.96

D

Q.97

B

Q.98

B

Q.99

A

Quest Q.100 D

Q.101 D

Q.102 A

Q.103 C

Q.104 D

Q.105 D

Q.107 B

Q.108 C

Q.109 C

Q.110 D

Q.111 B

Q.112 A

Q.113 A

Q.114 B

Q.115 B

Q.116 C

Q.117 A

Q.118 A

Q.119 A

Q.120 B

Q.121 D

Q.122 B

Q.123 D

Q.124 A

Q.125 B

Q.126 C

Q.127 A

Q.128 A

Q.106 C

Select the correct alternatives : (More than one are correct) Q.129 B,D

Q.130 A,B,D

Q.131 A,D

Q.132 A,B,C,D

Q.133 A,B,D

Q.134 A,C

Q.135 A,B,C

Q.136 C,D

Q.137 A,B

Q.138 B,D

Q.139 B,C

Q.140 B,C

Q.141 A,B,C,D

Q.142 A,C,D

Q.143 A,B,C,D

Q.144 C,D

Q.145 A,B

Q.146 B,C

Q.147 B,C

Q.148 A,B

Q.149 A,B,C,D

Q.150 A,B

Q.151 B,C

Q.152 A,B

Q.153 B,C

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[16]

TARGET IIT JEE

MATHEMATICS

STRAIGHT LINES & CIRCLES.

Circle & Straight line There are 125 questions in this question bank. Select the correct alternative : (Only one is correct) Q.1

Coordinates of the centre of the circle which bisects the circumferences of the circles x2 + y2 = 1 ; x2 + y2 + 2x – 3 = 0 and x2 + y2 + 2y – 3 = 0 is (A) (–1, –1) (B) (3, 3) (C) (2, 2) (D) (– 2, – 2)

Q.2

One side of a square is inclined at an acute angle α with the positive x-axis, and one of its extremities is at the origin. If the remaining three vertices of the square lie above the x-axis and the side of a square is 4, then the equation of the diagonal of the square which is not passing through the origin is (A) (cos α + sin α) x + (cos α – sin α) y = 4 (B) (cos α + sin α) x – (cos α – sin α) y = 4 (C) (cos α – sin α) x + (cos α + sin α) y = 4 (D) (cos α – sin α) x – (cos α + sin α) y = 4 cos 2α

Q.3

The line 2x – y + 1 = 0 is tangent to the circle at the point (2, 5) and the centre of the circles lies on x – 2y = 4. The radius of the circle is (A) 3 5

Q.4

(B) 5 3

(C) 2 5

(D) 5 2

Quest

Given the family of lines, a (2x + y + 4) + b (x − 2y − 3) = 0 . Among the lines of the family, the number of lines situated at a distance of 10 from the point M (2, − 3) is : (A) 0 (B) 1 (C) 2

(D) ∞

Q.5

The co-ordinate of the point on the circle x² + y² − 12x − 4y + 30 = 0, which is farthest from the origin are : (A) (9 , 3) (B) (8 , 5) (C) (12 , 4) (D) none

Q.6

The area of triangle formed by the lines x + y – 3 = 0 , x – 3y + 9 = 0 and 3x – 2y + 1= 0 (A)

16 sq. units 7

(B)

10 sq. units 7

(C) 4 sq. units

(D) 9 sq. units

Q.7

The number of common tangent(s) to the circles x² + y² + 2x + 8y − 23 = 0 and x² + y² − 4x − 10y + 19 = 0 is : (A) 1 (B) 2 (C) 3 (D) 4

Q.8

The four points whose co−ordinates are (2, 1), (1, 4), (4, 5), (5, 2) form : (A) a rectangle which is not a square (B) a trapezium which is not a parallelogram (C) a square (D) a rhombus which is not a square.

Q.9

From the point A (0 , 3) on the circle x² + 4x + (y − 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is : (A) x² + 8x + y² = 0 (B) x² + 8x + (y − 3)² = 0 (C) (x − 3)² + 8x + y² = 0 (D) x² + 8x + 8y² = 0

Q.10

A ray of light passing through the point A (1, 2) is reflected at a point B on the x − axis and then passes through (5, 3) . Then the equation of AB is : (A) 5x + 4y = 13 (B) 5x − 4y = − 3 (C) 4x + 5y = 14 (D) 4x − 5y = − 6

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.11

Two circles of radii 4 cms & 1 cm touch each other externally and θ is the angle contained by their direct common tangents. Then sin θ = (A)

Q.12

24 25

(B)

(C)

3 4

(D) none

If A & B are the points (− 3, 4) and (2, 1), then the co−ordinates of the point C on AB produced such that AC = 2 BC are : (A) (2, 4)

Q.13

12 25

(B) (3, 7)

(C) (7, −2)

 1 5  2 2

(D)  − , 

The locus of the mid points of the chords of the circle x2 + y2 − ax − by = 0 which subtend a right angle  a b  2 2

at  ,  is : (B) ax + by = a2 + b2

(A) ax + by = 0 (C) x2 + y2 − ax − by +

a 2 +b 2 =0 8

(D) x2 + y2 − ax − by −

a 2 +b 2 =0 8

Quest

Q.14

The base BC of a triangle ABC is bisected at the point (p, q) and the equation to the side AB & AC are px + qy = 1 & qx + py = 1 . The equation of the median through A is : (A) (p − 2q) x + (q − 2p) y + 1 = 0 (B) (p + q) (x + y) − 2 = 0 (C) (2pq − 1) (px + qy − 1) = (p2 + q2 − 1) (qx + py − 1) (D) none

Q.15

From (3 , 4) chords are drawn to the circle x² + y² − 4x = 0 . The locus of the mid points of the chords is : (A) x² + y² − 5x − 4y + 6 = 0 (B) x² + y² + 5x − 4y + 6 = 0 (C) x² + y² − 5x + 4y + 6 = 0 (D) x² + y² − 5x − 4y − 6 = 0

Q.16

The lines y − y1 = m (x − x1) ± a 1 + m 2 are tangents to the same circle . The radius of the circle is : (A) a/2 (B) a (C) 2a (D) none

Q.17

The centre of the smallest circle touching the circles x² + y² − 2y − 3 = 0 and x² + y² − 8x − 18y + 93 = 0 is : (B) (4 , 4) (C) (2 , 7) (D) (2 , 5) (A) (3 , 2)

Q.18

If a, b, c are in harmonical progression then the line, bcx + cay + ab = 0 passes through a fixed point whose co−ordinates are : (A) (1, 2) (B) (− 1, 2) (C) (− 1, − 2) (D) (1, − 2)

Q.19

A rhombus is inscribed in the region common to the two circles x2 + y2 − 4x − 12 = 0 and x2 + y2 + 4x − 12 = 0 with two of its vertices on the line joining the centres of the circles. The area of the rhombous is : (A) 8 3 sq.units

(B) 4 3 sq.units

(C) 16 3 sq.units

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[3]

Q.20

A variable straight line passes through the points of intersection of the lines, x + 2y = 1 and 2x − y = 1 and meets the co−ordinate axes in A & B . The locus of the middle point of AB is : (A) x + 3y − 10xy = 0 (B) x − 3y + 10xy = 0 (C) x + 3y + 10xy = 0 (D) none

Q.21

In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to AB ⋅ AD (A)

AB 2 + AD 2

(B)

AB ⋅ AD AB + AD

(C)

AB ⋅ AD

AB ⋅ AD (D)

AB2 − AD2

Q.22

A variable straight line passes through a fixed point (a, b) intersecting the co−ordinates axes at A & B. If 'O' is the origin then the locus of the centroid of the triangle OAB is : (A) bx + ay − 3xy = 0 (B) bx + ay − 2xy = 0 (C) ax + by − 3xy = 0 (D) none

Q.23

The equation of the circle having the lines y2 − 2y + 4x − 2xy = 0 as its normals & passing through the point (2 , 1) is : (A) x2 + y2 − 2x − 4y + 3 = 0 (B) x2 + y2 − 2x + 4y − 5 = 0 2 2 (C) x + y + 2x + 4y − 13 = 0 (D) none

Q.24

If P = (1, 0) ; Q = (−1, 0) & R = (2, 0) are three given points, then the locus of the points S satisfying the relation, SQ2 + SR2 = 2 SP2 is : (A) a straight line parallel to x−axis (B) a circle passing through the origin (C) a circle with the centre at the origin (D) a straight line parallel to y−axis .

Q.25

If a circle passes through the point (a , b) & cuts the circle x² + y² = K² orthogonally, then the equation of the locus of its centre is : (A) 2ax + 2by − (a² + b² + K²) = 0 (B) 2ax + 2by − (a² − b² + K²) = 0 (C) x² + y² − 3ax − 4by + (a² + b² − K²) = 0 (D) x² + y² − 2ax − 3by + (a² − b² − K²) = 0

Q.26

The co−ordinates of the orthocentre of the triangle bounded by the lines, 4x − 7y + 10 = 0; x + y=5 and 7x + 4y = 15 is : (A) (2, 1) (B) (− 1, 2) (C) (1, 2) (D) (1, − 2)

Q.27

The distance between the chords of contact of tangents to the circle ; x2+ y2 + 2gx+2fy+ c=0 from the origin & the point (g , f) is :

Quest

(A) g2 + f 2

(B)

g2 + f 2 − c 2

(C)

g2 + f 2 − c 2 g2 + f 2

(D)

g2 + f 2 + c 2 g2 + f 2

Q.28

The equation of the pair of bisectors of the angles between two straight lines is, 12x2 − 7xy − 12y2 = 0 . If the equation of one line is 2y − x = 0 then the equation of the other line is : (A) 41x − 38y = 0 (B) 38x − 41y = 0 (C) 38x + 41y = 0 (D) 41x + 38y = 0

Q.29

The points A (a , 0) , B (0 , b) , C (c , 0) & D (0 , d) are such that ac = bd & a, b, c, d are all non-zero. Then the points : (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.30

The line joining two points A (2, 0) ; B (3, 1) is rotated about A in the anticlock wise direction through an angle of 15º . The equation of the line in the new position is : (A) x −

3 y − 2 = 0 (B) x − 2y − 2 = 0

(C)

3x−y−2 3 =0

(D) none

Q.31

The locus of the centers of the circles which cut the circles x2 + y2 + 4x − 6y + 9 = 0 and x2 + y2 − 5x + 4y − 2 = 0 orthogonally is (A) 9x + 10y − 7 = 0 (B) x − y + 2 = 0 (C) 9x − 10y + 11 = 0 (D) 9x + 10y + 7 = 0

Q.32

Area of the rhombus bounded by the four lines, ax ± by ± c = 0 is : (A)

Q.33

Q.34

c2 2 ab

(B)

2 c2 ab

(C)

4 c2 ab

(D)

ab 4 c2

Given A ≡ (1, 1) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid- point P of BC is (A) x + y = 1 (B) x + y = 2 (C) x + y = 2xy (D) 2x + 2y = 1 The locus of the centers of the circles such that the point (2 , 3) is the mid point of the chord 5x + 2y = 16 is : (A) 2x − 5y + 11 = 0 (B) 2x + 5y − 11 = 0 (C) 2x + 5y + 11 = 0 (D) none

Quest

Q.35

A stick of length 10 units rests against the floor & a wall of a room . If the stick begins to slide on the floor then the locus of its middle point is : (A) x2 + y2 = 2.5 (B) x2 + y2 = 25 (C) x2 + y2 = 100 (D) none

Q.36

The locus of the mid points of the chords of the circle x² + y² + 4x − 6y − 12 = 0 which subtend an angle of

π radians at its circumference is : 3

(A) (x − 2)² + (y + 3)² = 6.25 (C) (x + 2)² + (y − 3)² = 18.75 Q.37

(B) (x + 2)² + (y − 3)² = 6.25 (D) (x + 2)² + (y + 3)² = 18.75

Through a given point P (a, b) a straight line is drawn to meet the axes at Q & R. If the parallelogram OQSR is completed then the equation of the locus of S is (given 'O' is the origin) : (A)

a b + =1 x y

(B)

a b + =1 y x

(C)

a b + =2 x y

(D)

a b + =2 y x

Q.38

The points (x1, y1) , (x2, y2) , (x1, y2) & (x2, y1) are always : (A) collinear (B) concyclic (C) vertices of a square (D) vertices of a rhombus

Q.39

The number of possible straight lines , passing through (2, 3) and forming a triangle with coordinate axes, whose area is 12 sq. units , is (A) one (B) two (C) three (D) four

Q.40

Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line 2x + y = 5 . Then the area of the triangle is : (A) 5 (B) 3 (C) 5/2 (D) 1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.41

The angle at which the circles (x – 1)2 + y2 = 10 and x2 + (y – 2)2 = 5 intersect is π π π π (A) (B) (C) (D) 6 4 3 2

Q.42

A pair of straight lines x2 – 8x + 12 = 0 and y2 – 14y + 45 = 0 are forming a square. Co-ordinates of the centre of the circle inscribed in the square are (A) (3, 6) (B) (4, 7) (C) (4, 8) (D) none

Q.43

The value of 'c' for which the set, {(x, y)x2 + y2 + 2x ≤ 1} ∩ {(x, y)x − y + c ≥ 0} contains only one point in common is : (A) (− ∞, − 1] ∪ [3, ∞) (B) {− 1, 3} (C) {− 3} (D) {− 1 }

Q.44

Co-ordinates of the orthocentre of the triangle whose vertices are A(0, 0) , B(3, 4) and C(4, 0) is (A) (3, 1)

(B) (3, 4)

 3 (D)  3,   4

(C) (3, 3)

Quest

Q.45

Three lines x + 2y + 3 = 0 ; x + 2y – 7 = 0 and 2x – y – 4 = 0 form the three sides of two squares. The equation to the fourth side of each square is (A) 2x – y + 14 = 0 & 2x – y + 6 = 0 (B) 2x – y + 14 = 0 & 2x – y – 6 = 0 (C) 2x – y – 14 = 0 & 2x – y – 6 = 0 (D) 2x – y – 14 = 0 & 2x – y + 6 = 0

Q.46

P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the co-ordinate axes cut at right angles, then : (B) a2 + 2ab − b2 = 0 (A) a2 − 6ab + b2 = 0 (C) a2 − 4ab + b2 = 0 (D) a2 − 8ab + b2 = 0

Q.47

If the vertices P and Q of a triangle PQR are given by (2, 5) and (4, –11) respectively, and the point R moves along the line N: 9x + 7y + 4 = 0, then the locus of the centroid of the triangle PQR is a straight line parallel to (A) PQ (B) QR (C) RP (D) N

Q.48

The range of values of 'a' such that the angle θ between the pair of tangents drawn from the point (a, 0) to the circle x2 + y2 = 1 satisfies (A) (1, 2)

(

(B) 1 , 2

)

π < θ < π is : 2

(

)

(C) − 2 , − 1

(

) (

(D) − 2 , − 1 ∪ 1 , 2

)

Q.49

The points A(a, 0), B(0, b), C(c, 0) & D(0, d) are such that ac = bd & a, b, c, d are all non−zero. The the points : (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic

Q.50

If (α, β) is a point on the circle whose centre is on the x -axis and which touches the line x + y = 0 at (2, –2), then the greatest value of α is (A) 4 –

2

(B) 6

(C) 4 + 2 2

(D) 4 +

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

2

[6]

Q.51

Distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x − 4y + 8 = 0 is (A) 15/2 (B) 9/2 (C) 5 (D) None

Q.52

Three concentric circles of which the biggest is x2 + y2 = 1, have their radii in A.P. If the line y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is  

1 4

(A)  0 , 

Q.53

Given



(B)  0 , 

1   2 2



(C)  0 , 

2 − 2  4 

(D) none

x y + = 1 and ax + by = 1 are two variable lines, 'a' and 'b' being the parameters connected by a b

the relation a2 + b2 = ab. The locus of the point of intersection has the equation (A) x2 + y2 + xy − 1 = 0 (B) x2 + y2 – xy + 1 = 0 2 2 (C) x + y + xy + 1 = 0 (D) x2 + y2 – xy – 1 = 0 Q.54

The chord of contact of the tangents drawn from a point on the circle, x2 + y2 = a2 to the circle x2 + y2 = b2 touches the circle x2 + y2 = c2 then a, b, c are in : (A) A.P. (B) G.P. (C) H.P. (D) A.G.P.

Q.55

A light beam emanating from the point A(3, 10) reflects from the line 2x + y - 6 = 0 and then passes through the point B(5, 6) . The equation of the incident and reflected beams are respectively : (A) 4 x − 3 y + 18 = 0 & y = 6 (B) x − 2 y + 8 = 0 & x = 5 (C) x + 2 y − 8 = 0 & y = 6 (D) none of these

Q.56

If the two circles, x2 + y2 + 2 g1x + 2 f1y = 0 & x2 + y2 + 2 g2x + 2 f2y = 0 touch each then:

Quest

(A) f1 g1 = f2 g2 Q.57

 1  , p ; Q =  xp 

If P ≡ 

k ∈ N, then :

(B)

f1 f = 2 g1 g2

(C) f1 f2 = g1 g2

(D) none

 1   1   , q ; R =  , r where xk ≠ 0, denotes the kth term of an H.P. for  xr   xq 

p2 q2 r2 (A) Ar. (∆ PQR) = 2

( p − q) 2 + (q − r ) 2 + ( r − p ) 2

(B) ∆ PQR is a right angled triangle (C) the points P, Q, R are collinear (D) none Q.58

Tangents are drawn to the circle x2 + y2 = 1 at the points where it is met by the circles, x2 + y2 − (λ + 6) x + (8 − 2 λ) y − 3 = 0 . λ being the variable . The locus of the point of intersection of these tangents is : (A) 2x − y + 10 = 0 (B) x + 2y − 10 = 0 (C) x − 2y + 10 = 0 (D) 2x + y − 10 = 0

Q.59

The acute angle between two straight lines passing through the point M(− 6, − 8) and the points in which the line segment 2x + y + 10 = 0 enclosed between the co-ordinate axes is divided in the ratio 1 : 2 : 2 in the direction from the point of its intersection with the x − axis to the point of intersection with the y − axis is : (A) π/3 (B) π/4 (C) π/6 (D) π/12

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[7]

Q.60

B & C are fixed points having co−ordinates (3, 0) and (− 3, 0) respectively . If the vertical angle BAC is 90º, then the locus of the centroid of the ∆ ABC has the equation : (A) x2 + y2 = 1 (B) x2 + y2 = 2 (C) 9 (x2 + y2) = 1 (D) 9 (x2 + y2) = 4

Q.61

Chords of the curve 4x2 + y2 − x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose co-ordinates are : 1 5

4 5

 1 4  5 5

(A)  , −  Q.62

 

1 a

(B)  − , 

 

1 b

 

1 c

 

 1 4  5 5

(C)  , 

 1  5

4 5

(D)  − , − 

1 d

If  a ,  ,  b ,  ,  c ,  &  d ,  are four distinct points on a circle of radius 4 units then, abcd is equal to (A) 4

(B) 1/4

(C) 1

(D) 16

Q.63

The pair of straight lines x2 − 4xy + y2 = 0 together with the line x + y + 4 6 = 0 form a triangle which is : (A) right angled but not isosceles (B) right isosceles (C) scalene (D) equilateral

Q.64

If two chords, each bisected by the x − axis can be drawn to the circle, 2 (x2 + y2) − 2ax − by = 0 (a ≠ 0 , b ≠ 0) from the point (a, b/2) then : (A) a2 > 8b2 (B) b2 > 2a2 (C) a2 > 2b2 (D) a2 = 2b2

Q.65

If the line y = mx bisects the angle between the lines ax2 + 2h xy + by2 = 0 then m is a root of the quadratic equation : (A) hx2 + (a − b) x − h = 0 (B) x2 + h (a − b) x − 1 = 0 (C) (a − b) x2 + hx − (a − b) = 0 (D) (a − b) x2 − hx − (a − b) = 0

Q.66

Tangents are drawn to a unit circle with centre at the origin from each point on the line 2x + y = 4. Then the equation to the locus of the middle point of the chord of contact is (A) 2 (x2 + y2) = x + y (B) 2 (x2 + y2) = x + 2 y (C) 4 (x2 + y2) = 2x + y (D) none An equilateral triangle has each of its sides of length 6 cm . If (x1, y1) ; (x2, y2) & (x3, y3) are its vertices then the value of the determinant,

Q.67

Quest x1 x2 x3

(A) 192 Q.68

2

is equal to : (B) 243

(C) 486

(D) 972

Two circles whose radii are equal to 4 and 8 intersect at right angles. The length of their common chord is (A)

Q.69

y1 1 y2 1 y3 1

16 5

(B) 8

(C) 4 6

(D)

8 5 5

Points A & B are in the first quadrant ; point 'O' is the origin . If the slope of OA is 1, slope of OB is 7 and OA = OB, then the slope of AB is : (A) − 1/5 (B) − 1/4 (C) − 1/3 (D) − 1/2 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

Q.70

can be seen from their centres at the angles of 90º and 60º respectively . If the distance between their centres is equal to

T

h e

c o

m

m

o

n

c h o

r d

o

f

t w

o

i n

t e r s e c t i n g

c i r c l e s

c

1 & c2

3 + 1 then the radii of c1 & c2 are :

(A)

3 &3

(B)

2 & 2 2

(C)

2 &2

(D) 2 2 & 4

Q.71

The co-ordinates of a point P on the line 2x − y + 5 = 0 such that PA − PB is maximum where A is (4, − 2) and B is (2, − 4) will be : (A) (11, 27) (B) (− 11, − 17) (C) (− 11, 17) (D) (0, 5)

Q.72

Three circles lie on a plane so that each of them externally touches the other two. Two of them has radius 3, the third having radius unity . If A, B & C are the points of tangency of the circles then the area of the triangle ABC is (A)

9 7 4

(B)

9 7 8

(C)

9 3 16

(D) none

Quest

Q.73

Let the co-ordinates of the two points A & B be (1, 2) and (7, 5) respectively. The line AB is rotated through 45º in anti clockwise direction about the point of trisection of AB which is nearer to B. The equation of the line in new position is : (A) 2x − y − 6 = 0 (B) x − y − 1 = 0 (C) 3x − y − 11 = 0 (D) none of these

Q.74

A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is (A)

2 π – 3 6

(B)

π 3 – 3

(C)

π 3 – 3 6

π  (D) 3 1 −  6 

Q.75

The true set of real values of λ for which the point P with co-ordinate (λ, λ2) does not lie inside the triangle formed by the lines, x − y = 0 ; x + y − 2 = 0 & x + 3 = 0 is : (A) (− ∞, − 2] (B) [0, ∞] (C) [− 2, 0] (D) (− ∞, − 2] ∪ [0, ∞]

Q.76

If the line x cos θ + y sin θ = 2 is the equation of a transverse common tangent to the circles

Q.77

x2 + y2 = 4 and x2 + y2 − 6 3 x − 6y + 20 = 0, then the value of θ is : (A) 5π/6 (B) 2π/3 (C) π/3 (D) π/6 The graph of the function, cos x cos (x + 2) − cos2 (x + 1) is : (A) a straight line passing through (0 , − sin2 1) with slope 2 (B) a straight line passing through (0 , 0) (C) a parabola with vertex (1 , − sin2 1) π (D) a straight line passing through the point  , − sin 2 1 & parallel to the x−axis .

2

Q.78



A circle is drawn with y-axis as a tangent and its centre at the point which is the reflection of (3, 4) in the line y = x. The equation of the circle is (A) x2 + y2 – 6x – 8y + 16 = 0 (B) x2 + y2 – 8x – 6y + 16 = 0 (C) x2 + y2 – 8x – 6y + 9 = 0 (D) x2 + y2 – 6x – 8y + 9 = 0

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[9]

Q.79

Let PQR be a right angled isosceles triangle, right angled at P (2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is (A) 3x2 − 3y2 + 8xy + 20x + 10y + 25 = 0 (B) 3x2 − 3y2 + 8xy − 20x − 10y + 25 = 0 2 2 (C) 3x − 3y + 8xy + 10x + 15y + 20 = 0 (D) 3x2 − 3y2 − 8xy − 10x − 15y − 20 = 0

Q.80

A circle of constant radius ' a ' passes through origin ' O ' and cuts the axes of co−ordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is :

Q.81

(A) (x2 + y2)  12 + 12  = 4 a2 x y 

(B) (x2 + y2)2  12 + 12  = a2 x y 

(C) (x2 + y2)2  12 + 12  = 4 a2 x y 

(D) (x2 + y2)  12 + 12  = a2 x y  4

A is a point on either of two lines y + 3 x = 2 at a distance of units from their point of intersection. 3 The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are  

(A)  −

 , 2 3 

2

 2  , 2  3 

(C) 

(B) (0, 0)

(D) (0, 4)

Quest

Q.82

The circle passing through the distinct points (1, t) , (t, 1) & (t, t) for all values of ' t ' , passes through the point : (B) (− 1, 1) (C) (1, − 1) (D) (1, 1) (A) (− 1, − 1)

Q.83

In a triangle ABC, side AB has the equation 2 x + 3 y = 29 and the side AC has the equation , x + 2 y = 16 . If the mid − point of BC is (5, 6) then the equation of BC is : (A) x − y = − 1 (B) 5 x − 2 y = 13 (C) x + y = 11 (D) 3 x − 4 y = − 9

Q.84

If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and B then the locus of the centroid of the triangle OAB is (A) x2 + y2 = (2k)2 (B) x2 + y2 = (3k)2 (C) x2 + y2 = (4k)2 (D) x2 + y2 = (6k)2

Q.85

The circumcentre of the triangle formed by the lines , x y + 2 x + 2 y + 4 = 0 and x + y + 2 = 0 is (A) (− 2, − 2) (B) (− 1, − 1) (C) (0, 0) (D) (− 1, − 2)

Q.86

The locus of the mid−points of the chords of the circle x2 + y2 − 2x − 4y − 11 = 0 which subtend 600 at the centre is (B) x2 + y2 + 4x + 2y − 7 = 0 (A) x2 + y2 − 4x − 2y − 7 = 0 2 2 (C) x + y − 2x − 4y − 7 = 0 (D) x2 + y2 + 2x + 4y + 7 = 0

Q.87

ABC is an isosceles triangle . If the co-ordinates of the base are (1, 3) and (− 2, 7) , then co-ordinates of vertex A can be :

(

)

(A) − 1 , 5 2 Q.88

(

)

(B) − 1 , 5 8

(C)

( 65 , − 5)

(

(D) − 7 , 1 8

)

Tangents are drawn from (4, 4) to the circle x2 + y2 − 2x − 2y − 7 = 0 to meet the circle at A and B. The length of the chord AB is (A) 2 3

(B) 3 2

(C) 2 6

(D) 6 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

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Q.89

The line x + y = p meets the axis of x & y at A & B respectively . A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q . P and Q lie respectively on OB and AB . If the area of the triangle APQ is 3/8th of the area of the triangle OAB, then

AQ is equal to : BQ

(A) 2

(D) 3

(B) 2/3

(C) 1/3

Q.90

The equation of the image of the circle x2 + y2 + 16x − 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0 is: (A) x2 + y2 + 32x − 4y + 235 = 0 (B) x2 + y2 + 32x + 4y − 235 = 0 (C) x2 + y2 + 32x − 4y − 235 = 0 (D) x2 + y2 + 32x + 4y + 235 = 0

Q.91

If in triangle ABC , A ≡ (1, 10) , circumcentre ≡ − 13 , 23 co-ordinates of mid-point of side opposite to A is : (B) (1, 5) (C) (1, − 3) (A) (1, − 11/3)

(

)

(

)

,4 and orthocentre ≡ 11 3 3 then the

(D) (1, 6)

Q.92

Let x & y be the real numbers satisfying the equation x2 − 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are M & m respectively, then the numerical value of M − m is : (A) 2 (B) 8 (C) 15 (D) none of these

Q.93

If the straight lines , ax + amy + 1 = 0 , b x + (m + 1) b y + 1 = 0 and cx + (m + 2)cy + 1 = 0, m ≠ 0 are concurrent then a, b, c are in : (A) A.P. only for m = 1 (B) A.P. for all m (C) G.P. for all m (D) H.P. for all m.

Q.94

A line meets the co-ordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d1 & d2 are the distances of the tangent to the circle at the origin O from the points A and B respectively, the diameter of the circle is :

Quest

(A)

2d1 + d 2 2

(B)

d1 + 2d 2 2

(C) d1 + d2

d1d 2

(D) d + d 1 2

Q.95

If x1 , y1 are the roots of x2 + 8 x − 20 = 0 , x2 , y2 are the roots of 4 x2 + 32 x − 57 = 0 and x3 , y3 are the roots of 9 x2 + 72 x − 112 = 0 , then the points, (x1 , y1) , (x2 , y2) & (x3 , y3) (A) are collinear (B) form an equilateral triangle (C) form a right angled isosceles triangle (D) are concyclic

Q.96

Two concentric circles are such that the smaller divides the larger into two regions of equal area. If the radius of the smaller circle is 2 , then the length of the tangent from any point ' P ' on the larger circle to the smaller circle is : (A) 1

Q.97

(B) 2

(C) 2

(D) none

Triangle formed by the lines x + y = 0 , x – y = 0 and lx + my = 1. If l and m vary subject to the condition l 2 + m2 = 1 then the locus of its circumcentre is (A) (x2 – y2)2 = x2 + y2 (B) (x2 + y2)2 = (x2 – y2) (C) (x2 + y2) = 4x2 y2 (D) (x2 – y2)2 = (x2 + y2)2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[11]

π to the axis X, such that the two circles 4 x2 + y2 = 4, x2 + y2 – 10x – 14y + 65 = 0 intercept equal lengths on it, is (A) 2x – 2y – 3 = 0 (B) 2x – 2y + 3 = 0 (C) x – y + 6 = 0 (D) x – y – 6 = 0

Q.98

The equation of a line inclined at an angle

Q.99

The co−ordinates of three points A(−4, 0) ; B(2, 1) and C(3, 1) determine the vertices of an equilateral trapezium ABCD . The co−ordinates of the vertex D are : (A) (6, 0) (B) (− 3, 0) (C) (− 5, 0) (D) (9, 0)

Q.100 Tangents are drawn from any point on the circle x2 + y2 = R2 to the circle x2 + y2 = r2. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals (A)

2r

(B) 2r

(C)

2r 2− 3

(D)

4r 3− 5

Q.101 The image of the pair of lines represented by ax2 + 2h xy + by2 = 0 by the line mirror y = 0 is (A) ax2 − 2h xy − by2 = 0 (B) bx2 − 2h xy + ay2 = 0 (C) bx2 + 2h xy + ay2 = 0 (D) ax2 − 2h xy + by2 = 0

Quest

Q.102 Pair of tangents are drawn from every point on the line 3x + 4y = 12 on the circle x2 + y2 = 4. Their variable chord of contact always passes through a fixed point whose co-ordinates are

4 3 (A)  ,  3 4

3 3 (B)  ,  4 4

(C) (1, 1)

 4 (D) 1,   3

Q.103 The set of values of 'b' for which the origin and the point (1, 1) lie on the same side of the straight line, a2x + a by + 1 = 0 ∀ a ∈ R, b > 0 are : (A) b ∈ (2, 4) (B) b ∈ (0, 2) (C) b ∈ [0, 2] (D) (2, ∞) Q.104 The equation of the circle symmetric to the circle x2 + y2 – 2x – 4y + 4 = 0 about the line x – y = 3 is (A) x2 + y2 – 10x + 4y + 28 = 0 (B) x2 + y2 + 6x + 8 = 0 (C) x2 + y2 – 14x – 2y + 49 = 0 (D) x2 + y2 + 8x + 2y + 16 = 0 Q.105 Which one of the following statement is True ? (A) The lines 2x + 3y + 19 = 0 and 9x + 6y − 17 = 0 cut the coordinate axes in concyclic points. (B) The circumcentre, orthocentre, incentre and centroid of the triangle formed by the points A(1, 2) , B(4, 6) , C(− 2, − 1) are colinear . (C) The mid point of the sides of a triangle are (1, 2) , (3, 1) & (5, 5) . The orthocentre of the triangle has the co−ordinates (3, 1) . (D) Equation of the line pair through the origin and perpendicular to the line pair x y − 3 y2 + y − 2 x + 10 = 0 is 3 y2 + x y = 0 Q.106 The locus of the centre of a circle which touches externally the circle , x² + y² − 6x − 6y + 14 = 0 & also touches the y-axis is given by the equation : (A) x² − 6x − 10y + 14 = 0 (B) x² − 10x − 6y + 14 = 0 (C) y² − 6x − 10y + 14 = 0 (D) y² − 10x − 6y + 14 = 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]

Q.107 The co-ordinates of the vertices P, Q, R & S of square PQRS inscribed in the triangle ABC with vertices A ≡ (0, 0) , B ≡ (3, 0) & C ≡ (2, 1) given that two of its vertices P, Q are on the side AB are respectively 1 4

 3  8

  3 1   8 8

 1 1  4 8

1 3 3 1 1 1 (B)  , 0 ,  , 0 ,  ,  &  , 

1 2

(D)  , 0 ,  , 0 ,  ,  &  , 

(A)  , 0 ,  , 0 ,  ,  &  ,  3 2

  3 1   3 2

 

(C) (1, 0) ,  , 0 ,  ,  &  1 , 

2

 4

  4 4

 2 4

3 2

 9  4

  9 3   4 4

 3 3  2 4

Q.108 The equation of the locus of the mid points of the chords of the circle 2π 4x2 + 4y2 − 12x + 4y + 1 = 0 that subtend an angle of at its centre is 3 (A) 16(x² + y²) − 48x + 16y + 31 = 0 (B) 16(x² + y²) − 48x – 16y + 31 = 0 (C) 16(x² + y²) + 48x + 16y + 31 = 0 (D) 16(x² + y²) + 48x – 16y + 31 = 0 Q.109 The line 2x + 3y = 12 meets the x - axis at A and the y - axis at B . The line through (5, 5) perpendicular to AB meets the x - axis, y - axis & the line AB at C, D, E respectively. If O is the origin, then the area of the OCEB is : (A)

Quest

20 sq. units 3

(B)

23 sq. units 3

(C)

26 sq. units 3

(D)

5 52 sq. units 9

Q.110 In the xy plane, the segment with end points (3, 8) and (–5, 2) is the diameter of the circle. The point (k, 10) lies on the circle for (A) no value of k (B) exactly one integral k (C) exacly one non integral k (D) two real values of k Q.111 Let A ≡ (3, 2) and B ≡ (5, 1). ABP is an equilateral triangle is constructed on the side of AB remote from the origin then the orthocentre of triangle ABP is

1 3   3, − 3 (A)  4 −   2 2

1 3   3, + 3 (B)  4 +   2 2

1 3 1   3, − 3 (C)  4 −  6 2 3 

1 3 1   3, + 3 (D)  4 +  6 2 3 

Q.112 The vertex of a right angle of a right angled triangle lies on the straight line 2x + y – 10 = 0 and the two other vertices, at points (2, –3) and (4, 1) then the area of triangle in sq. units is (A) 10

(B) 3

(C)

33 5

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 11

[13]

Select the correct alternatives : (More than one are correct) Q.113 Let u ≡ ax + by + a 3 b = 0 v ≡ bx − ay + b 3 a = 0 a, b ∈ R be two straight lines. The equation of the bisectors of the angle formed by k1u − k2v = 0 & k1u + k2v = 0 for non zero real k1 & k2 are: (A) u = 0 (B) k2u + k1v = 0 (C) k2u − k1v = 0 (D) v = 0 Q.114 A tangent drawn from the point (4, 0) to the circle x2 + y2 = 8 touches it at a point A in the first quadrant. The co−ordinates of another point B on the circle such that l (AB) = 4 are : (A) (2, − 2)

(

(C) − 2 2 , 0

(B) (− 2, 2)

)

(

(D) 0 , − 2 2

)

Q.115 Consider the equation y − y1 = m (x − x1) . If m & x1 are fixed and different lines are drawn for different values of y1, then : (A) the lines will pass through a fixed point (B) there will be a set of parallel lines (C) all the lines intersect the line x = x1 (D) all the lines will be parallel to the line y = x1. Q.116 A circle passes through the points (− 1, 1) , (0, 6) and (5, 5) . The point(s) on this circle, the tangent(s) at which is/are parallel to the straight line joining the origin to its centre is/are : (A) (1, − 5) (B) (5, 1) (C) (− 5, − 1) (D) (− 1, 5)

Quest

Q.117 If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x − 3 y = 0 then the co-ordinates of the third vertex are :  3a

a

(B)  2 , − 2   

(A) (0, a)



3a a

(D)  − 2 , 2   

(C) (0, − a)

Q.118 Equation of a line through (7, 4) and touching the circle, x2 + y2 − 6x + 4y − 3 = 0 is : (A) 5x − 12y + 13 = 0 (B) 12x − 5y − 64 = 0 (C) x − 7 = 0 (D) y = 4

Q.119 Three vertices of a triangle are A(4, 3) ; B(1, − 1) and C(7, k) . Value(s) of k for which centroid, orthocentre, incentre and circumcentre of the ∆ ABC lie on the same straight line is/are : (A) 7 (B) − 1 (C) − 19/8 (D) none Q.120 Point M moved along the circle (x − 4)2 + (y − 8)2 = 20 . Then it broke away from it and moving along a tangent to the circle, cuts the x−axis at the point (− 2, 0) . The co−ordinates of the point on the circle at which the moving point broke away can be : (A)  − 3 , 46   5

5

 2 44    5 5

(B)  − ,

(C) (6, 4)

(D) (3, 5)

Q.121 Straight lines 2x + y = 5 and x − 2y = 3 intersect at the point A . Points B and C are chosen on these two lines such that AB = AC . Then the equation of a line BC passing through the point (2, 3) is (A) 3x − y − 3 = 0 (B) x + 3y − 11 = 0 (C) 3x + y − 9 = 0 (D) x − 3y + 7 = 0 Q.122 The centre(s) of the circle(s) passing through the points (0, 0) , (1, 0) and touching the circle x2 + y2 = 9 is/are :  3 1 2 2

(A)  , 

 1 3 2 2

(B)  , 

1



1



1/ 2 1/ 2 (C)  , 2  (D)  , − 2  2 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[14]

Q.123 The x − co-ordinates of the vertices of a square of unit area are the roots of the equation x2 − 3x + 2 = 0 and the y − co-ordinates of the vertices are the roots of the equation y2 − 3y + 2 = 0 then the possible vertices of the square is/are : (A) (1, 1), (2, 1), (2, 2), (1, 2) (B) (− 1, 1), (− 2, 1), (− 2, 2), (− 1, 2) (C) (2, 1), (1, − 1), (1, 2), (2, 2) (D) (− 2, 1), (− 1, − 1), (− 1, 2), (− 2, 2) 

Q.124 A circle passes through the point  3 , 

7  and touches the line pair x2 − y2 − 2x + 1 = 0. The 2 

co-ordinates of the centre of the circle are : (A) (4, 0) (B) (5, 0)

(C) (6, 0)

(D) (0, 4)

Q.125 P (x, y) moves such that the area of the triangle formed by P, Q (a , 2 a) and R (− a, − 2 a) is equal to the area of the triangle formed by P, S (a, 2 a) & T (2 a, 3 a). The locus of 'P' is a straight line given by : (A) 3x − y = a (B) 5x − 3y + a = 0 (C) y = 2ax (D) 2y = ax

Quest Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

[16] Q.1 Q.6 Q.11 Q.16 Q.21 Q.26 Q.31 Q.36 Q.41 Q.46 Q.51 Q.56 Q.61 Q.66 Q.71 Q.76 Q.81 Q.86

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

B

Q.85

C

Q.80

D

Q.75

C

Q.70

A

Q.65

A

Q.60

B

A

Q.55

B

Q.54

C

Q.50

D

Q.49

Q.48 D

D

Q.45

D

Q.44

Q.43 D

B

A

Q.40

C

Q.39

Q.38 B

A

Q.37

B

Q.35

A

Q.34

Q.33 A

B

Q.32

C

C

Q.30

D

Q.29

Q.28 A

C

Q.27

C

A

Q.25

D

Q.24

Q.23 A

A

Q.22

D

A

Q.20

A

Q.19

Q.18 D

D

Q.17

B

A

Q.15

C

Q.14

Q.13 C

C

Q.12

A

A

Q.10

B

Q.9

C

Q.8

C

Q.7

B

A

Q.5

B

Q.4

A

Q.3

C

Q.2

D

A

Q.84

Q.83 C

D

Q.82

B

B

Q.79

Q.78 C

D

Q.77

D

B

Q.74

Q.73 C

C

Q.72

B

D

Q.69

Q.68 A

D

Q.67

C

C

Q.64

Q.63 D

C

Q.62

A

Q.59

Q.58 A

C

Q.57

B

Q.53 A

C

Q.52

C

D

Q.47

C

Q.42

B

B

Quest Q.124 A, C

Q.123 A, B

Q.122 C, D

Q.121 A, B

Q.119 B, C

Q.118 A, C

Q.117 A, B, C, D

Q.116 B, D

Q.114 A, B

Q.113 A, D

Q.112 B

Q.111 D

Q.109 B

Q.108 A

Q.107 D

Q.106 D

Q.104 A

Q.103 B

Q.102 D

Q.101 D

D

Q.99

Q.98 A

A

Q.97

C

C

Q.94

Q.93 D

B

Q.92

A

D

Q.89

Q.88 B

D

Q.87

C

Q.91 Q.96

A

Q.95

D

Q.90

Q.100 B Q.105 A Q.110 B Q.115 B, C Q.120 B, C Q.125 A, B

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

CONIC SECTION (Parabola, Ellipse & Hyperbola)

Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q.1

Two mutually perpendicular tangents of the parabola y2 = 4ax meet the axis in P1 and P2. If S is the focus of the parabola then

(A) Q.2

4 a

1 1 + is equal to l (SP1 ) l (SP2 ) (B)

2 a

(C)

Q.4

x = tan t ;

y = sec t

(B) x2 − 2 = − 2 cos t ; y = 4 cos2 (D) x = 1 − sin t ; y = sin

t 2

t t + cos 2 2

x2 y2 + = 1 & 'C' be the circle x2 + y2 = 9. Let P & Q be the points (1 , 2) and Let 'E' be the ellipse 9 4

(B) Q lies outside both C & E (D) P lies inside C but outside E.

Let S be the focus of y2 = 4x and a point P is moving on the curve such that it's abscissa is increasing at the rate of 4 units/sec, then the rate of increase of projection of SP on x + y = 1 when P is at (4, 4) is

2

(B) – 1

(C) –

2

(D) –

3 2

x 2 y2 Eccentricity of the hyperbola conjugate to the hyperbola − = 1 is 4 12 (A)

Q.7

1 4a

Quest

(A)

Q.6

(D)

x 2 y2 The magnitude of the gradient of the tangent at an extremity of latera recta of the hyperbola 2 − 2 = 1 a b is equal to (where e is the eccentricity of the hyperbola) (A) be (B) e (C) ab (D) ae

(2, 1) respectively. Then : (A) Q lies inside C but outside E (C) P lies inside both C & E Q.5

1 a

Which one of the following equations represented parametrically, represents equation to a parabolic profile ? (A) x = 3 cos t ; y = 4 sin t

Q.3

(C)

2 3

(B) 2

(C)

3

(D)

4 3

The points of contact Q and R of tangent from the point P (2, 3) on the parabola y2 = 4x are (A) (9, 6) and (1, 2)

(B) (1, 2) and (4, 4)

(C) (4, 4) and (9, 6)

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) (9, 6) and (

1 , 1) 4

[2]

Q.8

The eccentricity of the ellipse (x – 3)2 + (y – 4)2 = (A)

Q.9

3 2

(B)

1 3

y2 is 9

(C)

1

(D)

3 2

1 3

x2 y2 The asymptote of the hyperbola 2 − 2 = 1 form with any tangent to the hyperbola a triangle whose a b

area is a2tan λ in magnitude then its eccentricity is : (A) secλ (B) cosecλ (C) sec2λ

(D) cosec2λ

Q.10

A tangent is drawn to the parabola y2 = 4x at the point 'P' whose abscissa lies in the interval [1,4]. The maximum possible area of the triangle formed by the tangent at 'P' , ordinate of the point 'P' and the x-axis is equal to (A) 8 (B) 16 (C) 24 (D) 32

Q.11

From an external point P, pair of tangent lines are drawn to the parabola, y2 = 4x. If θ1 & θ2 are the

(A) x − y + 1 = 0 Q.12

Q.13

Q.14

(B) x + y − 1 = 0

(C) x − y − 1 = 0

(D) x + y + 1 = 0

x2 y2 + = 1 (p ≠ 4, 29) represents 29 − p 4 − p (A) an ellipse if p is any constant greater than 4. (B) a hyperbola if p is any constant between 4 and 29. (C) a rectangular hyperbola if p is any constant greater than 29. (D) no real curve if p is less than 29. The equation

x 2 y2 + = 1 with vertices A and A', tangent drawn at the point P in the first quadrant meets 9 4 the y-axis in Q and the chord A'P meets the y-axis in M. If 'O' is the origin then OQ2 – MQ2 equals to (A) 9 (B) 13 (C) 4 (D) 5 For an ellipse

Length of the normal chord of the parabola, y2 = 4x, which makes an angle of (A) 8

Q.15

π , then the locus of P is : 4

Quest

inclinations of these tangents with the axis of x such that, θ1 + θ2 =

(B) 8 2

(C) 4

π with the axis of x is: 4

(D) 4 2

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If e1, e2 be their eccentricities respectively, then e1−2 + e 2−2 equals (A) 1

Q.16

(B) 2

(C) 3

(D) 4

The coordiantes of the ends of a focal chord of a parabola y2 = 4ax are (x1, y1) and (x2, y2) then x1x2 + y1y2 has the value equal to (A) 2a2 (B) – 3a2 (C) – a2 (D) 4a2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[3]

Q.17

The line, lx + my + n = 0 will cut the ellipse π/2 if : (A) a2l2 + b2n2 = 2 m2 (C) a2l2 + b2m2 = 2 n2

x2 y2 + = 1 in points whose eccentric angles differ by a 2 b2

(B) a2m2 + b2l2 = 2 n2 (D) a2n2 + b2m2 = 2 l2

Q.18

Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola 16y2 – 9x2 = 1 is (A) x2 + y2 = 9 (B) x2 + y2 = 1/9 (C) x2 + y2 =7/144 (D) x2 + y2 = 1/16

Q.19

If the normal to a parabola y2 = 4ax at P meets the curve again in Q and if PQ and the normal at Q makes angles α and β respectively with the x-axis then tan α (tan α + tan β) has the value equal to (A) 0

Q.20

(B) – 2

Quest (B) 2 < t 22 < 4

(C) t 22 > 4

(D) t 22 > 8

The locus of the point of instruction of the lines 3 x − y − 4 3 t = 0 & 3 tx + ty − 4 3 = 0 (where t is a parameter) is a hyperbola whose eccentricity is (A)

Q.22

(D) – 1

If the normal to the parabola y2 = 4ax at the point with parameter t1 , cuts the parabola again at the point with parameter t2 , then (A) 2 < t 22 < 8

Q.21

1 2

(C) –

3

(B) 2

(C)

2 3

(D)

4 3

The equation to the locus of the middle point of the portion of the tangent to the ellipse

x2 y2 + =1 16 9

included between the co-ordinate axes is the curve : (A) 9x2 + 16y2 = 4 x2y2 (B) 16x2 + 9y2 = 4 x2y2 2 2 2 2 (C) 3x + 4y = 4 x y (D) 9x2 + 16y2 = x2y2 Q.23

A parabola y = ax2 + bx + c crosses the x − axis at (α , 0) (β , 0) both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is : (A)

bc a

(B) ac2

(C)

b a

(D)

c a

Q.24

Two parabolas have the same focus. If their directrices are the x − axis & the y − axis respectively, then the slope of their common chord is : (A) ± 1 (B) 4/3 (C) 3/4 (D) none

Q.25

The locus of a point in the Argand plane that moves satisfying the equation, z − 1 + i − z − 2 − i = 3 (A) is a circle with radius 3 & centre at z = 3/2 (B) is an ellipse with its foci at 1 − i and 2 + i and major axis = 3 (C) is a hyperbola with its foci at 1 − i and 2 + i and its transverse axis = 3 (D) is none of the above .

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.26

A circle has the same centre as an ellipse & passes through the foci F1 & F2 of the ellipse, such that the two curves intersect in 4 points. Let 'P' be any one of their point of intersection. If the major axis of the ellipse is 17 & the area of the triangle PF1F2 is 30, then the distance between the foci is : (A) 11 (B) 12 (C) 13 (D) none

Q.27

The straight line joining any point P on the parabola y2 = 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equaiton of the locus of R is (A) x2 + 2y2 – ax = 0 (B) 2x2 + y2 – 2ax = 0 2 2 (C) 2x + 2y – ay = 0 (D) 2x2 + y2 – 2ay = 0

Q.28

A normal chord of the parabola y2 = 4x subtending a right angle at the vertex makes an acute angle θ with the x-axis, then θ equals to (A) arc tan 2

Q.29

(B) arc sec 3

(C) arc cot 2

If the eccentricity of the hyperbola x2 − y2 sec2 α = 5 is x2 sec2 α + y2 = 25, then a value of α is : (A) π/6 (B) π/4 (C) π/3

(D) none

3 times the eccentricity of the ellipse

(D) π/2

Quest

Q.30

Point 'O' is the centre of the ellipse with major axis AB & minor axis CD. Point F is one focus of the ellipse. If OF = 6 & the diameter of the inscribed circle of triangle OCF is 2, then the product (AB) (CD) is equal to (A) 65 (B) 52 (C) 78 (D) none

Q.31

Locus of the feet of the perpendiculars drawn from vertex of the parabola y2 = 4ax upon all such chords of the parabola which subtend a right angle at the vertex is (B) x2 + y2 – 2ax = 0 (A) x2 + y2 – 4ax = 0 2 2 (C) x + y + 2ax = 0 (D) x2 + y2 + 4ax = 0

Q.32

For all real values of m, the straight line y = mx + 9 m2 − 4 is a tangent to the curve : (C) 9x2 − 4y2 = 36 (D) 4x2 − 9y2 = 36 (A) 9x2 + 4y2 = 36 (B) 4x2 + 9y2 = 36

Q.33

C is the centre of the circle with centre (0, 1) and radius unity. P is the parabola y = ax2. The set of values of 'a' for which they meet at a point other than the origin, is (A) a > 0

Q.34

 1 (B) a ∈  0,   2

A tangent having slope of −

1 1 (C)  ,  4 2

1  (D)  , ∞  2 

4 x2 y2 to the ellipse + = 1 intersects the major & minor axes in points A 3 18 32

& B respectively. If C is the centre of the ellipse then the area of the triangle ABC is : (A) 12 sq. units (B) 24 sq. units (C) 36 sq. units (D) 48 sq. units

x 2 y2 x 2 y2 1 + 2 = 1 and the hyperbola − = Q.35 The foci of the ellipse coincide. Then the value of b2 is 16 b 144 81 25 (A) 5 (B) 7 (C) 9 (D) 4

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.36

TP & TQ are tangents to the parabola, y2 = 4ax at P & Q. If the chord PQ passes through the fixed point (− a, b) then the locus of T is : (A) ay = 2b (x − b) (B) bx = 2a (y − a) (C) by = 2a (x − a) (D) ax = 2b (y − b)

Q.37

Through the vertex O of the parabola, y2 = 4ax two chords OP & OQ are drawn and the circles on OP & OQ as diameters intersect in R. If θ1, θ2 & φ are the angles made with the axis by the tangents at P & Q on the parabola & by OR then the value of, cot θ1 + cot θ2 = (A) − 2 tan φ (B) − 2 tan (π − φ) (C) 0 (D) 2 cot φ

Q.38

Locus of the middle points of the parallel chords with gradient m of the rectangular hyperbola xy = c2 is (A) y + mx = 0 (B) y − mx = 0 (C) my − x = 0 (D) my + x = 0

Q.39

If the chord through the point whose eccentric angles are θ & φ on the ellipse, (x2/a2) + (y2/b2) = 1 passes through the focus, then the value of (1 + e) tan(θ/2) tan(φ/2) is (A) e + 1 (B) e − 1 (C) 1 − e (D) 0

Q.40

The given circle x2 + y2 + 2px = 0, p ∈ R touches the parabola y2 = 4x externally, then (A) p < 0 (B) p > 0 (C) 0 < p < 1 (D) p < – 1

Q.41

The locus of the foot of the perpendicular from the centre of the hyperbola xy = c2 on a variable tangent is : (A) (x2 − y2)2 = 4c2 xy (B) (x2 + y2)2 = 2c2 xy 2 2 2 (C) (x + y ) = 4x xy (D) (x2 + y2)2 = 4c2 xy

Q.42

The tangent at P to a parabola y2 = 4ax meets the directrix at U and the latus rectum at V then SUV (where S is the focus) : (A) must be a right triangle (B) must be an equilateral triangle (C) must be an isosceles triangle (D) must be a right isosceles triangle.

Q.43

Given the base of a triangle and sum of its sides then the locus of the centre of its incircle is (A) straight line (B) circle (C) ellipse (D) hyperbola

Q.44

P is a point on the hyperbola

Quest

x2 y2 − = 1, N is the foot of the perpendicular from P on the transverse a 2 b2

axis. The tangent to the hyperbola at P meets the transverse axis at T . If O is the centre of the hyperbola, the OT. ON is equal to : (A) e2 (B) a2 (C) b2 (D)b2/a2 Q.45

Two parabolas y2 = 4a(x - l1) and x2 = 4a (y – l2) always touch one another, the quantities l1 and l2 are both variable. Locus of their point of contact has the equation (A) xy = a2 (B) xy = 2a2 (C) xy = 4a2 (D) none

Q.46

If a normal to a parabola y2 = 4ax make an angle φ with its axis, then it will cut the curve again at an angle (A) tan–1(2 tanφ)

1 2

 

(B) tan−1  tan φ

1 2

 

(C) cot–1  tan φ

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[6]

Q.47

If PN is the perpendicular from a point on a rectangular hyperbola x2 − y2 = a2 on any of its asymptotes, then the locus of the mid point of PN is : (A) a circle (B) a parabola (C) an ellipse (D) a hyperbola

Q.48

Which one of the following is the common tangent to the ellipses,

x2 y2 x2 y2 + + = 1 & =1? a 2 + b2 b2 a 2 a 2 + b2

(A) ay = bx + a 4 − a 2 b 2 + b 4

(B) by = ax − a 4 + a 2 b 2 + b 4

(C) ay = bx −

(D) by = ax + a 4 − a 2 b 2 + b 4

a 4 + a 2 b2 + b 4

Q.49

The vertex of a parabola is (2,2) and the co-ordinates of its two extrimities of the latus rectum are (–2,0) and (6,0). The equation of the parabola is (A) y2 – 4y + 8x – 12 = 0 (B) x2 + 4x – 8y – 12 = 0 (C) x2 – 4x + 8y – 12 = 0 (D) x2 – 8y – 4x + 20 = 0

Q.50

The equation to the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyperbola xy = c2 is

Q.51

Q.52

Quest

(A)

y x + =1 y + y2 x1 + x 2 1

(B)

y x + =1 y − y2 x1 − x 2 1

(C)

y x + =1 x1 + x 2 y1 + y 2

(D)

y x + =1 x1 − x 2 y1 − y 2

The length of the chord of the parabola y2 = x which is bisected at the point (2, 1) is (A) 2 3

(B) 4 3

(C) 3 2

(D) 2 5

The normal at a variable point P on an ellipse

x2 y2 + = 1 of eccentricity e meets the axes of the ellipse a 2 b2

in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e ′ such that : (A) e ′ is independent of e (B) e ′ = 1 (C) e ′ = e (D) e ′ = 1/e Q.53

If the tangents & normals at the extremities of a focal chord of a parabola intersect at (x1, y1) and (x2, y2) respectively, then : (A) x1 = x2 (B) x1 = y2 (C) y1 = y2 (D) x2 = y1

Q.54

If P(x1, y1), Q(x2, y2), R(x3, y3) & S(x4, y4) are 4 concyclic points on the rectangular hyperbola x y = c2, the co-ordinates of the orthocentre of the triangle PQR are : (A) (x4, − y4) (B) (x4, y4) (C) (− x4, − y4) (D) (− x4, y4)

Q.55

If the chord of contact of tangents from a point P to the parabola y2 = 4ax touches the parabola x2 = 4by, the locus of P is : (A) circle (B) parabola (C) ellipse (D) hyperbola

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[7]

Q.56

An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is (A)

(B) 2

3

(D)

(C) 2 2

5

Q.57

The latus rectum of a parabola whose focal chord PSQ is such that SP = 3 and SQ = 2 is given by (A) 24/5 (B) 12/5 (C) 6/5 (D) none of these

Q.58

The chord PQ of the rectangular hyperbola xy = a2 meets the axis of x at A ; C is the mid point of PQ & 'O' is the origin. Then the ∆ ACO is : (A) equilateral (B) isosceles (C) right angled (D) right isosceles.

Q.59

The circle x2 + y2 = 5 meets the parabola y2 = 4x at P & Q. Then the length PQ is equal to (A) 2

(B) 2 2

(C) 4

(D) none

Q.60

A common tangent to 9x2 + 16y2 = 144 ; y2 − x + 4 = 0 & x2 + y2 − 12x + 32 = 0 is (A) y = 3 (B) x = − 4 (C) x = 4 (D) y = − 3

Q.61

A conic passes through the point (2, 4) and is such that the segment of any of its tangents at any point contained between the co-ordinate axes is bisected at the point of tangency. Then the foci of the conic are

(

Quest ) (

(A) 2 2 , 0 & − 2 2 , 0

( (D) ( 4

)

) ( 2) & (− 4

) 2)

(B) 2 2 , 2 2 & − 2 2 , − 2 2

(C) (4, 4) & (− 4, − 4)

2,4

2 , −4

Q.62

If two normals to a parabola y2 = 4ax intersect at right angles then the chord joining their feet passes through a fixed point whose co-ordinates are (B) (a, 0) (C) (2a, 0) (D) none (A) (− 2a, 0)

Q.63

The equation of a straight line passing through the point (3, 6) and cutting the curve y = x orthogonally is (A) 4x + y – 18 =0 (B) x + y – 9 = 0 (C) 4x – y – 6 = 0 (D) none

Q.64

Latus rectum of the conic satisfying the differential equation, x dy + y dx = 0 and passing through the point (2, 8) is (A) 4 2

Q.65

(B) 8

(C) 8 2

(D) 16

The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is π/4 is

(a (A)

2

)

− b 2 ab

a 2 + b2

(a

)

− b2 (B) 2 a + b 2 ab

(

2

)

(a − b ) (C) ab (a + b ) 2

2

2

2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

(a

a 2 + b2 2

)

− b 2 ab

[8]

Q.66

PQ is a normal chord of the parabola y2 = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x−axis in R. Then the length of AR is : (A) equal to the length of the latus rectum (B) equal to the focal distance of the point P (C) equal to twice the focal distance of the point P (D) equal to the distance of the point P from the directrix.

Q.67

If the normal to the rectangular hyperbola xy = c2 at the point 't' meets the curve again at 't1' then t3 t1 has the value equal to (A) 1 (B) – 1 (C) 0 (D) none

Q.68

Locus of the point of intersection of the perpendicular tangents of the curve y2 + 4y − 6x − 2 = 0 is : (A) 2x − 1 = 0 (B) 2x + 3 = 0 (C) 2y + 3 = 0 (D) 2x + 5 = 0

Q.69

If tan θ1. tan θ2 = −

a2 x2 y2 then the chord joining two points θ1 & θ2 on the ellipse 2 + 2 = 1 will subtend b2 a b

a right angle at : (A) focus (C) end of the major axis

Q.70

Quest

x2 y2 − = 1 as the centre , a circle is drawn which is tangent to the 9 16 hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is With one focus of the hyperbola

(A) less than 2 Q.71

(B) 2

(C)

11 3

(D) none

Length of the focal chord of the parabola y2 = 4ax at a distance p from the vertex is : (A)

Q.72

(B) centre (D) end of the minor axis

2a2 p

(B)

a3 p2

(C)

4a3 p2

(D)

p2 a

The locus of a point such that two tangents drawn from it to the parabola y2 = 4ax are such that the slope of one is double the other is : (A) y2 =

9 ax 2

(B) y2 =

9 ax 4

(C) y2 = 9 ax

x2

y2

Q.73

= 1 such that ∆AOB (where 'O' is the origin) is an a 2 b2 equilateral triangle, then the eccentricity e of the hyperbola satisfies 2 2 2 (A) e > 3 (B) 1 < e < (C) e = (D) e > 3 3 3

Q.74

An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is 2/3 then the eccentricity of the ellipse is :

AB is a double ordinate of the hyperbola

(A)

2 2 3

(B)

5 3

−

(D) x2 = 4 ay

(C)

8 9

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

2 3

[9]

Q.75

The triangle PQR of area 'A' is inscribed in the parabola y2 = 4ax such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is : (A)

Q.76

A 2a

(B)

A a

(C)

2A a

(D)

If the product of the perpendicular distances from any point on the hyperbola

4A a

x2

−

a2

y2 b2

=1 of eccentricity

e = 3 from its asymptotes is equal to 6, then the length of the transverse axis of the hyperbola is (A) 3 (B) 6 (C) 8 (D) 12 Q.77

The point(s) on the parabola y2 = 4x which are closest to the circle, x2 + y2 − 24y + 128 = 0 is/are : (A) (0, 0)

(

(B) 2 , 2 2

)

(C) (4, 4)

(D) none

Q.78

A point P moves such that the sum of the angles which the three normals makes with the axis drawn from P on the standard parabola, is constant. Then the locus of P is : (A) a straight line (B) a circle (C) a parabola (D) a line pair

Q.79

If x + iy = φ + iψ where i = − 1 and φ and ψ are non zero real parameters then φ = constant and ψ = constant, represents two systems of rectangular hyperbola which intersect at an angle of π π π π (A) (B) (C) (D) 3 6 2 4 Three normals drawn from any point to the parabola y2 = 4ax cut the line x = 2a in points whose ordinates are in arithmetical progression. Then the tangents of the angles which the normals make the axis of the parabola are in : (A) A.P. (B) G.P. (C) H.P. (D) none

Q.80

Q.81

Quest

A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola y2 = 4ax. If PQ is the common chord of the circle and the parabola and L1 L2 is the latus rectum, then the area of the trapezium PL1 L2Q is : (A) 3 2 a2

Q.82

 2 +1 2 (B)  2  a  

(C) 4a2

 2 + 2  a2  2 

(D) 

The tangent to the hyperbola xy = c2 at the point P intersects the x-axis at T and the y-axis at T′. The normal to the hyperbola at P intersects the x-axis at N and the y-axis at N′. The areas of the triangles 1 1 + is ∆ ∆' (C) depends on c

PNT and PN'T' are ∆ and ∆' respectively, then (A) equal to 1 Q.83

(B) depends on t

(D) equal to 2

If y = 2 x − 3 is a tangent to the parabola y2 = 4a  x − 1  , then ' a ' is equal to : 

(A)

22 3

(B) − 1

(C)

3

14 3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

− 14 3

[10]

Q.84

Q.85

An ellipse having foci at (3, 3) and (– 4, 4) and passing through the origin has eccentricity equal to 3 5 3 2 (A) (B) (C) (D) 7 5 7 7 and the hyperbola 4x2 – y2 = 4 have the same foci and they intersect at right angles then the equation of the circle through the points of intersection of two conics is

T

h

e

e l l i p

s e

4

x

2 + 9y2 = 36

(A) x2 + y2 = 5 5 (x2 + y2) + 3x + 4y = 0

(C) Q.86

5 (x2 + y2) – 3x – 4y = 0

(B)

(D) x2 + y2 = 25

Tangents are drawn from the point (− 1, 2) on the parabola y2 = 4 x. The length , these tangents will intercept on the line x = 2 is : (A) 6

(B) 6 2

(C) 2 6

(D) none of these

Q.87

The curve describes parametrically by x = t2 – 2t + 2, y = t2 + 2t + 2 represents (A) straight line (B) pair of straight lines (C) circle (D) parabola

Q.88

At the point of intersection of the rectangular hyperbola xy = c2 and the parabola y2 = 4ax tangents to the rectangular hyperbola and the parabola make an angle θ and φ respectively with the axis of X, then (B) φ = tan–1(– 2 tanθ ) (A) θ = tan–1(– 2 tanφ )

Quest

(C) θ = Q.89

Q.90

1 tan–1(– tanφ ) 2

1 tan–1(– tanθ ) 2

The tangent and normal at P(t), for all real positive t, to the parabola y2 = 4ax meet the axis of the parabola in T and G respectively, then the angle at which the tangent at P to the parabola is inclined to the tangent at P to the circle passing through the points P, T and G is (A) cot–1t (B) cot–1t2 (C) tan–1t (D) tan–1t2 Area of the quadrilateral formed with the foci of the hyperbola (A) 4(a2 + b2)

Q.91

(D) φ =

(B) 2(a2 + b2)

x2 a2

−

y2 b2

(C) (a2 + b2)

= 1 and (D)

x2 a2

−

y2 b2

= −1 is

1 2 (a + b2) 2

A bar of length 20 units moves with its ends on two fixed straight lines at right angles. A point P marked on the bar at a distance of 8 units from one end describes a conic whose eccentricity is (A)

5 9

(B)

2 3

(C)

4 9

(D)

5 3

Q.92

In a square matrix A of order 3, ai i = mi + i where i = 1, 2, 3 and mi's are the slopes (in increasing order of their absolute value) of the 3 normals concurrent at the point (9, – 6) to the parabola y2 = 4x. Rest all other entries of the matrix are one. The value of det. (A) is equal to (A) 37 (B) – 6 (C) – 4 (D) – 9

Q.93

An equation for the line that passes through (10, –1) and is perpendicular to y = (A) 4x + y = 39

(B) 2x + y = 19

(C) x + y = 9

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

x2 − 2 is 4 (D) x + 2y = 8

[11]

Direction for Q.94 to Q.97. (4 questions together) A quadratic polynomial y = f (x) with absolute term 3 neither touches nor intersects the abscissa axis and is symmetric about the line x = 1. The coefficient of the leading term of the polynomial is unity. A point A(x1, y1) with abscissa x1 = 1 and a point B(x2, y2) with ordinate y2 = 11 are given in a cartisian rectangular system of co-ordinates OXY in the first quadrant on the curve y = f (x) where 'O' is the origin. Now answer the following questions: Q.94

Vertex of the quadratic polynomial is (A) (1, 1) (B) (2, 3)

(C) (1, 2)

→

Q.95 Q.96 Q.97

(D) none

→

The scalar product of the vectors OA and OB is (A) –18 (B) 26 (C) 22

(D) –22

The area bounded by the curve y = f(x) and a line y = 3 is (A) 4/3 (B) 5/3 (C) 7/3

(D) 28/3

The graph of y = f(x) represents a parabola whose focus has the co-ordinates (A) (1, 7/4) (B) (1, 5/4) (C) (1, 5/2) (D) (1, 9/4)

Direction for Q.98 to Q.66. (3 questions together) The graph of the conic x2 – (y – 1)2 = 1 has one tangent line with positive slope that passes through the origin. the point of tangency being (a, b). Then Q.98

5π 12

(A) Q.99

Quest

a The value of sin–1   is b

π 6

(B)

(C)

π 3

(D)

π 4

Length of the latus rectum of the conic is (A) 1

(B)

2

(C) 2

(D) none

(B) 3

(C) 2

(D) none

Q.100 Eccentricity of the conic is 4 3

(A)

Select the correct alternatives : (More than one are correct) Q.101 Consider a circle with its centre lying on the focus of the parabola, y2 = 2 px such that it touches the directrix of the parabola. Then a point of intersection of the circle & the parabola is : p 2

p 2

 

 

 p  , p  2 

(B)  , − p

(A)  , p

 p  2

(C)  −

 

(D)  − , − p

Q.102 Identify the statements which are True. (A) the equation of the director circle of the ellipse, 5x2 + 9y2 = 45 is x2 + y2 = 14. (B) the sum of the focal distances of the point (0 , 6) on the ellipse

x2 y2 + = 1 is 10. 25 36

(C) the point of intersection of any tangent to a parabola & the perpendicular to it from the focus lies on the tangent at the vertex. ( D

)

P

&

Q

a r e

t h e

p

o

i n t s

w

i t h

e c c e n t r i c

a n g

l e s

θ & θ + α on the ellipse

x2 y2 + = 1, then the area of the a 2 b2

triangle OPQ is independent of θ. Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]

Q.103 For the hyperbola

x2 y2 − = 1 the incorrect statement is : 9 3

(A) the acute angle between its asymptotes is 60º (B) its eccentricity is 4/3 (C) length of the latus rectum is 2 (D) product of the perpendicular distances from any point on the hyperbola on its asymptotes is less than the length of its latus rectum . Q.104 The locus of the mid point of the focal radii of a variable point moving on the parabola, y2 = 4ax is a parabola whose (A) Latus rectum is half the latus rectum of the original parabola (B) Vertex is (a/2, 0) (C) Directrix is y-axis (D) Focus has the co-ordinates (a, 0) Q.105 P is a point on the parabola y2 = 4ax (a > 0) whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a diameter then it intersects the x−axis at a point whose co−ordinates are : (B) (− a, 0) (C) (− 2a, 0) (D) (a, 0) (A) (− 3a, 0)

Quest

Q.106 If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then (A) x1 + x2 + x3 + x4 = 0 (B) y1 + y2 + y3 + y4 = 0 (C) x1 x2 x3 x4 = c4 (D) y1 y2 y3 y4 = c4

Q.107 Extremities of the latera recta of the ellipses (A) x2 = a(a – y)

(B) x2 = a (a + y)

x2 a2

+

y2

=1 (a > b) having a given major axis 2a lies on b2 (C) y2 = a(a + x) (D) y2 = a (a – x)

Q.108 Let y2 = 4ax be a parabola and x2 + y2 + 2 bx = 0 be a circle. If parabola and circle touch each other externally then : (A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0 Q.109 The tangent to the hyperbola, x2 − 3y2 = 3 at the point

(

)

3 , 0 when associated with two asymptotes

constitutes : (A) isosceles triangle

(B) an equilateral triangle

(C) a triangles whose area is 3 sq. units

(D) a right isosceles triangle .

Q.110 Let P, Q and R are three co-normal points on the parabola y2 = 4ax. Then the correct statement(s) is/are (A) algebraic sum of the slopes of the normals at P, Q and R vanishes (B) algebraic sum of the ordinates of the points P, Q and R vanishes (C) centroid of the triangle PQR lies on the axis of the parabola (D) circle circumscribing the triangle PQR passes through the vertex of the parabola

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Q.111 A variable circle is described to pass through the point (1, 0) and tangent to the curve y = tan (tan −1 x). The locus of the centre of the circle is a parabola whose : (A) length of the latus rectum is 2 2 (B) axis of symmetry has the equation x + y = 1 (C) vertex has the co-ordinates (3/4, 1/4) (D) none of these Q.112 Which of the following equations in parametric form can represent a hyperbola, where 't' is a parameter. (A) x =

a  b  1 1 t +  & y = t −  2  2  t t

(C) x = et + e−t & y = et − e−t

(B)

tx y x ty − +t=0 & + −1=0 a b a b

(D) x2 − 6 = 2 cos t & y2 + 2 = 4 cos2

t 2

Q.113 The equations of the common tangents to the ellipse, x2 + 4y2 = 8 & the parabola y2 = 4x can be (A) x + 2y + 4 = 0 (B) x – 2y + 4 = 0 (C) 2x + y – 4 = 0 (D) 2x – y + 4 = 0

Quest

Q.114 Variable chords of the parabola y2 = 4ax subtend a right angle at the vertex. Then : (A) locus of the feet of the perpendiculars from the vertex on these chords is a circle (B) locus of the middle points of the chords is a parabola (C) variable chords passes through a fixed point on the axis of the parabola (D) none of these

Q.115 Equations of a common tangent to the two hyperbolas (A) y = x +

(C) y = − x +

x2 y2 y2 x2 − − = 1 & = 1 is : a 2 b2 a 2 b2

(B) y = x − a 2 − b 2

a 2 − b2

(D) − x − a 2 − b 2

a 2 − b2

Q.116 The equation of the tangent to the parabola y = (x − 3)2 parallel to the chord joining the points (3, 0) and (4, 1) is : (A) 2 x − 2 y + 6 = 0 (B) 2 y − 2 x + 6 = 0 (C) 4 y − 4 x + 13 = 0 (D) 4 x − 4 y = 13 Q.117 Let A be the vertex and L the length of the latus rectum of the parabola, y2 − 2 y − 4 x − 7 = 0. The equation of the parabola with A as vertex, 2L the length of the latus rectum and the axis at right angles to that of the given curve is : (A) x2 + 4 x + 8 y − 4 = 0 (B) x2 + 4 x − 8 y + 12 = 0 2 (C) x + 4 x + 8 y + 12 = 0 (D) x2 + 8 x − 4 y + 8 = 0

Q.118 The differential equation

dx 3y = represents a family of hyperbolas (except when it represents a pair dy 2x

of lines) with eccentricity : (A)

3 5

(B)

5 3

(C)

2 5

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

5 2

[14]

Q.119 If a number of ellipse be described having the same major axis 2a but a variable minor axis then the tangents at the ends of their latera recta pass through fixed points which can be (A) (0, a) (B) (0, 0) (C) (0, – a) (D) (a, a) Q.120 The straight line y + x = 1 touches the parabola : (A) x2 + 4 y = 0 (B) x2 − x + y = 0 (C) 4 x2 − 3 x + y = 0 (D) x2 − 2 x + 2 y = 0 Q.121 Circles are drawn on chords of the rectangular hyperbola xy = c2 parallel to the line y = x as diameters. All such circles pass through two fixed points whose co-ordinates are : (A) (c, c) (B) (c, − c) (C) (− c, c) (D) (− c, − c)

Quest Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[16]

Select the correct alternative : (Only one is correct) Q.1 Q.7 Q.13 Q.19 Q.25 Q.31 Q.37 Q.43 Q.49 Q.55 Q.61 Q.67 Q.73 Q.79 Q.85 Q.91 Q.97

C B C B D A A C C D C B D D A D D

Q.2 Q.8 Q.14 Q.20 Q.26 Q.32 Q.38 Q.44 Q.50 Q.56 Q.62 Q.68 Q.74 Q.80 Q.86 Q.92 Q.98

B B B D C D A B A B B D A B B C D

Q.3 Q.9 Q.15 Q.21 Q.27 Q.33 Q.39 Q.45 Q.51 Q.57 Q.63 Q.69 Q.75 Q.81 Q.87 Q.93 Q.99

B A B B B D B C D A A B C D D D C

Q.4 Q.10 Q.16 Q.22 Q.28 Q.34 Q.40 Q.46 Q.52 Q.58 Q.64 Q.70 Q.76 Q.82 Q.88 Q.94 Q.100

D B B A B B B B C B C B B C A C D

Q.5 Q.11 Q.17 Q.23 Q.29 Q.35 Q.41 Q.47 Q.53 Q.59 Q.65 Q.71 Q.77 Q.83 Q.89 Q.95

C C C D B B D D C C A C C D C B

Q.6 Q.12 Q.18 Q.24 Q.30 Q.36 Q.42 Q.48 Q.54 Q.60 Q.66 Q.72 Q.78 Q.84 Q.90 Q.96

Quest

A B D A A C C B C C C A A C B A

Select the correct alternatives : (More than one are correct) Q.101 Q.105 Q.109 Q.113 Q.117 Q.121

Q.102 Q.106 Q.110 Q.114 Q.118

A,B A,D B,C A,B A,B A,D

A,C,D A,B,C,D A,B,C,D A,B,C B,D

Q.103 Q.107 Q.111 Q.115 Q.119

B,D A,B B,C A,B,C,D A,C

Q.104 Q.108 Q.112 Q.116 Q.120

A,B,C,D A,D A,C,D C,D A,B,C

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

BINOMIAL THEOREM (Parabola, Ellipse & Hyperbola)

Question bank on Compound angles, Trigonometric eqn and ineqn, Solutions of Triangle & Binomial

There are 142 questions in this question bank. Select the correct alternative : (Only one is correct) Q.1

Q.2

If x + y = 3 – cos4θ and x – y = 4 sin2θ then (A) x4 + y4 = 9

(B)

x + y =16

(C) x3 + y3 = 2(x2 + y2)

(D)

x + y =2

If in a triangle ABC, b cos2 (A) in A.P.

Q.3

Q.4

(B) in G.P.

sin A (1 − n ) cos A

(B)

(n − 1) cos A sin A

s2

3 3

The exact value of cos (A) – 1/2

sin A (n − 1) cos A

(D)

sin A (n + 1) cos A

x ∈I 2 (D) a , x are finite but not possible to find (B) a = –1 ;

(B) A =

s2 2

(C) A >

s2

3

(D) None

2π 3π 6π 9π 18π 27 π cos ec + cos cos ec + cos cos ec is equal to 28 28 28 28 28 28 (B) 1/2 (C) 1 (D) 0

In any triangle ABC, (a + b)2 sin2 (A) c (a + b)

Q.8

(C)

If A is the area and 2s the sum of the 3 sides of a triangle, then : (A) A ≤

Q.7

None

FG π (a + x)IJ = 0 then, which of the following holds good? H2 K

x ∈I 2 (C) a ∈ R ; x ∈φ

Q.6

(D)

Quest

Given a2 + 2a + cosec2

(A) a = 1 ;

Q.5

(C) in H.P.

n sin A cos A then tan(A + B) equals 1 − n cos2 A

If tanB = (A)

A B 3 + a cos2 = c then a, b, c are : 2 2 2

C C + (a − b)2 cos2 = 2 2

(B) b (c + a)

(C) a (b + c)

(D) c2

) . cos ( 32π + x) − sin3 ( 72π − x) when simplified reduces to : cos ( x − π2 ) . tan ( 32π + x)

tan ( x −

π 2

(A) sin x cos x

(B) − sin2 x

(C) − sin x cos x

(D) sin2x

Q.9

If in a ∆ ABC, sin3A + sin3B + sin3C = 3 sinA · sinB · sinC then (A) ∆ ABC may be a scalene triangle (B) ∆ ABC is a right triangle (C) ∆ ABC is an obtuse angled triangle (D) ∆ ABC is an equilateral triangle

Q.10

In a triangle ABC, CH and CM are the lengths of the altitude and median to the base AB. If a = 10, b = 26, c = 32 then length (HM) (A) 5 (B) 7 (C) 9 (D) none Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.11

sin 2 θ sin θ + cos θ − The value of for all permissible vlaues of θ sin θ − cos θ tan 2 θ − 1 (A) is less than – 1 (B) is greater than 1 (C) lies between – 1 and 1 including both (D) lies between – 2 and

Q.12

sin 3θ = 4 sin θ sin 2θ sin 4θ in 0 ≤ θ ≤ π has : (A) 2 real solutions (B) 4 real solutions (C) 6 real solutions (D) 8 real solutions.

Q.13

In a triangle ABC, CD is the bisector of the angle C. If cos

2

C 1 has the value and l (CD) = 6, then 2 3

 1 1  +  has the value equal to  a b

(A) Q.14

1 9

(B)

1 12

(C)

Quest

RS π , 5π , 19π , 23π UV T12 12 12 12 W R 5π , 13π , 19π UV (C) S T 12 12 12 W

π 7π 17π 23π  (B)  , , , 

Q.17

RS T

1 2

(B) 2

If cos (α + β) = 0 then sin (α + 2β) = (A) sin α (B) − sin α

UV W

tan A has the value equal to tan B

1 2

(C) − 2

(D) −

(C) cos β

(D) − cos β

With usual notations, in a triangle ABC, a cos(B – C) + b cos(C – A) + c cos(A – B) is equal to (A)

Q.18

12 12 12 12  π 7 π 19 π 23π , , , (D) 12 12 12 12

If the median of a triangle ABC through A is perpendicular to AB then (A)

Q.16

(D) none

The set of angles btween 0 & 2π satisfying the equation 4 cos2 θ − 2 2 cos θ − 1 = 0 is (A)

Q.15

1 6

abc R2

(B)

sin 3 θ − cos 3 θ − sin θ − cos θ

 

π 2

(A) θ ∈  0 , 

abc 4R 2

cos θ 1 + cot 2 θ

(C)

4abc R2

(D)

abc 2R 2

− 2 tan θ cot θ = − 1 if : π 2

 

(B) θ ∈  , π

 

(C) θ ∈  π ,

3π   2

 3π  , 2 π  2 

(D) θ ∈ 

Q.19

With usual notations in a triangle ABC, ( I I1 ) · ( I I2 ) · ( I I3 ) has the value equal to (A) R2r (B) 2R2r (C) 4R2r (D) 16R2r

Q.20

In a triangle ABC, angle B < angle C and the values of B & C satisfy the equation 2 tan x - k (1 + tan2 x) = 0 where (0 < k < 1) . Then the measure of angle A is : (A) π/3 (B) 2π/3 (C) π/2 (D) 3π/4

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[3]

Q.21

If cos α =

α β 2 cos β − 1 then tan cot has the value equal to, where(0 < α < π and 0 < β < π) 2 2 2 − cos β

(A) 2 Q.22

2π π π , , 3 4 12

(B)  

k −1 k +1

(B)

The equation, sin2 θ − (A) no root

Q.25

Q.27

(C)

π 3π π , , 2 8 8

(D)

π 3π π , , 2 10 5

k +1 k −1

(C)

k k+1

(D)

k+1 k

4 4 =1− has : 3 sin θ − 1 sin θ − 1 3

(C) two roots

(D) infinite roots

Quest

3  1 1  1 1  1 1 +   +   +  = KR where K has the value  r1 r2   r2 r3   r3 r1  a 2 b2 c2

(B) 16

(C) 64

(D) 128

1 − sin x + 1 + sin x 5π < x < 3π , then the value of the expression If is 2 1 − sin x − 1 + sin x x x x x (B) cot (C) tan (D) –tan (A) –cot 2 2 2 2  

If x sin θ = y sin  θ +

2π 4 π   = z sin  θ +  then :   3 3

(B) xy + yz + zx = 0

In a ∆ ABC, the value of (A)

Q.29

3

C A B C  = k sin , then tan tan = 2 2 2 2

(B) one root

(A) x + y + z = 0

Q.28

(D)

With usual notation in a ∆ ABC  equal to : (A) 1

Q.26

π π π , , 2 3 6

If A + B + C = π & sin  A + (A)

Q.24

(C) 3

2

In a ∆ ABC, if the median, bisector and altitude drawn from the vertex A divide the angle at the vertex into four equal parts then the angles of the ∆ ABC are : (A)

Q.23

(B)

r R

The value of cos (A)

1 32

(C) xyz + x + y + z = 1 (D) none

a cos A + b cos B + c cos C is equal to : a+b+c

(B)

R 2r

(C)

R r

(D)

2r R

π 2π 4π 8π 16 π cos cos cos cos is : 10 10 10 10 10

(B)

1 16

(C)

cos ( π / 10 ) 16

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) −

10 + 2 5 64

[4]

Q.30

With usual notation in a ∆ ABC, if R = k (A) 1

(B) 2

(r1 + r2 ) ( r2 + r3 ) ( r3 + r1 ) r1 r2 + r2 r3 + r3 r1

where k has the value equal to:

(C) 1/4

(D) 4

Q.31

If a cos3 α + 3a cos α sin2 α = m and a sin3 α + 3a cos2 α sin α = n . Then (m + n)2/3 + (m − n)2/3 is equal to : (A) 2 a2 (B) 2 a1/3 (C) 2 a2/3 (D) 2 a3

Q.32

In a triangle ABC , AD is the altitude from A . Given b > c , angle C = 23° & AD = then angle B = (A) 157°

Q.33

(B) 113°

(C) 147°

[JEE ’94, 2] (D) none

The value of cot x + cot (60º + x) + cot (120º + x) is equal to : (A) cot 3x

(B) tan 3x

(C) 3 tan 3x

3 − 9 tan 2 x (D) 3 tan x − tan 3 x

Quest

Q.34

In a ∆ ABC, cos 3A + cos 3B + cos 3C = 1 then : (A) ∆ ABC is right angled (B) ∆ ABC is acute angled (C) ∆ ABC is obtuse angled (D) nothing definite can be said about the nature of the ∆.

Q.35

The value of

3 + cot 76° cot 16° is : cot 76° + cot 16°

(A) cot 44º Q.36

(B) tan 44º

(C) tan 2º

(D) cot 46º

If the incircle of the ∆ ABC touches its sides respectively at L, M and N and if x, y, z be the circumradii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz is equal to : (A) R r2

(B) r R2

(C)

1 R r2 2

(D)

1 r R2 2

Q.37

The number of solutions of tan (5π cos θ) = cot (5 π sin θ) for θ in (0, 2π) is : (A) 28 (B) 14 (C) 4 (D) 2

Q.38

If A = 3400 then 2 sin

Q.39

a bc b 2 − c2

A is identical to 2

(A)

1 + sin A + 1 − sin A

(B) − 1 + sin A − 1 − sin A

(C)

1 + sin A − 1 − sin A

(D) − 1 + sin A + 1 − sin A

AD, BE and CF are the perpendiculars from the angular points of a ∆ ABC upon the opposite sides. The perimeters of the ∆ DEF and ∆ ABC are in the ratio : (A)

2r R

(B)

r 2R

(C)

r R

(D)

r 3R

where r is the in radius and R is the circum radius of the ∆ ABC

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.40

The value of cosec

π – 18

π 3 sec 18 is a

(A) surd (C) negative natural number Q.41

(B) rational which is not integral (D) natural number

In a ∆ ABC if b + c = 3a then cot (A) 4

B C · cot has the value equal to : 2 2

(B) 3

(C) 2

(D) 1

Q.42

The set of values of ‘a’ for which the equation, cos 2x + a sin x = 2a − 7 possess a solution is : (A) (− ∞, 2) (B) [2, 6] (C) (6, ∞) (D) (− ∞, ∞)

Q.43

In a right angled triangle the hypotenuse is 2 2 times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are (A)

Q.44

π π & 3 6

(B)

π 3π & 8 8

(C)

(A) 1/4

Quest (B) 1/2

(C) 1

In ∆ ABC, the minimum value of

(A) 1

is

(C) 3

(D) non existent

(B)

1 3

(C) – 3

(D) –

The general solution of sin x + sin 5x = sin 2x + sin 4x is : (A) 2nπ (B) nπ (C) nπ/3 where n ∈ I

1 3

(D) 2 nπ/3

The product of the distances of the incentre from the angular points of a ∆ ABC is : (A) 4 R2 r

Q.49

A 2

If the orthocentre and circumcentre of a triangle ABC be at equal distances from the side BC and lie on the same side of BC then tanB tanC has the value equal to : (A) 3

Q.48

(B) 2

(D) 2

B

∑ cot 2 2 . cot 2 2 ∏ cot 2

Q.47

π 3π & 5 10

abc a b c then the value of λ is : + + =λ fgh f g h

A

Q.46

(D)

Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ∆ ABC on the sides a, b and c respectively . If

Q.45

π π & 4 4

(B) 4 Rr2

2 Number of roots of the equation cos x +

[−π, π] is (A) 2

(B) 4

(C)

(a b c) R s

(D)

(a bc )s R

3 +1 3 sin x − − 1 = 0 which lie in the interval 2 4 (C) 6

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 8

[6]

Q.50

sec 8θ − 1 sec 4θ − 1 is equal to (A) tan 2θ cot 8θ

(B) tan 8θ tan 2θ

(

(C) cot 8θ cot 2θ

(D) tan 8θ cot 2θ

)

Q.51

In a ∆ABC if b = a 3 − 1 and ∠C = 300 then the measure of the angle A is (A) 150 (B) 450 (C) 750 (D) 1050

Q.52

Number of values of θ ∈ [ 0 , 2 π] satisfying the equation cotx – cosx = 1 – cotx. cosx (A) 1 (B) 2 (C) 3 (D) 4

Q.53

The exact value of cos273º + cos247º + (cos73º . cos47º) is (A) 1/4 (B) 1/2 (C)3/4

Q.54

I n

a

∆ABC, a = a1 = 2 , b = a2 , c = a3 such that ap+1 =

where p = 1,2 then (A) r1 = r2

Q.55

(

) (

3π

3π

(C) r2 = 2r1

)



(D) none

 3π   5π  − α + cos  − α  when simplified  2   2 



(B) − sin 2α

(C) 1 − sin 2α

(D) 1 + sin 2α

If ‘O’ is the circumcentre of the ∆ ABC and R1, R2 and R3 are the radii of the circumcircles of triangles

(A)

a bc 2 R3

(B)

a b c + + has the value equal to: R1 R2 R 3

R3 a bc

(C)

4∆ R2

(D)

∆ 4R 2

The maximum value of ( 7 cosθ + 24 sinθ ) × ( 7 sinθ – 24 cosθ ) for every θ ∈ R . (A) 25

Q.59

(C) − 1

The expression [1 − sin (3π − α) + cos (3π + α)] 1 − sin 

OBC, OCA and OAB respectively then

Q.58

(D) r2 = 3r1

tan 2 − α cos 2 − α π π   The expression, + cos  α −  sin (π − α) + cos (π + α) sin  α −  when 2 2 cos(2 π − α )

reduces to : (A) sin 2α Q.57

5p 4p − 2   a 22 − p − ap  2−p p  3 5p  

Quest (B) r3 = 2r1

simplified reduces to : (A) zero (B) 1 Q.56

(D) 1

(B) 625

(C)

625 2

(D)

625 4

4 sin50 sin550 sin650 has the values equal to (A)

3 +1 2 2

(B)

3 −1 2 2

(C)

3 −1 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

d

i

3 3−1 2 2

[7]

Q.60

If x, y and z are the distances of incentre from the vertices of the triangle ABC respectively then a bc x y z is equal to (A)

A

∏ tan 2

(B)

A

∑ cot 2

(C)

A

∑ tan 2

(D)

A

∑ sin 2

Q.61

The medians of a ∆ ABC are 9 cm, 12 cm and 15 cm respectively . Then the area of the triangle is (A) 96 sq cm (B) 84 sq cm (C) 72 sq cm (D) 60 sq cm

Q.62

If x =

nπ x x x π 3π , satisfies the equation sin − cos = 1 − sin x & the inequality − ≤ , then: 2 2 2 2 2 4

(A) n = −1, 0, 3, 5 (C) n = 0, 2, 4 Q.63

FG H

The value of 1 + cos (A)

9 16

(B) n = 1, 2, 4, 5 (D) n = −1, 1, 3, 5

π 9

IJ FG1 + cos 3π IJ FG1+ cos 5π IJ FG1+ cos 7π IJ is KH 9K H 9KH 9K

Quest (B)

10 16

(C)

12 16

(D)

5 16

Q.64

The number of all possible triplets (a1 , a2 , a3) such that a1+ a2 cos 2x + a3 sin² x = 0 for all x is (A) 0 (B) 1 (C) 3 (D) infinite

Q.65

In a ∆ABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius of the semicircle is 2∆ 2∆ (B) a+b a +b−c Where ∆ is the area of the triangle ABC. (A)

(C)

2∆ s

(D)

c 2

Q.66

For each natural number k , let Ck denotes the circle with radius k centimeters and centre at the origin. On the circle Ck , a particle moves k centimeters in the counter- clockwise direction. After completing its motion on Ck , the particle moves to Ck+1 in the radial direction. The motion of the particle continues in this manner .The particle starts at (1, 0).If the particle crosses the positive direction of the x- axis for the first time on the circle Cn then n equal to (A) 6 (B) 7 (C) 8 (D) 9

Q.67

If in a ∆ ABC,

cos A cos B cos C = = a b c

(A) right angled Q.68

Q.69

then the triangle is

(B) isosceles

(C) equilateral

If cos A + cosB + 2cosC = 2 then the sides of the ∆ ABC are in (A) A.P. (B) G.P (C) H.P.

(D) obtuse

(D) none

If A and B are complimentary angles, then :  

A  2 

B 2

(A) 1 + tan   1 + tan  = 2

 

A  2 

B 2

(B) 1 + cot  1 + cot  = 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[8]

 

A B (C) 1 + sec  1 + cos ec  = 2



Q.70

2 

2

B 2

The value of , 3 cosec 20° − sec 20° is : 2 sin 20°

(A) 2 Q.71

A  2 

(D)  1 − tan   1 − tan  = 2

(B) sin 40°

4 sin 20°

(C) 4

(D) sin 40°

If in a ∆ ABC, cosA·cosB + sinA sinB sin2C = 1 then, the statement which is incorrect, is (A) ∆ ABC is isosceles but not right angled (B) ∆ ABC is acute angled (C) ∆ ABC is right angled

(D) least angle of the triangle is

π 4

2 π ( ) − 2 (0.25)sincos(x2−x 4 ) + 1 = 0, is

tan x − π 4

Q.72

The set of values of x satisfying the equation, 2 (A) an empty set (B) a singleton (C) a set containing two values (D) an infinite set

:

Q.73

The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to : (A) ∆ (B) 2 ∆ (C) 3 ∆ (D) 4 ∆ [ where ∆ is the area of the triangle ABC ]

Q.74

If in a triangle sin A : sin C = sin (A − B) : sin (B − C) then a2 : b2 : c2 (A) are in A.P. (B) are in G.P. (C) are in H.P. (D) none of these

Quest

[ Y G ‘99 Tier - I ]

5

Q.75

The number of solution of the equation,

∑ cos(r x) = 0

lying in (0, p) is :

r =1

(A) 2

Q.76

If θ = 3 α and sin θ = 1

(A)

Q.78

(B) 3

2

a +b

a 2

a + b2

If in a triangle ABC (A)

π 8

(C) a + b

(D) none

10 10 10 10 + tan 67 – cot 67 – tan7 is : 2 2 2 2

(A) a rational number (B) irrational number Q.79

(D) more than 5

. The value of the expression , a cosec α − b sec α is

(B) 2 a 2 + b 2

2

The value of cot 7

(C) 5

(C) 2(3 + 2 3 )

(D) 2 (3 – 3 )

2 cos A cos B 2 cos C a b + + = + then the value of the angle A is : a b c b c ca

(B)

π 4

(C)

π 3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

π 2

[9]

Q.80

The value of the expression (sinx + cosecx)2 + (cosx + secx)2 – ( tanx + cotx)2 wherever defined is equal to (A) 0 (B) 5 (C) 7 (D) 9

Q.81

If A = 5800 then which one of the following is true

Q.82

A (A) 2 sin   = 1 + sin A − 1 − sin A 2

A (B) 2 sin   = − 1 + sin A + 1 − sin A 2

A (C) 2 sin   = − 1 + sin A − 1 − sin A 2

A (D) 2 sin   = 2

1 + sin A + 1 − sin A

With usual notations in a triangle ABC, if r1 = 2r2 = 2r3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a

(D) 2a = 3b

1 π x2 − x Q.83 If tan α = 2 and tan β = (x ≠ 0, 1), where 0 < α, β < , then tan 2 2 2x − 2x + 1 x − x +1 (α + β) has the value equal to :

Quest

(A) 1

Q.84

Q.86

(C) 2

(D)

If r1, r2, r3 be the radii of excircles of the triangle ABC, then (A)

Q.85

(B) – 1

A

∑ cot 2

(B)

A

B

∑ cot 2 cot 2

(C)

∑ r1 ∑ r1r2

A

∑ tan 2

3 4

is equal to :

(D)

A

∏ tan 2

Minimum value of 8cos2x + 18sec2x ∀ x ∈ R wherever it is defined, is : (A) 24 (B) 25 (C) 26 (D) 18

In a ∆ABC

 a2 b2 c 2  A B C  + +  sin A sin B sin C  . sin 2 sin 2 sin 2 simplifies to  

(B) ∆

(A) 2∆

(C)

∆ 2

(D)

∆ 4

where ∆ is the area of the triangle Q.87

If θ is eliminated from the equations x = a cos(θ – α) and y = b cos (θ – β) then

x2

y2

2xy cos(α − β) is equal to ab a b (A) cos2 ( α – β) (B) sin2 (α – β) 2

Q.88

+

2

−

(C) sec2 ( α – β)

The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x · tan 3x is π (A) x = nπ (B) nπ ± (C) x = 2nπ 3 where n ∈ I Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) cosec2 (α – β)

(D) x =

nπ 3

[10]

Q.89

If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (logbc)3 + (logca)3 is (A) an odd prime (B) an even prime (C) an odd composite (D) an irrational number

Q.90

If the arcs of the same length in two circles S1 and S2 subtend angles 75° and 120° respectively at the S centre. The ratio 1 is equal to S2 (A)

Q.91

Q.92

1 5

81 16

(B)

(C)

Number of principal solution of the equation tan 3x – tan 2x – tan x = 0, is (A) 3 (B) 5

(D)

(C) 7

25 64

(D) more than 7

tan 2 20° − sin 2 20° simplifies to tan 2 20° · sin 2 20°

The expression

Quest

(A) a rational which is not integral (C) a natural which is prime Q.93

64 25

(B) a surd (D) a natural which is not composite

The value of x that satisfies the relation x = 1 – x + x2 – x3 + x4 – x5 + ......... ∞ (A) 2 cos36° (B) 2 cos144° (C) 2 sin18°

(D) none

Select the correct alternatives : (More than one are correct) Q.94

If sin θ = sin α then sin (A) sin

Q.95

θ = 3

π 3

α 3

π 3

α 3

(C) sin  + 

π 3

α 3

(D) − sin  + 

Choose the INCORRECT statement(s). °

Q.96

α 3

(B) sin  − 

°

°

°

(A

1 1 1 1 sin 82 . cos 37 and sin 127 . sin 97 have the same value. 2 2 2 2

(B)

If tan A =

(C) (D)

The sign of the product sin 2 . sin 3 . sin 5 is positive. There exists a value of θ between 0 & 2 π which satisfies the equation ; sin4 θ – sin2 θ – 1 = 0.

3 4− 3

& tan B =

3 4+ 3

then tan (A − B) must be irrational.

Which of the following functions have the maximum value unity ? (A) sin2 x − cos2 x (C) −

sin 2x − cos 2x 2

(B)

(D)

sin 2x − cos 2x 2

6  1 sin x + 1 cos x    3 5  2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[11]

Q.97

If the sides of a right angled triangle are {cos2α + cos2β + 2cos(α + β)} and {sin2α + sin2β + 2sin(α + β)}, then the length of the hypotenuse is : (A) 2[1+cos(α − β)] (B) 2[1 − cos(α + β)] (C) 4 cos2

Q.98

Q.99

α−β 2

(D) 4sin2

α+β 2

An extreme value of 1 + 4 sin θ + 3 cos θ is : (A) − 3 (B) − 4 (C) 5

The sines of two angles of a triangle are equal to (A) 245/1313

(B) 255/1313

(D) 6

5 99 & . The cosine of the third angle is : 13 101

(C) 735/1313

4 Q.100 It is known that sin β = & 0 < β < π then the value of 5

(A) independent of α for all β in (0, π/2) (C)

5

(B)

(D) 765/1313

3 sin (α + β) − cos2 π cos(α + β) 6 is: sin α

for tan β > 0

Quest

3 (7 + 24 cot α ) for tan β < 0 15

3

(D) none

Q.101 If x = sec φ − tan φ & y = cosec φ + cot φ then : (A) x =

y +1 y −1

(B) y =

1+ x 1− x

(C) x =

y−1 y+1

(D) xy + x − y + 1 = 0

Q.102 If 2 cosθ + sinθ = 1, then the value of 4 cosθ + 3sinθ is equal to (A) 3 Q.103 If sin t + cos t = (A) −1

(B) –5

(C)

7 5

(D) –4

1 t then tan is equal to : 5 2

(*B) –

1 3

(C) 2

(D) −

1 6

BINOMIAL There are 39 questions in this question bank. Q.104 Given that the term of the expansion (x1/3 − x−1/2)15 which does not contain x is 5 m where m ∈ N , then m = (A) 1100 (B) 1010 (C) 1001 (D) none Q.105 In the binomial (21/3 + 3−1/3)n, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is 1/6 , then n = (A) 6 (B) 9 (C) 12 (D) 15

N o rth

D e lh i :

Quest Tutorials E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]



x





n

Q.106 If the coefficients of x7 & x8 in the expansion of 2 +  are equal , then the value of n is : 3 (A) 15

(B) 45

(C) 55

(D) 56

1  1  1   Q.107 The coefficient of x49 in the expansion of (x – 1)  x −   x − 2  .....  x − 49  is equal to 2 2  2    1 (A) – 2 1 − 50   2 

(B) + ve coefficient of x

(C) – ve coefficient of x

1   (D) – 2 1 − 49   2 

Q.108 The last digit of (3P + 2) is : (A) 1 (B) 2 4n where P = 3 and n ∈ N

(C) 4

(D) 5

Quest 

1





n

Q.109 The sum of the binomial coefficients of 2 x +  is equal to 256 . The constant term in the x expansion is (A) 1120

Q.110 The coefficient of (A)

405 256

(B) 2110

x4

3 x in  − 2  2 x 

(B)

(C) 1210

(D) none

10

is :

504 259

(C)

450 263

(D)

Q.111 The remainder, when (1523 + 2323) is divided by 19, is (A) 4 (B) 15 (C) 0

405 512

(D) 18

Q.112 Let (7 + 4 3 ) n = p + β when n and p are positive integers and β ∈ (0, 1) then (1 – β) (p + β) is (A) rational which is not an integer (B) a prime (C) a composite (D) none of these Q.113 If (11)27 + (21)27 when divided by 16 leaves the remainder (A) 0 (B) 1 (C) 2

(D) 14

Q.114 Last three digits of the number N = 7100 – 3100 are (A) 100 (B) 300 (C) 500

(D) 000

Q.115 The last two digits of the number 3400 are : (A) 81 (B) 43 (C) 29

(D) 01

Q.116 If (1 + x + x²)25 = a0 + a1x + a2x² + ..... + a50 . x50 then a0 + a2 + a4 + ..... + a50 is : (A) even (B) odd & of the form 3n (C) odd & of the form (3n − 1) (D) odd & of the form (3n + 1) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Q.117 The sum of the series (1² + 1).1! + (2² + 1).2! + (3² + 1). 3! + ..... + (n² + 1). n! is : (A) (n + 1). (n+2)! (B) n.(n+1)! (C) (n + 1). (n+1)! (D) none of these Q.118 Let Pm stand for nPm . Then the expression 1 . P1 + 2 . P2 + 3 . P3 + ..... + n . Pn = (A) (n + 1) ! − 1 (B) (n + 1) ! + 1 (C) (n + 1) ! (D) none of these Q.119 The expression

  1 + 4x + 1  7  1 − 4x + 1  7    −   is a polynomial in x of degree 2 2 4x + 1         1

(A) 7

(B) 5

(C) 4

 Q.120 If the second term of the expansion a 1/13 + 

(A) 4

(D) 3

n

n a  C3 5/2 is :  is 14a then the value of n −1 C2 a 

(B) 3

(C) 12

(D) 6

Q.121 If (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + x3 + ...... + xn) m

≡ a0 + a1x + a2x2 + a3x3 + ...... + amxm then

∑

a r has the value equal to

Quest

(A) n!

(B) (n + 1) !

r=0

(C) (n – 1)!

(D) none

Q.122 The value of 4 {nC1 + 4 . nC2 + 42 . nC3 + ...... + 4n − 1} is : (A) 0 (B) 5n + 1 (C) 5n

(D) 5n − 1

Q.123 If n be a positive integer such that n ≥ 3, then the value of the sum to n terms of the series 1.n−

( n −1) 1!

(A) 0

(n − 1) +

( n − 1) ( n − 2) 2!

(n − 1) ( n − 2) ( n − 3)

(n − 2) –

(B) 1

3!

(n − 3) + ...... is :

(C) – 1

(D) none of these

Q.124 In the expansion of (1 + x)43 if the co−efficients of the (2r + 1)th and the (r + 2)th terms are equal, the value of r is : (A) 12 (B) 13 (C) 14 (D) 15  

Q.125 The positive value of a so that the co−efficient of x5 is equal to that of x15 in the expansion of  x 2 + (A)

1 2 3

1

(B)

3

 x+1 x −1  −  Q.126 In the expansion of  2/ 3 1/ 3  x − x + 1 x − x1/ 2 

(A)

10C 0

(B)

10C 7

(C) 1

a  x3 

10

is

(D) 2 3

10

, the term which does not contain x is : (C)

10C 4

(D) none 8

Q.127

If the 6th (A) 5

 1  2 term in the expansion of the binomial  8/ 3 + x log10 x is 5600, then x equals to x 

(B) 8

(C) 10

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(D) 100

[14]

Q.128 Co-efficient of αt in the expansion of, (α + p)m − 1 + (α + p)m − 2 (α + q) + (α + p)m − 3 (α + q)2 + ...... (α + q)m − 1 where α ≠ − q and p ≠ q is : m

(A)

(

Ct pt − qt

)

m

(B)

p−q m

(C)

(

Ct pt + q t

)

)

p−q m

(D)

p−q

(

Ct pm− t − qm− t

(

Ct p m− t + q m− t

)

p−q

Q.129 (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + ...... + x100) when written in the ascending power of x then the highest exponent of x is ______ . (A) 4950 (B) 5050 (C) 5150 (D) none

(

Q.130 Let 5 + 2 6

)

n

= p + f where n ∈ N and p ∈ N and 0 < f < 1 then the value of, f2 − f + pf − p is

(A) a natural number (C) a prime number

(B) a negative integer (D) are irrational number

Quest

Q.131 Number of rational terms in the expansion of (A) 25

(B) 26

(

2 + 43

)

100

is :

(C) 27

(D) 28

10

cos θ    is Q.132 The greatest value of the term independent of x in the expansion of  x sin θ + x   10

(A)

10C 5

(B) 25

(C)

25

·

10C 5

(D)

C5

25

Q.133 If (1 + x – 3x2)2145 = a0 + a1x + a2x2 + ......... then a0 – a1 + a2 – a3 + ..... ends with (A) 1 (B) 3 (C) 7 (D) 9 9

 4x 2 3  6 − Q.134 Coefficient of x in the binomial expansion  is 2x   3 (A) 2438 (B) 2688 (C) 2868  1   Q.135 The term independent of ' x ' in the expansion of  9 x − 3 x 

(D) none 18

, x > 0 , is α times the corresponding

binomial co-efficient . Then ' α ' is : (A) 3

(B)

1 3

(C) −

1 3

(D) 1

Q.136 The expression [x + (x3−1)1/2]5 + [x − (x3−1)1/2]5 is a polynomial of degree : (A) 5 (B) 6 (C) 7 (D) 8 [JEE’92, 6 + 2]

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

Q.137 Given (1 – 2x + 5x2 – 10x3) (1 + x)n = 1 + a1x + a2x2 + .... and that a12 = 2a2 then the value of n is (A) 6 (B) 2 (C) 5 (D) 3 Q.138 The sum of the series aC0 + (a + b)C1 + (a + 2b)C2 + ..... + (a + nb)Cn is where Cr's denotes combinatorial coefficient in the expansion of (1 + x)n, n ∈ N (A) (a + 2nb)2n (B) (2a + nb)2n (C) (a +nb)2n – 1 (D) (2a + nb)2n – 1 Q.139 The coefficient of the middle term in the binomial expansion in powers of x of (1 + αx)4 and of (1 – αx)6 is the same if α equals 5 3

(A) –

(B)

10 3

(C) –

3 10

(D)

3 5

Q.140 (2n + 1) (2n + 3) (2n + 5) ....... (4n − 1) is equal to : (A)

(4 n) ! 2 . (2n) ! (2n) ! n

n

Q.141 If Sn = (A)

n 2

n

n

(C)

( 4 n) ! n ! (2n) ! (2n) !

(D)

(4 n) ! n ! 2 n ! (2n) !

Tn then S is equal to n

Quest 1

∑ nC r =0

(4 n) ! n ! 2 . (2n) ! (2n) !

(B)

r

and Tn =

r

∑ nC r =0

(B)

n −1 2

r

(C) n – 1

(D)

2n − 1 2

Q.142 The coefficient of xr (0 ≤ r ≤ n − 1) in the expression : (x + 2)n−1 + (x + 2)n−2. (x + 1) + (x + 2)n−3 . (x + 1)² + ...... + (x + 1)n−1 is : (A) nCr (2r − 1) (B) nCr (2n−r − 1) (C) nCr (2r + 1) (D) nCr (2n−r + 1)

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[16]

Answers Select the correct alternative : (Only one is correct) Q.1 D Q.2 D Q.3 A Q.4 B Q.5 Q.8 D Q.9 D Q.10 C Q.11 D Q.12 Q.15 C Q.16 A Q.17 A Q.18 B Q.19 Q.22 C Q.23 A Q.24 D Q.25 C Q.26 Q.29 D Q.30 C Q.31 C Q.32 B Q.33 Q.36 C Q.37 A Q.38 D Q.39 C Q.40 Q.43 B Q.44 A Q.45 A Q.46 A Q.47 Q.50 D Q.51 D Q.52 B Q.53 C Q.54 Q.57 C Q.58 C Q.59 B Q.60 B Q.61 Q.64 D Q.65 A Q.66 B Q.67 C Q.68 Q.71 C Q.72 A Q.73 B Q.74 A Q.75 Q.79 D Q.80 B Q.81 C Q.82 C Q.83 Q.86 B Q.87 B Q.88 D Q.89 A Q.90 Q.93 C Select the correct alternatives : (More than one are correct) Q.94 ABD Q.95 BCD Q.96 ABCD Q.97 AC Q.100 ABC Q.101 BCD Q.102 AC Q.103 BC BINOMIAL Select the correct alternative : (Only one is correct) Q.104 C Q.105 B Q.106 C Q.107 A Q.108 Q.111 C Q.112 D Q.113 A Q.114 D Q.115 Q.118 A Q.119 D Q.120 A Q.121 B Q.122 Q.125 A Q.126 C Q.127 C Q.128 B Q.129 Q.132 D Q.133 B Q.134 B Q.135 D Q.136 Q.139 C Q.140 B Q.141 A Q.142 B

A D D D D D C D C A C A C

Q.6 Q.13 Q.20 Q.27 Q.34 Q.41 Q.48 Q.55 Q.62 Q.69 Q.76 Q.84 Q.91

D A C B C C B A B A B C C

Q.7 Q.14 Q.21 Q.28 Q.35 Q.42 Q.49 Q.56 Q.63 Q.70 Q.78 Q.85 Q.92

Q.98

BD

Q.99

BC

D D D B C

Q.109 Q.116 Q.123 Q.130 Q.137

A A A B A

Q.110 Q.117 Q.124 Q.131 Q.138

Quest Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

D B D A A B B B A C B C D

A B C B D

[17]

TARGET IIT JEE

MATHEMATICS

FUNCTION, LIMIT, CONTINUITY & DERIVABILITY

Question bank on function limit continuity & derivability There are 105 questions in this question bank. Select the correct alternative : (Only one is correct) Q.1 If both f(x) & g(x) are differentiable functions at x = x0 , then the function defined as, h(x) = Maximum {f(x), g(x)} (A) is always differentiable at x = x0 (B) is never differentiable at x = x0 (C) is differentiable at x = x0 provided f(x0) ≠ g(x0) (D) cannot be differentiable at x = x0 if f(x0) = g(x0) . Q.2

If Lim (x−3 sin 3x + ax−2 + b) exists and is equal to zero then : x→0

(A) a = − 3 & b = 9/2 (C) a = − 3 & b = − 9/2 Q.3

(B) a = 3 & b = 9/2 (D) a = 3 & b = − 9/2 x m sin x1

A function f(x) is defined as f(x) =  

0

continuous at x = 0 is (A) 1 (B) 2

x ≠ 0, m ∈ N if x = 0

. The least value of m for which f ′ (x) is

(C) 3

(D) none

Quest

 1 if x = q where p & q > 0 are relatively prime integers For x > 0, let h(x) =  q 0 if x is irrational then which one does not hold good? (A) h(x) is discontinuous for all x in (0, ∞) (B) h(x) is continuous for each irrational in (0, ∞) (C) h(x) is discontinuous for each rational in (0, ∞) (D) h(x) is not derivable for all x in (0, ∞) . p

Q.4

1

Q.5

The value of Limit

 2 

x →∞

2 (A) ln   3 Q.6

1

x

 e −  3x n  e x    n x

(where n∈ N ) is 2 (C) n ln   3

(B) 0

(D) not defined

For a certain value of c, Lim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the value x →−∞

of the limit is (A) 1/5, 7/5 Q.7

xn

(B) 0, 1

(C) 1, 7/5

(D) none

Consider the piecewise defined function

  f (x) =  

−x 0 x−4

if

x4

choose the answer which best describes the continuity of this function (A) The function is unbounded and therefore cannot be continuous. (B) The function is right continuous at x = 0 (C) The function has a removable discontinuity at 0 and 4, but is continuous on the rest of the real line. (D) The function is continuous on the entire real line Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[2]

Q.8

If α, β are the roots of the quadratic equation (A) 0

Q.9

(B)

(C)

(B)

 Lim   x →1

4 1 − 3x + x 2  −     x 2 − x −1 1 − x3  

1 3

−1

+

(

2

) equals

a2 (α − β)2 2

2 tan −1 (nx ) π

(D)

)

3 . x4 − 1   = x 3 − x −1  

Quest (B) 3

(C)

1 2

(D) none

ABC is an isosceles triangle inscribed in a circle of radius r . If AB = AC & h is the altitude from A to BC

(A)

1 32r

(B)

1 64 r

∆ equals (where ∆ is the area of the triangle) P3 (C)

1 128r

(D) none

Let the function f, g and h be defined as follows :

1 x sin    x f (x) =   0

1 x 2 sin   x  g (x) =   0

for − 1 ≤ x ≤ 1 and x ≠ 0 for

x=0

for − 1 ≤ x ≤ 1 and x ≠ 0 for

x=0

h (x) = | x |3 for – 1 ≤ x ≤ 1 Which of these functions are differentiable at x = 0? (A) f and g only (B) f and h only (C) g and h only Q.13

(x − α )

(D) −

(C)

and P be the perimeter of ABC then Lim h →0

Q.12

a2 (α − β)2 2

(

1 − cos ax 2 + bx + c

n →∞

(A) Q.11

1 (α − β)2 2

then Lim x →α

Which one of the following best represents the graph of the function f(x) = Lim

(A)

Q.10

ax2 + bx + c = 0

1 If [x] denotes the greatest integer ≤ x, then Limit 4 n→∞ n

(A) x/2

(B) x/3

(D) none

([1 x] + [2 x] + ...... + [n x]) equals 3

3

(C) x/6

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

3

(D) x/4

[3]

Q.14

g (x ) Let f (x) = h ( x ) , where g and h are cotinuous functions on the open interval (a, b). Which of the

following statements is true for a < x < b? (A) f is continuous at all x for which x is not zero. (B) f is continuous at all x for which g (x) = 0 (C) f is continuous at all x for which g (x) is not equal to zero. (D) f is continuous at all x for which h (x) is not equal to zero.

Q.15

The period of the function f (x) = (A) π/2

(B) π/4

| sin x | + | cos x | is | sin x − cos x | (C) π

(D) 2π

x − e x + cos 2x Q.16 If f(x) = , x ≠ 0 is continuous at x = 0, then x2 5 (B) [f(0)] = – 2 (C) {f(0)} = –0.5 (D) [f(0)] . {f(0)} = –1.5 2 where [x] and {x} denotes greatest integer and fractional part function

Quest

(A) f (0) =

∞

Q.17

The value of the limit

∏ 1 − n

1  is 2 

n =2

(A) 1

Q.18

(B)

1 4

(D)

1 2

(B) if b is not equal to zero (D) for no value of b

Let f be differentiable at x = 0 and f ' (0) = 1. Then Lim h →0

(A) 3 Q.20

1 3

x + b, x < 0  The function g (x) = can be made differentiable at x = 0.  cos x , x ≥ 0

(A) if b is equal to zero (C) if b takes any real value Q.19

(C)

f (h ) − f (−2h ) = h (B) 2

(C) 1

If f (x) = sin–1(sinx) ; x ∈ R then f is (A) continuous and differentiable for all x (B) continuous for all x but not differentiable for all x = (2k + 1)

(D) – 1

π , k∈I 2

π ; k∈I 2 (D) neither continuous nor differentiable for x ∈ R − [ −1,1]

(C) neither continuous nor differentiable for x = (2k – 1)

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[4]

Q.21

Limit x→

(A) Q.22

−1  1

 cos  (3 sin x − sin 3x ) 4 

2 π

where [ ] denotes greatest integer function , is

(B) 1

If Lim x→ 0

(A) Q.23

π 2

sin x

(C)

4 π

(D) does not exist

ln (3 + x ) − ln (3 − x ) = k , the value of k is x

2 3

1 3

(B) –

(C) –

2 3

(D) 0

x 2n − 1 is identical with the function n →∞ x 2 n + 1 (A) g (x) = sgn(x – 1) (B) h (x) = sgn (tan–1x) (C) u (x) = sgn( | x | – 1) (D) v (x) = sgn (cot–1x) The function f (x) = Lim

Q.24

The functions defined by f(x) = max {x2, (x − 1)2, 2x (1 − x)}, 0 ≤ x ≤ 1 (A) is differentiable for all x (B) is differentiable for all x excetp at one point (C) is differentiable for all x except at two points (D) is not differentiable at more than two points.

Q.25

f (x) = (A)

Quest x lnx and g (x) = . Then identify the CORRECT statement lnx x

1 and f (x) are identical functions g(x )

(C) f (x) . g (x) = 1

Q.26

(D)

1 = 1 ∀ x>0 f (x ) . g( x )

x−3

(B) 4

(C) 0

(D) none of these

Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable? (A) f (x) = x1/3

Q.28

1 and g (x) are identical functions f (x)

x f (3) − 3 f (x ) If f(3) = 6 & f ′ (3) = 2, then Limit is given by : x→3

(A) 6 Q.27

∀ x >0

(B)

(B) f (x) =

|x| x

The limiting value of the function f(x) = (A) 2

(B)

1 2

(C) f (x) = e–x

(D) f (x) = tan x

π 2 2 − (cos x + sin x )3 when x → is 4 1 − sin 2 x

(C) 3 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

3 2

[5]

Q.29

Let

  f (x) =    

2 x + 23 − x − 6 2 − x − 21− x

if x > 2 then

x2 − 4 if x < 2 x − 3x − 2 (A) f (2) = 8 ⇒ f is continuous at x = 2 (C) f (2–) ≠ f (2+) ⇒ f is discontinuous

(B) f (2) = 16 ⇒ f is continuous at x = 2 (D) f has a removable discontinuity at x = 2 

Q.30

 On the interval I = [− 2, 2], the function f(x) = (x + 1) e

[

− 1 + x1 |x |

 0

]

(x ≠ 0) (x = 0)

then which one of the following does not hold good? (A) is continuous for all values of x ∈ I (B) is continuous for x ∈ I − (0) (C) assumes all intermediate values from f(− 2) & f(2) (D) has a maximum value equal to 3/e . Q.31

Quest

Which of the following function is surjective but not injective (B) f : R → R f (x) = x3 + x + 1 (A) f : R → R f (x) = x4 + 2x3 – x2 + 1 (C) f : R → R+ f (x) = 1+ x 2

Q.32

x  [x]  Consider the function f (x) =  1   6−x

(D) f : R → R f (x) = x3 + 2x2 – x + 1

if 1 ≤ x < 2

if x = 2

if 2 < x ≤ 3

where [x] denotes step up function then at x = 2 function (A) has missing point removable discontinuity (B) has isolated point removable discontinuity (C) has non removable discontinuity finite type (D) is continuous Q.33

Suppose that f is continuous on [a, b] and that f (x) is an integer for each x in [a, b]. Then in [a, b] (A) f is injective (B) Range of f may have many elements (C) {x} is zero for all x ∈ [a, b] where { } denotes fractional part function (D) f (x) is constant

Q.34

The graph of function f contains the point P (1, 2) and Q(s, r). The equation of the secant line through  s 2 + 2s − 3  P and Q is y =  s − 1  x – 1 – s. The value of f ' (1), is   (A) 2 (B) 3 (C) 4

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) non existent

[6]

e x ln x 5( x + 2 ) ( x 2 − 7 x + 10) The range of the function f(x) = is 2 x 2 − 11x + 12 2

Q.35

(A) ( −∞ , ∞)

Q.36

C

o

n

(A) (B) (C) (D)

s i d

e r

f ( x

(B) [0 , ∞)

)

=

( (

3  (D)  , 4  2 

3  (C)  , ∞  2 

) )

 2 sin x − sin 3 x + sin x − sin 3 x  π   , x≠ 3 3 2  2 sin x − sin x − sin x − sin x   

for x ∈ (0, π)

f(π/2) = 3 where [ ] denotes the greatest integer function then, f is continuous & differentiable at x = π/2 f is continuous but not differentiable at x = π/2 f is neither continuous nor differentiable at x = π/2 none of these

Q.37

The number of points at which the function, f(x) = x – 0.5 + x – 1 + tan x does not have a derivative in the interval (0, 2) is : (A) 1 (B) 2 (C) 3 (D) 4

Q.38

Let [x] denote the integral part of x ∈ R. g(x) = x − [x]. Let f(x) be any continuous function with f(0) = f(1) then the function h(x) = f(g(x)) : (A) has finitely many discontinuities (B) is discontinuous at some x = c (C) is continuous on R (D) is a constant function .

Q.39

Given the function f(x) = 2x x 3 − 1 + 5 x 1 − x 4 + 7x2 x − 1 + 3x + 2 then : (A) the function is continuous but not differentiable at x = 1 (B) the function is discontinuous at x = 1 (C) the function is both cont. & differentiable at x = 1 (D) the range of f(x) is R+.

Q.40

If f (x + y) = f (x) + f (y) + | x | y + xy2, ∀ x, y ∈ R and f ' (0) = 0, then (A) f need not be differentiable at every non zero x (B) f is differentiable for all x ∈ R (C) f is twice differentiable at x = 0 (D) none

Q.41

For Lim x →8

Q.42

Lim

Quest

sin{x − 10} (where { } denotes fractional part function) {10 − x} (A) LHL exist but RHL does not exist (B) RHL exist but LHL does not exist. (C) neither LHL nor RHL does not exist (D) both RHL and LHL exist and equals to 1

n →∞

(A)

12 n +22 (n−1)+32 (n−2)+.....+n 2.1 is equal to : 13 +23 +33 +......+n 3

1 3

(B)

2 3

(C)

1 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

1 6

[7]

Q.43

T

h

e

d

o

m

a i n

o

f

d

e f i n

i t i o

n

o

f

t h

e

f u

n

c t i o

n

f

( x

)

=

3  (B)  , ∞  − {2, 3} 4  Where [x] denotes greatest integer function.

(A) {2}

Q.44

If f (x) =

x 2 − bx + 25 x 2 − 7 x + 10

(A) 0 Q.45

Q.46

 1  x + x 

| x 2 − x − 6 | + 16–xC2x–1 + 20–3xP2x–5 is  1  (D)  − , ∞   4 

(C) {2, 3}

for x ≠ 5 and f is continuous at x = 5, then f (5) has the value equal to (B) 5

(C) 10

(D) 25

Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true? I. f is continuous on the closed interval [a, b] II. f is bounded on the open interval (a, b) III. If a 0 then f ′ (x) is equal to (A)

f ' (1) x

Given

Q.68

Let f(x) =

Q.70

Limit

Quest

x →0

(sin x − tan x ) 2 − (1 − cos 2x ) 4 + x 5 7.(tan −1 x ) 7 + (sin −1 x ) 6 + 3 sin 5 x (B)

1 7

(D) 1

(

π

[ x ]2 Limit Let Limit x→0 2 =l & x→0 x

The value of Limit x→0



[ x2 ] x2



If f (x) =

is :

π

(C)  0 ,   4 

 

π 2

(D)  0 , 

= m , where [ ] denotes greatest integer , then:

sin {x }

{x } ( {x } − 1 )

(B) is tan 1 2
n  e x + 2 x   

)

(B) m exists but l does not (D) neither l nor m exists .

( tan ( {x } − 1 ) )

(A) is 1

Q.74

1 3

2 Range of the function , f (x) = cot −1 log 4 / 5 (5 x − 8 x + 4)

(B)  , π 4 

(D) does not exist

is equal to (C)

(A) l exists but m does not (C) l & m both exist Q.73

(D) f ′(1).(lnx)

Lim cos  π n 2 + n  when n is an integer : n →∞   (C) is equal to zero (A) is equal to 1 (B) is equal to − 1

(A) (0 , π) Q.72

(C) f ′ (1)

ln ( x 2 + e x ) . If Limit f(x) = l and Limit f(x) = m then : x→∞ x → −∞ ln ( x 4 + e 2 x ) (A) l = m (B) l = 2m (C) 2 l = m (D) l + m = 0

(A) 0 Q.71

1 x

f(x) = b ([x]2 + [x]) + 1 for x ≥ −1 = Sin (π (x+a) ) for x < −1 where [x] denotes the integral part of x, then for what values of a, b the function is continuous at x = −1? (A) a = 2n + (3/2) ; b ∈ R ; n ∈ I (B) a = 4n + 2 ; b ∈ R ; n ∈ I + (D) a = 4n + 1 ; b ∈ R+ ; n ∈ I (C) a = 4n + (3/2) ; b ∈ R ; n ∈ I

Q.67

Q.69

(B)

where { x } denotes the fractional part function: (C) is sin 1

(D) is non existent

is continuous at x = 0 , then f (0) must be equal to :

tan x

(A) 0

(B) 1

(C) e2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 2

[11]

Q.75

2 + 2 x + sin 2 x is : (2 x + sin 2 x )esin x (A) equal to zero (B) equal to 1 Lim

x →∞

Q.76 The value of

(A)

e

lim x→ 0

(C) equal to − 1

(D) non existent

( cos ax) cos ec bx is 2

 8b2  −     a2 

(B)

e

 8a 2  −     b2 

(C)

e

 a2  −     2 b2 

(D)

e

 b2  −     2a 2 

Select the correct alternative : (More than one are correct) Q.77

Lim f(x) does not exist when : x →c

(A) f(x) = [[x]] − [2x − 1], c = 3

(B) f(x) = [x] − x, c = 1

(C) f(x) = {x}2 − {−x}2, c = 0

(D) f(x) =

tan (sgn x) ,c =0. sgn x

where [x] denotes step up function & {x} fractional part function.

Q.78

Quest

  Let f (x) =   

tan2 {x } x2 − [ x ]2 1

for x > 0 for x = 0

where [ x ] is the step up function and { x } is the fractional

{x } cot {x } for x < 0

part function of x , then : (A) xLimit f (x) = 1 → 0+ (C)

Q.79

cot -1

(B) xLimit f (x) = 1 → 0−

   Limit f (x) − x → 0 

 x .
n (cos x ) If f(x) = 
n (1 + x2 )  0

2

=1

x≠0 x=0

then :

(A) f is continuous at x = 0 (C) f is differentiable at x = 0 Q.80

Q.81

(D) f is continuous at x = 1 .

(B) f is continuous at x = 0 but not differentiable at x=0 (D) f is not continuous at x = 0.

Which of the following function (s) is/are Transcidental? 2 sin 3x x + 2x − 1

(A) f (x) = 5 sin x

(B) f (x) =

(C) f (x) =

(D) f (x) = (x2 + 3).2x

x2 + 2x + 1

2

Which of the following function(s) is/are periodic? (A) f(x) = x − [x] (B) g(x) = sin (1/x) , x ≠ 0 & g(0) = 0 (C) h(x) = x cos x (D) w(x) = sin−1 (sin x)

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[12]

Q.82

Which of following pairs of functions are identical : (A) (B) (C) (D)

Q.83

Q.84

Which of the following functions are homogeneous ? (A) x sin y + y sin x (B) x ey/x + y ex/y (C) x2 − xy

sin θ =1 θ

Let f(x) =

(B) θ < sin θ < tan θ

(D)

tan θ sin θ > θ θ

(B) Limit x → ∞ g(x) = ln 4

Quest

(C) Limit x → 0 f(x) = ln 4

Q.87

(C) sin θ < θ < tan θ

x . 2x − x 
n 2  & g(x) = 2x sin  x  then :  2  1 − cos x

(A) Limit x → 0 f(x) = ln 2

Q.86

(D) arc sin xy

If θ is small & positive number then which of the following is/are correct ? (A)

Q.85

−1

f(x) = e
n sec x & g(x) = sec−1 x f(x) = tan (tan−1 x) & g(x) = cot (cot1 x) f(x) = sgn (x) & g(x) = sgn (sgn (x)) f(x) = cot2 x.cos2 x & g(x)= cot2 x − cos2 x

(D) Limit x → ∞ g(x) = ln 2

x −1 . Then : 2 x − 7x + 5 1 1 (A) Limit f(x) = − (B) Limit f(x) = − x→1 x → 0 5 3

Let f(x) =

2

(C) Limit x → ∞ f(x) = 0

(D) Limit does not exist x → 5/ 2

Which of the following limits vanish? 1 1 4 sin (A) Limit x x→∞

(B) Limit x → π /2 (1 − sin x) . tan x

x

(C) Limit x→∞

2 x2 + 3 . sgn (x) x2 + x − 5

[x]2 − 9 (D) Limit + x→3 2 x −9

where [ ] denotes greatest integer function Q.88

2m (n ! π x)] is given by Limit If x is a real number in [0, 1] then the value of Limit m → ∞ n → ∞ [1 + cos

(A) 1 or 2 according as x is rational or irrational (B) 2 or 1 according as x is rational or irrational (C) 1 for all x (D) 2 for all x . Q.89

If f(x) is a polynomial function satisfying the condition f(x) . f(1/x) = f(x) + f(1/x) and f(2) = 9 then : (A) 2 f(4) = 3 f(6) (B) 14 f(1) = f(3) (C) 9 f(3) = 2 f(5) (D) f(10) = f(11)

Q.90

Which of the following function(s) not defined at x = 0 has/have removable discontinuity at x = 0 ? (A) f(x) =

1 1 + 2cot x

π (B) f(x)=cos  | sin x |  (C) f(x) = x sin x  x 

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) f(x) =

1
n x

[13]

Q.91

 x−3

( )−( )+( )

The function f(x) =  

2

3x 2

x 4

13 4

(A) continuous at x = 1 (C) continuous at x = 3

Q.92

Q.93

π

, x ≥1 , x 0; g(x) = a x , a > 0 where {x} and [x] denotes the fractional part & integral part functions.

1

Quest

Fill in the blanks:

 sin x if x ≤ c where c is a known quantity. If f is derivable ax + b if x > c

Q.101 A function f is defined as follows, f(x) = 

at x = c, then the values of 'a' & 'b' are _____ &______ respectively .

Q.102 A weight hangs by a spring & is caused to vibrate by a sinusoidal force . Its displacement s(t) at time t is given by an equation of the form, s(t) =

A (sin kt − sin ct) where A, c & k are positive constants c − k2 2

with c ≠ k, then the limiting value of the displacement as c → k is ______. π (cos α )x − (sin α )x − cos 2α Limit Q.103 x → 4 where 0 < α < is ______ . 2 x−4

3 / x2 cos 2x) Q.104 Limit has the value equal to ______ . x→0 ( Q.105 If f(x) = sin x, x ≠ nπ , n = 0, ±1, ±2, ±3,.... = 2, otherwise and g(x) = x² + 1, x ≠ 0, 2 = 4, x=0 = 5, x=2 then Limit g [f(x)] is ______ x→ 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[15]

[16] Q.1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.74

Q.73 D

B

Q.72

B

Q.71

Q.69

Q.68 A

A

Q.67

A

Q.66

Q.64

Q.63 A

C

Q.62

D

Q.61

Q.59

Q.58 B

C

Q.57

B

Q.56

Q.54

Q.53 D

D

Q.52

C

Q.51

Q.49

Q.48 D

B

Q.47

A

Q.46

Q.44

Q.43 A

A

Q.42

B

Q.41

Q.29 Q.34 Q.39

Q.28 D Q.33 D Q.38 C

A B C

Q.27 Q.32 Q.37

C D A

Q.26 Q.31 Q.36

Q.24

Q.23 C

A

Q.22

A

Q.21

Q.19

Q.18 D

D

Q.17

D

Q.16

Q.14

Q.13 D

C

Q.12

C

Q.11

Q.9

C

Q.8

D

Q.7

A

Q.4

C

Q.3

A

Q.2

C

Q.6

D

Q.75

C

Q.70

A

Q.65

D

Q.60

B

Q.55

A

Q.50

D

Q.45

A A B

Q.30 Q.35 Q.40

A

Q.25

C

B

Q.20

A

C

Q.15

D

B

Q.10

A

B

Q.5

A

C C B

Quest A

B

C

A

A

C

D

Q.76 C

Q.98

Q.97 B, D

Q.94

Q.93 A, B, C, D

Q.90

Q.89 B, C

Q.77 B, C Q.81 A, D Q.85 C, D

A,B, D

Q.100 A, B, C

Q.99 A, C

A, B, C

A, B, D

Q.96

Q.95 B, D

A, B, C, D

Q.92

Q.91 A, B, C

B, C, D

Q.84 Q.88

Q.79 A, C Q.83 B, C Q.87 A, B, D

B, C, D A, B, C, D

Q.82 Q.86

A, C

Q.78

Q.104 e-6

Q.103 cos4 α ln cos α − sin4 α ln sin α

Q.102

Q.101 cos c & sin c - c cos c

Q.80

C, D B, D B, C

Q.105 1

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

FUNCTION, LIMIT, CONTINUITY & DERIVABILITY METHOD OF DIFFERENTIATION INVERSE TRIGONOMETRIC FUNCTION

Definite, Indefinite Integration, MOD & ITF Select the correct alternative : (Only one is correct) Q.1

Q.2

Minimum period of the function, f (x) = | sin32x | + | cos32x | is π π (A) π (B) (C) 2 4 If Lim (x−3 sin 3x + ax−2 + b) exists and is equal to zero then : (B) a = 3 & b = 9/2 (D) a = 3 & b = − 9/2

1 then g ′ (x) = 1+ x 5 1 1 (B) (C) − 5 1 + [g(x)]5 1 + [g(x)]

If g is the inverse of f & f ′ (x) = (A) 1 + [g(x)]5

Q.4

x m sin x1 A function f(x) is defined as f(x) =   0

if x = 0

. The least value of m for which f ′ (x) is

(C) 3

(D) none

The number k is such that tan{arc tan(2) + arc tan(20k )} = k. The sum of all possible values of k is (A) –

Q.6

x ≠ 0, m ∈ N

(D) none

Quest

continuous at x = 0 is (A) 1 (B) 2 Q.5

3π 4

x→0

(A) a = − 3 & b = 9/2 (C) a = − 3 & b = − 9/2 Q.3

(D)

19 40

21 40

(B) –

(C) 0

(D)

1 5

x for 0 ≤ x ≤ 1   x >1 Let f1(x) =  1 for  0 for otherwise and

f2 (x) = f1 (– x) for all x f3 (x) = – f2(x) for all x f4 (x) = f3(– x) for all x Which of the following is necessarily true? (A) f4 (x) = f1 (x) for all x (C) f2 (–x) = f4 (x) for all x

Q.7

(B) f1 (x) = – f3 (–x) for all x (D) f1 (x) + f3 (x) = 0 for all x

dy  3x + 4  =  & f ′ (x) = tan x2 then dx  5x + 6 

If y = f  (A)

 3x + (B) − 2 tan   5x +

tan x3  3 tan x 2 + 4   tan x2  5 tan x 2 + 6 

(C) f 

2

4 1 .  6  (5x + 6) 2

(D) none 1

1

x

Q.8

 2 x n  e −  3x n  e x         The value of Limit n x →∞ x 2 (A) ln   3

(B) 0

( where n∈ N ) is

2 (C) n ln   3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) not defined

[2]

Q.9

Q.10

Which one of the following depicts the graph of an odd function?

(A)

(B)

(C)

(D)

If sinθ =

I. III.

12 5 , cosθ = – , 0 < θ < 2π. Consider the following statements. 13 13

Quest  5 θ = cos–1  −   13 

II.

 12  θ = sin–1    13 

 12  θ = π – sin–1    13 

IV.

 12  θ = tan–1  −   5

 12  θ = π – tan–1   5 then which of the following statements are true? (A) I, II and IV only (B) III and V only (C) I and III only (D) I, III and V only

V.

Q.11

Let g is the inverse function of f & f ′ (x) = 5 (A) 10 2

Q.12

Q.13

1 + a2 (B) a 10

x10

(1 + x ) 2

. If g(2) = a then g ′ (2) is equal to

a 10 (C) 1 + a2

1 + a 10 (D) a2

For a certain value of c, xLim [(x5 + 7x4 + 2)C - x] is finite & non zero. The value of c and the value of →−∞ the limit is (A) 1/5, 7/5

(B) 0, 1

(C) 1, 7/5

(A)

(B)

(C)

(D) none 2 −1 Which one of the following best represents the graph of the function f(x) = Lim tan (nx ) n →∞ π

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

[3]

Q.14

 d   3 d 2y    y . 2  equals :  dx   dx 

If y2 = P(x), is a polynomial of degree 3, then 2  (A) P ′′′ (x) + P ′ (x) ∞

tan −1

∑

(B) P ′′ (x) . P ′′′ (x)

(C) P (x) . P ′′′ (x)

(D) a constant

3 is equal to n + n −1

Q.15

The sum

Q.16

3π π π −1 −1 + cot −1 2 (B) + cot 3 (C) π (D) + tan 2 2 4 2 If f (x) is a diffrentiable function and f ′(2) = 6 , f ′(1) = 4, f ′(x) represents the diffrentiation of f (x)

n =1

2

(A)

Limit

w.r.t. x then

h→ 0

f (2 + 2h + h 2 ) − f (2) = f (1 + h 2 + h ) − f (1)

(A) 3 Q.17

Q.18

(B) 4

(C) 6

cos 2 − cos 2 x = x →−1 x2 − | x | (B) − 2 cos 2 (A) 2 cos 2 Lim

0

if x ≠ 0

if x = 0

(C) 2 sin 2

where g(x) is an even function differentiable at x = 0, passing

(C) is equal to 2

 3x2 − 7 x + The domain of definition of the function , f (x) = arc cos  1 + x2 

integer function, is : (A) (1, 6) Q.20

(D) − 2 sin 2

Quest g (x) . cos x1

Let f(x) = 

through the origin . Then f ′ (0) : (A) is equal to 1 (B) is equal to 0 Q.19

(D) 14

(B) [0, 6)

(D) does not exist

8  where [ *] denotes the greatest 

(C) [0, 1]

(D) (− 2, 5]

The sum of the infinite terms of the series  

3 4

 

3 4

 

3 4

2 2 2 cot −1 1 +  + cot −1  2 +  + cot −1  3 +  + ..... is equal to :

(A) tan–1 (1) Q.21

(B) tan–1 (2)

(C) tan–1 (3)

(D) tan–1 (4)

Let the function f, g and h be defined as follows : 1 x sin    x f (x) =   0

1 x 2 sin   x  g (x) =   0

for − 1 ≤ x ≤ 1 and x ≠ 0 for

x=0

for − 1 ≤ x ≤ 1 and x ≠ 0 for

x=0

h (x) = | x |3 for – 1 ≤ x ≤ 1 Which of these functions are differentiable at x = 0? (A) f and g only (B) f and h only (C) g and h only Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[4]

Q.22

 3x2 + 2 x − 1 for x ≠  Let f (x) =  6 x 2 − 5 x + 1  − 4 for x =

1 3

 1  3

then f ′   :

1 3

(B) is equal to − 27

(A) is equal to − 9

(C) is equal to 27

(D) does not exist

n

Q.23

Q.24

  n α 1 Lim    + sin  when α ∈ Q is equal to n →∞   n + 1  n   (B) – α (C) e1 – α (A) e–α

(D) e1 + α

g (x ) Let f (x) = h ( x ) , where g and h are cotinuous functions on the open interval (a, b). Which of the

following statements is true for a < x < b? (A) f is continuous at all x for which x is not zero. (B) f is continuous at all x for which g (x) = 0 (C) f is continuous at all x for which g (x) is not equal to zero. (D) f is continuous at all x for which h (x) is not equal to zero. Q.25

Quest 1 2

(A) 4 tan−1 (1) Q.26

If y = (A)

Q.27

Q.28

(B) 2 tan−1 (2)

1

1+ x

n− m

+x

emnp

Q.31

(C) 0

1

1+ x

m− n

+x

emn/p

8 8 + 1− x 1+ x (A) periodic with period π/2 (C) periodic with period 2π Given f (x) =

The period of the function f (x) = (B) π/4

p− n

+

1

1+ x

(C)

and

(D) none

m− p

+x

n− p

then

np dy at e m is equal to: dx

enp/m

(D) none

4 4 g (x) = f (sin x ) + f (cos x )

then g(x) is

(B) periodic with period π (D) aperiodic | sin x | + | cos x | is | sin x − cos x | (C) π

(D) 2π

cos x sin x cos x  π Let f(x) = cos 2x sin 2x 2 cos 2x then f ′   = 2 cos 3x sin 3x 3 cos 3x

(A) 0 Q.30

+

p− m

(B)

(A) π/2 Q.29

 

The value of tan−1  tan 2A + tan −1(cot A) + tan −1(cot3A) for 0 < A < (π/4) is

(B) – 12

( (

−1 α = sin −1 cos sin x

))

(C) 4

( (

−1 and β = cos −1 sin cos x

(D) 12

)) , then :

(A) tan α = cot β (B) tan α = − cot β (C) tan α = tan β x x − e + cos 2x If f(x) = , x ≠ 0 is continuous at x = 0, then x2

(D) tan α = − tan β

5 (B) [f(0)] = – 2 (C) {f(0)} = –0.5 (D) [f(0)] . {f(0)} = –1.5 2 where [x] and {x} denotes greatest integer and fractional part function (A) f (0) =

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.32

x + b, x < 0 The function g (x) =  can be made differentiable at x = 0.  cos x , x ≥ 0

(A) if b is equal to zero (C) if b takes any real value Q.33

(B) if b is not equal to zero (D) for no value of b

People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D*f(x) by the formula D*f(x) = Limit h→ 0

f 2 (x + h) − f 2 (x) where f2 (x) means [f(x)]2. If f(x) = x lnx then h

D * f ( x ) x = e has the value

(A) e Q.34

(B) 2e

(C) 4e

Which one of the following statement is meaningless?

  2e + 4     (A) cos−1  ln    3 

π (B) cosec−1   3

Quest

π (C) cot−1   2 Q.35

Limit x→

π 2

2 (A) π Q.36

(D) none

(D) sec−1 (π)

sin x

−1  1

 cos  (3 sin x − sin 3x ) 4 

where [ ] denotes greatest integer function , is

(B) 1

(C)

4 π

(D) does not exist

Which one of the following statement is true? (A) If Lim f ( x ) · g( x ) and Lim f ( x ) exist, then Lim g( x ) exists. x→c

x→c

x→c

(B) If Lim f ( x ) · g( x ) exists, then Lim f ( x ) and Lim g( x ) exist. x→c

x→c

x→c

(C) If Lim (f ( x ) + g( x ) ) and Lim f ( x ) exist, then Lim g( x ) exist. x →c

x→c

x→c

(D) If Lim (f ( x ) + g( x ) ) exists, then Lim f ( x ) and Lim g( x ) exist. x →c

Q.37

x→c

If f(4) = g(4) = 2 ; f ′ (4) = 9 ; g ′ (4) = 6 then Limit x→4 (A) 3 2

Q.38

x→c

f (x) = (A)

3

(B)

f (x) − g (x) x −2

(C) 0

2

is equal to : (D) none

x lnx and g (x) = . Then identify the CORRECT statement lnx x

1 and f (x) are identical functions g(x )

(C) f (x) . g (x) = 1

∀ x >0

N o rth

D e lh i :

(B)

1 and g (x) are identical functions f (x)

(D)

1 = 1 ∀ x>0 f (x ) . g( x )

Quest Tutorials E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[6]

Q.39

Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable? (A) f (x) = x1/3

Q.40

(B) f (x) =  1  2

(B) y = πx

(B) 5f ′ (x)

(C) 0

(B) 20

(D) none

(C) 40

(D) 60

Quest | x|

{e Let f (x) = e

Q.44

If y = x + ex then

sgn x}

| x|

[e and g (x) = e

sgn x ]

, x ∈ R where { x } and [ ] denotes the fractional part and

integral part functions respectively. Also h (x) = ln (f ( x ) ) + ln (g( x ) ) then for all real x, h (x) is (A) an odd function (B) an even function (C) neither an odd nor an even function (D) both odd as well as even function

(A)

ex

d 2x is : dy 2

(B) −

ex

(1+ e ) x

3

(C) −

ex

(1+ e ) x

2

(D)

−1

(1+ e ) x

3

  8π    8π  cos cos −1 cos  + tan −1 tan   has the value equal to  7  7   (A) 1

Let

(C) cos

(B) –1

  f (x) =    

2 x + 23 − x − 6 2 − x − 21− x

π 7

(D) 0

if x > 2 then

x2 − 4 if x < 2 x − 3x − 2 (A) f (2) = 8 ⇒ f is continuous at x = 2 (C) f (2–) ≠ f (2+) ⇒ f is discontinuous Q.47

(D) tan x = (4/3) y

f (x) for all positive real numbers x and y. If f (30) = 20, then the y

Let f be a function satisfying f (xy) =

Q.43

Q.46

(C) tan x = − (4/3) y h

value of f (40) is (A) 15

Q.45

(D) f (x) = tan x

f (x + 3h) − f (x − 2h) If f(x) is a differentiable function of x then Limit = h→0

(A) f ′ (x) Q.42

(C) f (x) = e–x

1 1  1  ; y = cos  cos −1    then : 2  8  2

If x = tan−1 1 − cos−1  −  + sin−1 (A) x = πy

Q.41

|x| x

(B) f (2) = 16 ⇒ f is continuous at x = 2 (D) f has a removable discontinuity at x = 2

Which of the following function is surjective but not injective (A) f : R → R f (x) = x4 + 2x3 – x2 + 1 (B) f : R → R f (x) = x3 + x + 1 (C) f : R → R+ f (x) = 1+ x 2

(D) f : R → R f (x) = x3 + 2x2 – x + 1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[7]

Q.48

If f is twice differentiable such that f ′′ (x) = − f (x), f ′ (x) = g(x)

h ′ (x ) = [ f (x)] + [ g(x )] and h (0 ) = 2 , h (1) = 4 2

then the equation y = h(x) represents : (A) a curve of degree 2 (C) a straight line with slope 2 Q.49

Q.50

2

(B) a curve passing through the origin (D) a straight line with y intercept equal to − 2 .

The graph of function f contains the point P (1, 2) and Q(s, r). The equation of the secant line through

 s 2 + 2s − 3   x – 1 – s. The value of f ' (1), is P and Q is y =   s − 1   (A) 2 (B) 3 (C) 4

(D) non existent

If f (x) = 2x3 + 7x – 5 then f–1(4) is (A) equal to 1 (B) equal to 2

(D) non existent

(C) equal to 1/3

e x ln x 5( x + 2) ( x 2 − 7 x + 10) is Q.51 The range of the function f(x) = 2x 2 −11x + 12 3  (A) ( −∞ , ∞ ) (B) [0 , ∞ ) (C)  , ∞  2  2

Q.52

Q.53

Quest

If f(x) is a twice differentiable function, then between two consecutive roots of the equation f ′ (x) = 0, there exists : (A) atleast one root of f(x) = 0 (B) atmost one root of f(x) = 0 (C) exactly one root of f(x) = 0 (D) atmost one root of f ′′ (x) = 0

(

)(

)

n Limit (1+ x ) 1+ x 2 1+ x 4 ......1+ x 2  if x < 1 has the value equal to : n→∞



(A) 0 Q.54

3  (D)  , 4  2 



(B) 1



(C) 1 − x

(D) (1 − x) −1

2 2 2 2 Lim 1 n + 2 (n−1)+3 (n−2)+.....+n .1 is equal to : n →∞ 13 +23 +33 +......+ n 3

(A)

1 3

(B)

2 3

(C)

1 2

(D)

1 6

Q.55

If x = cos–1 (cos 4) ; y = sin–1 (sin 3) then which of the following holds ? (A) x – y = 1 (B) x + y + 1 = 0 (C) x + 2y = 2 (D) tan (x + y) = – tan7

Q.56

Let f (x) = (A) 4

Q.57

If f (x) = (A) 0

Q.58

tan 6 x + 9 tan 4 x − 9 tan 2 x − 1 , if f ' (x) = λ cosec4(2x) then the value of λ equals 3 tan 3 x (B) 9 (C) 16 (D) 64

x 2 − bx + 25 for x ≠ 5 and f is continuous at x = 5, then f (5) has the value equal to x 2 − 7 x + 10 (B) 5 (C) 10 (D) 25

Let f be a differentiable function on the open interval (a, b). Which of the following statements must be true? I. f is continuous on the closed interval [a, b] II. f is bounded on the open interval (a, b) III. If a 0) then k =

(A) –

1 R2

(C) 0 y ′′

(1 + y ′ )

(B) –

1 R

(D) – 3

where k in terms of R alone is equal to

2 R

(D) –

2 R2

1+ x − tan−1 x is : 1− x

(B) {− (π/4) , 3π/4}

The domain of the function f(x) =

3

(C)

The range of the function, f(x) = tan−1 (A) {π/4}

Q.76

(D) none

1 , which of the following holds? 1 + n sin 2 (πx ) (A) The range of f is a singleton set (B) f is continuous on R (C) f is discontinuous for all x ∈ I (D) f is discontinuous for some x ∈ R

2

Q.75

π 3

For the function f (x) = Lim n →∞

(A) 6 Q.74

b ab + 2 ay

 5−2 6 1  is equal : − tan −1 The value of  tan −1

(A) Q.72

(D)

arc cot x

[ ]

x2 − x2

(C) {π/4 , − (3π/4)}

(D) {3π/4}

, where [x] denotes the greatest integer not greater than x, is (B) R − {0}

(A) R

{

}

(C) R − ± n : n ∈ I + ∪ {0}

(D) R − {n : n ∈ I}

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[10]

Q.77

 1  7π 2π  − sin  cos  is equal to cos–1  5 5   2 (A)

Q.78

23π 20

Given f(x) = −

(B)

13π 20

33π 20

(D)

17π 20

x3 + x2 sin 1.5 a − x sin a . sin 2a − 5 arc sin (a2 − 8a + 17) then : 3

(A) f(x) is not defined at x = sin 8 (C) f ′ (x) is not defined at x = sin 8 Q.79

(C)

(B) f ′ (sin 8) > 0 (D) f ′ (sin 8) < 0

  1 1 + Range of the function f (x) =  is , where [*] denotes the greatest integer 2 1+ x2  ln ( x + e )  (1 + α )1 / α function and e = Limit α →0

 e +1  ∪{2} (A)  0, e   Q.80

Q.83

Q.84

 π 3π  (D)  ,  2 4 

(

)

(B) f ′ (1/2) < 0 (D) f ′ (1/2) > 0

The period of the function f (x) = sin (x + 3 – [x + 3 ] ), where [ ] denotes the greatest integer function is (A) 2π + 3 (B) 2π (C) 1 (D) 3 π Sum of the roots of the equation, arc cot x – arc cot (x + 2) = is 12 (A) 3 (B) 2 (C) – 2 (D) – 3 Which one of the following functions best represent the graph as shown adjacent?

1 1 + x2 (C) f(x) = e–|x|

(B) f(x) =

If y = (A + Bx) emx + (m − 1)−2 ex then

 3 Limit  x→0  1 + 4 +

(A) e −1/12

1

1 + | x| (D) f(x) = a|x| (a > 1)

(A) ex Q.86

 3π  (C)  , π  4

Given: f(x) = 4x3 − 6x2 cos 2a + 3x sin 2a . sin 6a +
n 2 a − a 2 then

(A) f(x) =

Q.85

(D) (0, 1) ∪ {2}

Quest  3π  (B)  0,  4 

(A) f(x) is not defined at x = 1/2 (C) f ′(x) is not defined at x = 1/2 Q.82

(C) (0, 1] ∪ {2}

The range of the function, f(x) = cot–1 log 0.5 (x 4 − 2x 2 + 3) is: (A) (0, π)

Q.81

(B) (0, 1)

(B) emx   x 

d 2y dy + m2y is equal to : 2 − 2m dx dx

(C) e−mx

(D) e(1 − m) x

cos ecx

has the value equal to : (B) e −1/6

(C) e −1/4

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) e −1/3

[11]

Q.87

Q.88

Q.89

Q.90

(

)

cot −1 x + 1 − x is equal to x x →∞  2 x + 1     sec −1    x − 1    (A) 1 (B) 0 Limit

The solution set for [x] {x} = 1 where {x} and [x] are fractional part & integral part of x, is (A) R+ – (0, 1) (B) R+ – {1}

1   (D) m + / m ∈ N − {1} m  

Q.93

Quest (

Limit 1 + (arc cos x )1− x x →1−

(A) 4 Q.92

(B) 2

(C) 1

(B) – 1

(D) 0

(C) 0

The range of values of p for which the equation 1 1  ,  2 2

(B) [0, 1)

(D) non existent

sin cos–1 (cos(tan −1 x) ) = p has a solution is:  1



(C)  , 1  2

(D) (– 1, 1)

Let f(x) be a differentiable function which satisfies the equation f(xy) = f(x) + f(y) for all x > 0, y > 0 then f ′ (x) is equal to (A)

f ' (1) x

(B)

1 x

(C) f ′ (1)

(D) f ′(1).(lnx)

Let ef(x) = ln x . If g(x) is the inverse function of f(x) then g ′ (x) equals to : (A) ex

Q.97

has the value equal to

4 2 1 Lim x sin x + x is equal to x →−∞ 1+ | x |3



Q.96

2

Let n being a non-negative integer . The number of values of n for which f ′ (p + q) = f ′ (p) + f ′ (q) is valid for all p, q > 0 is : (A) 0 (B) 1 (C) 2 (D) none of these

(A)  − Q.95

)

f(x) = xn ,

(A) 1 Q.94

(D) non existent

Suppose f (x) = eax + ebx, where a ≠ b, and that f '' (x) – 2 f ' (x) – 15 f (x) = 0 for all x. Then the product ab is equal to (A) 25 (B) 9 (C) – 15 (D) – 9  x2  There exists a positive real number x satisfying cos(tan–1x) = x. The value of cos–1   is  2  π π 2π 4π (A) (B) (C) (D) 10 5 5 5

1   (C) m + / m ∈ I − {0} m   Q.91

(C) π/2

(B) ex + x

(C) e ( x

+ ex )

(D) e(x + ln x)

The domain of definition of the function : f (x) = ln ( x 2 − 5x − 24 – x – 2) is (A) (–∞, –3]

28   (B) (–∞, –3 ] U [8, ∞) (C)  −∞, −   9

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[12]

Q.98 If f (x) = 2 tan 3x + 5 1 − cos 6x ; g(x) is a function having the same time period as that of f(x), then which

Q.99

of the following can be g(x). (A) (sec23x + cosec23x)tan23x

(B) 2 sin3x + 3cos3x

(C) 2 1 − cos 2 3x + cosec3x

(D) 3 cosec3x + 2 tan3x

(

)

2 cot cot −1 (3) + cot −1 (7) + cot −1 (13) + cot −1 ( 21) has the value equal to (A) 1 (B) 2 (C) 3 (D) 4

dy Q.100 The equation y2exy = 9e–3·x2 defines y as a differentiable function of x. The value of for dx x = – 1 and y = 3 is (A) – 15/2 (B) – 9/5 (C) 3 (D) 15 ln ( x 2 + e x )

. If Limit f(x) = l and Limit f(x) = m then : x→∞ x → −∞ ln ( x 4 + e 2 x ) (A) l = m (B) l = 2m (C) 2 l = m (D) l + m = 0

Q.101 Let f(x) =

Q.102 Which one of the following statements is NOT CORRECT ? (A) The derivative of a diffrentiable periodic function is a periodic function with the same period. (B) If f (x) and g (x) both are defined on the entire number line and are aperiodic then the function F(x) = f (x) . g (x) can not be periodic. (C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function. (D) Every function f (x) can be represented as the sum of an even and an odd function

Quest

Q.103 Lim cos  π n 2 + n  when n is an integer : n →∞   (A) is equal to 1 (B) is equal to − 1 (C) is equal to zero Q.104 The value of tan −1 (where a, b, c > 0) (A) π/4

a (a + b + c ) + tan −1 bc

(B) π/2

b (a + b + c) + tan −1 ca

(C) π

(D) does not exist c (a + b + c) is : ab

(D) 0

Q.105 The function f(x) = ex + x, being differentiable and one to one, has a differentiable inverse f–1(x). The d –1 value of (f ) at the point f(l n2) is dx 1 1 1 (A) (B) (C) (D) none
n 2 3 4 Q.106 Given the graphs of the two functions, y = f(x) & y = g(x). In the adjacent figure from point A on the graph of the function y = f(x) corresponding to the given value of the independent variable (say x0), a straight line is drawn parallel to the X-axis to intersect the bisector of the first and the third quadrants at point B . From the point B a straight line parallel to the Y-axis is drawn to intersect the graph of the function y = g(x) at C. Again a straight line is drawn from the point C parallel to the X-axis, to intersect the line NN ′ at D . If the straight line NN ′ is parallel to Y-axis, then the co-ordinates of the point D are (A) f(x0), g(f(x0)) (B) x0, g(x0) (C) x0, g(f(x0)) (D) f(x0), f(g (x0)) Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[13]

Q.107 A function f : R → R, f(x) =

2x is 1 + x2

(A) injective by not surjective (C) injective as well as surjective [ x ]2 Limit Q.108 Let Limit x→0 2 =l & x→0 x

(B) surjective but not injective (D) neither injective nor surjective

[x2 ] x2

= m , where [ ] denotes greatest integer , then:

(A) l exists but m does not (C) l & m both exist

(B) m exists but l does not (D) neither l nor m exists .

Q.109 Which of the following is the solution set of the equation sin–1x = cos–1x + sin–1(3x – 2)?

1  (A)  , 1 2 

1  (B)  , 1 2 

Q.110 If y is a function of x then

1  (C)  , 1 3 

1  (D)  , 1 3 

dy d2 y = 0 . If x is a function of y then the equation becomes 2 +y dx dx 3

 dx d2 x (B)  =0 2 +y  dy  dy

dx d2 x (A) =0 2 +x dy dy

Quest 2

2

d2 x  dx (C)  =0 2 −y   dy dy

  2 1 + logcos x cos x Q.111 Limit x→0   2

(A) is equal to 4 Q.112 If y =

x4 − x2 +1

x + 3x + 1

(A) cot

2

d2 x  dx (D)  =0 2 −x   dy dy

2

(B) is equal to 9

and

5π 8

5π 12

( tan ( {x } − 1 ) )

(C) tan sin {x }

{x } ( {x } − 1 )

(A) is 1

(D) is non existent

dy = ax + b then the value of a + b is equal to dx

(B) cot

Q.113 The value of Limit x→0

(C) is equal to 289

(B) is tan 1

5π 12

(D) tan

5π 8

where { x } denotes the fractional part function: (C) is sin 1

(D) is non existent

Q.114 If f(x) = cosec–1(cosecx) and cosec(cosec–1x) are equal functions then maximum range of values of x is  π   π (A) − ,−1 ∪ 1,   2   2

 π   π (B) − ,0 ∪ 0 ,   2   2

(C) (− ∞ , − 1]∪ [1, ∞ )

(D) [− 1, 0) ∪ [0 ,1)

Q.115 A function f (x) satisfies the condition, f (x) = f ′ (x) + f ′′ (x) + f ′′′ (x) + ...... ∞ where f (x) is a differentiable function indefinitely and dash denotes the order of derivative . If f (0) = 1, then f (x) is : (A) ex/2 (B) ex (C) e2x (D) e4x Q.116 Let f : R → R

f (x) =

x . Then f (x) is : 1 + |x|

(A) injective but not surjective (C) injective as well as surjective

(B) surjective but not injective (D) neither injective nor surjective .

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[14]

Q.117

The solution set of the equation sin–1 (A) [–1, 1] – {0}

1− x

(B) (0, 1] U {–1}

2

 1 − x2   x  – sin–1x  

+ cos–1x = cot–1 

(C) [–1, 0) U {1}

(D) [–1, 1]

Q.118 Suppose the function f (x) – f (2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The derivative of the function f (x) – f (4x) at x = 1, has the value equal to (A) 19 (B) 9 (C) 17 (D) 14 2 + 2 x + sin 2 x is : (2 x + sin 2 x )esin x (A) equal to zero (B) equal to 1

Q.119 Lim x →∞

−

 e y = f(x) =    0

Q.120 Let

1 x2

(C) equal to − 1

(D) non existent

if x ≠ 0 if x = 0

Then which of the following can best represent the graph of y = f(x) ?

(A)

Quest (B)

Q.121 The value of

(A)

e

 8b2  −     a2 

lim x→ 0

(C)

(D)

( cos ax) cos ec bx is 2

(B)

e

 8a 2  −     b2 

(C)

e

 a2  −     2 b2 

(D)

e

 b2  −   2  2a 

x 1  π 3 − 3x 2  = holds good is Q.122 The set of values of x for which the equation cos–1x + cos–1  + 2 2  3 (A) [0, 1] Q.123 Limit x → 0+

 1 (B) 0,   2

1  (C)  , 1 2 

(D) {–1, 0, 1}

 x x  a arc tan  has the value equal to − b arc tan a b  x x  1

a−b (a 2 − b 2 ) a 2 − b2 (B) 0 (C) (D) (A) 3 3a 2 b2 6a 2 b 2 Q.124 If f (x) is a function from R → R, we say that f (x) has property I if f (f (x) ) = x for all real number x, and we say that f (x) has property II if f (–f(x)) = – x for all real number x. How many linear functions, have both property I and II? (A) exactly one (B) exactly two (C) exactly three (D) infinite 1

1

1


+ m  n −
 m+ n 
− m  n +
 m − n m− n  .  x n−
 . x
− m Q.125 Diffrential coefficient of  x            (A) 1

(B) 0

(C) – 1

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

w.r.t. x is (D)

x
mn [15]

x rx . Let S be the set of all real numbers r such that f (g(x)) = g (f (x)) and let g(x)= 1+ x 1− x for infinitely many real number x. The number of elements in set S is (A) 1 (B) 2 (C) 3 (D) 5

Q.126 Let f (x) =

Q.127 Let f (x) be a linear function with the properties that f (1) ≤ f (2), f (3) ≥ f (4), and f (5) = 5. Which of the following statements is true? (A) f (0) < 0 (B) f (0) = 0 (C) f (1) < f (0) < f (–1) (D) f (0) = 5 Q.128 Let f (x) be diffrentiable at x = h then Lim x→ h (A) f(h) + 2hf '(h)

(B) 2 f(h) + hf '(h)

b x + hg f ( x)

− 2 h f ( h) is equal to x−h

(C) hf(h) + 2f '(h)

(D) hf(h) – 2f '(h)

2m (n ! π x)] is given by Limit Q.129 If x is a real number in [0, 1] then the value of Limit m → ∞ n → ∞ [1 + cos

(A) 1 or 2 according as x is rational or irrational (B) 2 or 1 according as x is rational or irrational (C) 1 for all x (D) 2 for all x .

Quest

Q.130 If y = at2 + 2bt + c and t = ax2 + 2bx + c, then (A) 24 a2 (at + b)

(B) 24 a (ax + b)2

d 3y equals dx 3

(C) 24 a (at + b)2

(D) 24 a2 (ax + b)

Direction for Q.131 and Q.132 The graph of a relation is (i) Symmetric with respect to the x-axis provided that whenever (a, b) is a point on the graph, so is (a, – b) (ii) Symmetric with respect to the y-axis provided that whenever (a, b) is a point on the graph, so is (– a, b) (iii) Symmetric with respect to the origin provided that whenever (a, b) is a point on the graph, so is (– a, – b) (iv) Symmetric with respect to the line y = x, provided that whenever (a, b) is a point on the graph, so is (b, a) Q.131 The graph of the relation x4 + y3 = 1 is symmetric with respect to (A) the x-axis (B) the y-axis (C) the origin (E) both the x-axis and y-axis

(D) the line y = x

Q.132 Suppose R is a relation whose graph is symmetric to both the x-axis and y-axis, and that the point (1, 2) is on the graph of R. Which one of the following points is NOT necessarily on the graph of R? (A) (–1, 2) (B) (1, – 2) (C) (–1, –2) (D) (2, 1) (E) all of these points are on the graph of R.

Select the correct alternatives : (More than one are correct) Q.133 If y = tan x tan 2x tan 3x then

dy has the value equal to : dx

(A) 3 sec2 3x tan x tan 2x + sec2 x tan 2x tan 3x + 2 sec2 2x tan 3x tan x (B) 2y (cosec 2x + 2 cosec 4x + 3 cosec 6x) (C) 3 sec2 3x − 2 sec2 2x − sec2 x (D) sec2 x + 2 sec2 2x + 3 sec2 3x Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[16]

Q.134 Lim f(x) does not exist when : x →c

(A) f(x) = [[x]] − [2x − 1], c = 3

(B) f(x) = [x] − x, c = 1

(C) f(x) = {x}2 − {−x}2, c = 0

(D) f(x) =

tan (sgn x) ,c =0. sgn x

where [x] denotes step up function & {x} fractional part function.   Q.135 Let f (x) =   

tan2 {x } x2 − [ x ]2 1

for x > 0 for x = 0

where [ x ] is the step up function and { x } is the fractional

{x } cot {x } for x < 0

part function of x , then : (A) xLimit f (x) = 1 → 0+ (C)

cot -1

(B) xLimit f (x) = 1 → 0−

   Limit f (x)  x → 0− 

2

=1

(D) f is continuous at x = 1 .

Quest

Q.136 Which of the following function (s) is/are Transcidental?

2 sin 3x x + 2x − 1

(A) f (x) = 5 sin x

(B) f (x) =

(C) f (x) =

(D) f (x) = (x2 + 3).2x

Q.137 If

2x +

x2 + 2x + 1

2y = 2x + y then

2y (A) − x 2

2

dy has the value equal to : dx

1 (B) 1 − 2x

(C) 1 −

2y

( (2

) − 1)

2x 1 − 2 y

(D)

2y

x

  −1  2 x 3 − 3   3 −  cot  x 2   for x > 0 Q.138 Given f(x) =  where { } & [ ] denotes the fractional part and the 2 1/ x x cos e for x < 0 

( )

{ }

integral part functions respectively, then which of the following statement does not hold good. −) = 0 (B) f(0+) = 3 (C) f(0) = 0 ⇒ continuity of f at x = 0 (D) irremovable discontinuity of f at x = 0

( A

)

f ( 0

Q.139 The graphs of which of the following pairs differ . (A) y =

sin x 1 + tan x 2

+

cos x 1 + cot 2 x

; y = sin 2x

(B) y = tan x cot x ; y = sin x cosec x (C) y = cos x + sin x ; y =

sec x + cos ecx sec x cos ecx

(D) none of these

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[17]

1

 14 π       is : 5  

 

Q.140 The value of cos  cos −1  cos  −  2

 π   10 

 7π   5

(A) cos  −

 2π (C) cos    5

(B) sin 

Q.141 Which of the following functions are homogeneous ? (A) x sin y + y sin x (B) x ey/x + y ex/y (C) x2 − xy Q.142 Let f (x) =

x −2 x −1 x −1 −1

 3π    5

(D) − cos 

(D) arc sin xy

. x then :

(A) f ′ (10) = 1 (C) domain of f (x) is x ≥ 1

(B) f ′ (3/2) = − 1 (D) none

Q.143 If θ is small & positive number then which of the following is/are correct ? (A)

sinθ =1 θ

(B) θ < sin θ < tan θ

(C) sin θ < θ < tan θ

(D)

tanθ sinθ > θ θ

Quest

x −1 . Then : 2 x − 7x + 5 1 1 f(x) = − (B) Limit f(x) = − (A) Limit x→1 x → 0 5 3

Q.144 Let f(x) =

2

(C) Limit x → ∞ f(x) = 0

(D) Limit does not exist x → 5/ 2

Q.145 If f(x) is a polynomial function satisfying the condition f(x) . f(1/x) = f(x) + f(1/x) and f(2) = 9 then : (A) 2 f(4) = 3 f(6) (B) 14 f(1) = f(3) (C) 9 f(3) = 2 f(5) (D) f(10) = f(11) Q.146 Two functions f & g have first & second derivatives at x = 0 & satisfy the relations, f(0) =

2 , f ′ (0) = 2 g ′ (0) = 4g (0) , g ′′ (0) = 5 f ′′ (0) = 6 f(0) = 3 then : g(0)

(A) if h(x) =

15 f (x) then h ′ (0) = 4 g(x)

(B) if k(x) = f(x) . g(x) sin x then k ′ (0) = 2

1 g′ (x) (C) Limit = x→0 f ′ (x)

(D) none

2

Q.147 Which of the following function(s) not defined at x = 0 has/have removable discontinuity at x = 0 ? (A) f(x) =

π (B) f(x)=cos  | sin x |  (C) f(x) = x sin x  x 

1 1 + 2cot x

(D) f(x) =

1
n x

Q.148 For the equation 2x = tan(2tan–1a) + 2tan(tan–1a + tan–1a3), which of the following is invalid? (A) a2x + 2a = x (B) a2 + 2ax + 1 = 0 (C) a ≠ 0 (D) a ≠ –1, 1  x−3

( )−( )+( )

Q.149 The function f(x) =  

2

x 4

(A) continuous at x = 1 (C) continuous at x = 3

3x 2

13 4

, x ≥1 , x 6  y  x  z 1 ln x

Q.157 The function f(x) = x (A) is a constant function (C) is such that Lim f(x) exist x →1

(B) has a domain (0, 1) U (e, ∞) (D) is aperiodic

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[19]

ANSWER KEY Q.1

C

Q.2

A

Q.3

A

Q.4

C

Q.5

A

Q.6

B

Q.7

B

Q.8

B

Q.9

D

Q.10

D

Q.11

B

Q.12

A

Q.13

A

Q.14

C

Q.15

A

Q.16

A

Q.17

C

Q.18

B

Q.19

A

Q.20

B

Q.21

C

Q.22

B

Q.23

C

Q.24

D

Q.25

A

Q.26

D

Q.27

A

Q.28

C

Q.29

C

Q.30

A

Q.31

D

Q.32

D

Q.33

C

Q.34

A

Q.35

A

Q.36

C

Q.37

A

Q.38

A

Q.39

A

Q.40

C

Q.41

B

Q.42

A

Q.43

A

Q.44

B

Q.45

B

Q.46

C

Q.47

D

Q.48

C

Q.49

C

Q.50

A

Q.51

A

Q.52

B

Q.53

D

Q.54

A

Q.55

D

Q.56

C

Q.57

A

Q.58

D

Q.59

C

Q.60

A

Q.61

C

Q.62

A

Q.63

C

Q.64

B

Q.65

A

Q.66

D

Q.67

D

Q.68

D

Q.69

C

Q.70

D

Q.71

A

Q.78

D

Q.85

A

Q.92

C

Q.99

C

Quest Q.72

C

Q.73

D

Q.74

B

Q.75

C

Q.76

C

Q.77

D

Q.79

D

Q.80

C

Q.81

D

Q.82

C

Q.83

C

Q.84

C

Q.86

A

Q.87

A

Q.88

C

Q.89

C

Q.90

D

Q.91

A

Q.93

B

Q.94

B

Q.95

A

Q.96

C

Q.97

A

Q.98

A

Q.100 D

Q.101 A

Q.102 B

Q.103 C

Q.104 C

Q.105 B

Q.107 D

Q.108 B

Q.109 A

Q.110 C

Q.111 C

Q.112 B

Q.113 D

Q.114 A

Q.115 A

Q.116 A

Q.117 C

Q.118 A

Q.119 D

Q.120 C

Q.121 C

Q.122 C

Q.123 D

Q.124 B

Q.125 B

Q.126 B

Q.127 D

Q.128 A

Q.129 B

Q.130 D

Q.131 B

Q.132 D

Q.106 C

Q.133 A,B,C

Q.134 B,C

Q.135 A,C

Q.136 A,B,D

Q.137 A,B,C,D

Q.138 B,D

Q.139 A,B,C

Q.140 B,C,D

Q.141 B,C

Q.142 A,B

Q.143 C,D

Q.144 A,B,C,D

Q.145 B,C

Q.146 A,B,C

Q.147 B,C,D

Q.148 B,C

Q.149 A,B,C

Q.150 A,B,C,D

Q.151 B,D

Q.152 A,B,C,D

Q.153 A,B,C,D

Q.154 B,C

Q.155 A,B,C

Q.156 A,B

Q.157 A,C

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[20]

TARGET IIT JEE

MATHEMATICS

FUNCTIONS AND INVERSE TRIGONOMETRIC FUNCTION

There are 95 questions in this question bank. Only one alternative is correct. Q.1 Let f be a real valued function such that  2002   = 3x f (x) + 2f   x  for all x > 0. Find f (2). (A) 1000 (B) 2000

(C) 3000

(D) 4000

Q.2

Solution set of the equation , cos−1 x – sin−1 x = cos−1(x 3 ) (A) is a unit set (B) consists of two elements (C) consists of three elements (D) is a void set

Q.3

If f (x) = 2 tan 3x + 5 1 − cos 6x ; g(x) is a function having the same time period as that of f(x), then which

Q.4

of the following can be g(x). (A) (sec23x + cosec23x)tan23x

(B) 2 sin3x + 3cos3x

(C) 2 1 − cos 2 3x + cosec3x

(D) 3 cosec3x + 2 tan3x

Quest

Which one of the following depicts the graph of an odd function?

(A)

(B)

(C)

Q.5

(D)

The sum of the infinite terms of the series  

3 4

 

3 4

 

3 4

2 2 2 cot −1 1 +  + cot −1  2 +  + cot −1  3 +  + ..... is equal to :

(A) tan–1 (1)

(B) tan–1 (2)

(C) tan–1 (3)

(D) tan–1 (4)

Q.6

Domain of definition of the function f (x) = log 10·3x − 2 − 9 x −1 − 1 + cos−1 (1 − x ) is (A) [0, 1] (B) [1, 2] (C) (0, 2) (D) (0, 1)

Q.7

The value of tan−1  tan 2A + tan −1(cot A) + tan −1(cot3A) for 0 < A < (π/4) is

1 2

(A) 4 tan−1 (1)

 

(B) 2 tan−1 (2)

(C) 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

Q.8

Let

f (x)

=

g(x)

=

max.{sin t : 0 ≤ t ≤ x}

min. {sin t : 0 ≤ t ≤ x}

[f (x) − g(x)]

and h(x) =

where [ ] denotes greatest integer function, then the range of h(x) is (A) {0, 1} (B) {1, 2} (C) {0, 1, 2} (D) {−3, −2, −1, 0, 1, 2, 3} Q.9

 πx   πx  The period of the function f(x) = sin 2πx + sin   + sin   is  3  5 (A) 2

Q.10

(B) 6

10 π 9

(B) sec

π 9

Q.14

Quest ( (

−1 α = sin −1 cos sin x

( (

(B) tan α = − cot β and

 1  2

If x = tan−1 1 − cos−1  −  + sin−1 (B) y = πx

8  where [ *] denotes the 

(D) (− 2, 5]

(C) [0, 1]

−1 and β = cos −1 sin cos x

8 8 + 1− x 1+ x (A) periodic with period π/2 (C) periodic with period 2π

)) , then :

(C) tan α = tan β

4 4 g (x) = f (sin x ) + f (cos x )

(D) tan α = − tan β then g(x) is

(B) periodic with period π (D) aperiodic 1 1  1  ; y = cos  cos −1    then : 2  8  2

(C) tan x = − (4/3) y

(D) tan x = (4/3) y

In the square ABCD with side AB = 2 , two points M & N are on the adjacent sides of the square such that MN is parallel to the diagonal BD. If x is the distance of MN from the vertex A and f (x) = Area (∆ AMN) , then range of f (x) is :

(

(A) 0 , 2 Q.16

))

Given f (x) =

(A) x = πy Q.15

(D) –1

 3x2 − 7 x + The domain of definition of the function , f (x) = arc cos  1 + x2 

(A) tan α = cot β Q.13

is equal to

(C) 1

greatest integer function, is : (A) (1, 6) (B) [0, 6) Q.12

(D) 30

 −1  50 π   31π   −1 The value of sec sin  − sin  + cos cos  −    9   9   (A) sec

Q.11

(C) 15

]

(B) (0 , 2 ]

(

(C) 0 , 2 2

]

(

(D) 0 , 2 3

  8π   8π   cos cos −1 cos  + tan −1 tan   has the value equal to  7  7  

(A) 1

(B) –1

(C) cos

π 7

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 0

]

Q.17

 x − 5  + [ log10 (6 − x)] −1 is :  2 

The domain of the definition of the function f(x) = sin−1 

(B) (− 7, − 3) ∪ (− 3, 7) (D) (− 3, 3) ∪ (5, 6)

(A) (7, 7) (C) [− 7, − 3] ∪ [3, 5) ∪ (5, 6)

−1

Q.18

  π 1 −1  a    π 1 −1  a   The value of  tan  + sin    + tan  − sin    , where (0 < a < b), is b  b  4 2  4 2

(A) Q.19

b 2a

(B)

(C)

Let f be a function satisfying f (xy) = value of f (40) is (A) 15

Q.20

a 2b

b2 − a 2 2b

(D)

b2 − a 2 2a

f (x) for all positive real numbers x and y. If f (30) = 20, then the y

(B) 20

(C) 40

(D) 60

Number of real value of x satisfying the equation, arc tan x(x + 1) + arc sin x(x + 1) + 1 = (A) 0

Quest (B) 1

(C) 2

π is 2

(D) more than 2

Q.21

Let f (x) = sin2x + cos4x + 2 and g (x) = cos (cos x) + cos (sin x) also let period of f (x) and g (x) be T1 and T2 respectively then (A) T1 = 2T2 (B) 2T1 = T2 (C) T1 = T2 (D) T1 = 4T2

Q.22

Number of solutions of the equation 2 cot–12 + cos–1(3/5) = cosec–1 x is (A) 0 (B) 1 (C) 2 (D) more than 2

Q.23

The domain of definition of the function : f (x) = ln ( x 2 − 5x − 24 – x – 2) is (A) (–∞, –3]

Q.24

28   (B) (–∞, –3 ]∪[8, ∞) (C)  −∞, −   9

x  The period of the function f(x) = sin  cos  + cos(sinx) equal  2 π (A) (B) 2π (C) π 2

(D) none

(D) 4π

Q.25

If x = cos–1 (cos 4) ; y = sin–1 (sin 3) then which of the following holds ? (A) x – y = 1 (B) x + y + 1 = 0 (C) x + 2y = 2 (D) tan (x + y) = – tan7

Q.26

and g (x) = e , x ∈ R where { x } and [ ] denotes the fractional part and Let f (x) = e integral part functions respectively. Also h (x) = ln (f ( x ) ) + ln (g( x ) ) then for all real x, h (x) is (A) an odd function (B) an even function (C) neither an odd nor an even function (D) both odd as well as even function

Q.27

{e| x| sgn x}

[ e| x| sgn x ]

 x  x  The number of solutions of the equation tan–1  3  + tan–1   = tan–1 x  2 (A) 3 (B) 2 (C) 1 (D) 0 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

is

Q.28

Which of the following is the solution set of the equation (A) (0, 1)

Q.29

(B) (–1, 1) – {0}

Q.30

(C) (–1, 0)

  ? 

1 and that f (2) = 5 and f (9 4) = 2 then 2

Suppose that f is a periodic function with period f (–3) – f (1 4) has the value equal to (A) 2 (B) 3

 2x 2 − 1   2x 1 − x 2  (D) [–1, 1]

2 cos–1 x = cot –1

(C) 5

(D) 7

 5−2 6 1  is equal : − tan −1 The value of  tan −1  

(A)

1+ 6  

2

π 6

(B)

π 4

(C)

π 3

(D) none

Q.31

Given f (x) = (x+1)C(2x– 8) ; g (x) = (2x – 8) C(x + 1) and h (x) = f (x) . g (x) , then which of the following holds ? (A) The domain of 'h' is φ (B) The range of 'h' is {– 1} (C) The domain of 'h' is {x / x > 3 or x < – 3 ; x∈ I (D) The range of 'h' is {1}

Q.32

The sum

Quest ∞

∑

n =1

tan −1

1 2 (A) tan−1 + tan−1 2 3 Q.33

4n is equal to : n − 2 n2 + 2 4

(B) 4 tan −1 1

Range of the function f (x) = tan–1

(C)

[ x ] + [ − x] +

where [*] is the greatest integer function. (A) Q.34

Q.35

LM 1 , ∞IJ N4 K

RS 1 UV ∪ T4 W

2 ,∞

g

(C)

(D) sec −1 − 2

2 − |x| +

)

1 is x2

RS 1 , 2UV T4 W

(D)

LM 1 , 2OP N4 Q

Let [x] denote the greatest integer in x . Then in the interval [0, 3] the number of solutions of the equation, x2 − 3x + [x] = 0 is : (A) 6 (B) 4 (C) 2 (D) 0 The range of values of p for which the equation, sin cos–1 (cos(tan −1 x ) ) = p has a solution is: 

(A)  − Q.36

(B)

(

π 2

1 1  ,  2 2

0  Let f (x) =   x

(B) [0, 1) if x is rational

 1

0  and g (x) =   x if x is irrational

Then the function (f – g) x is (A) odd (C) neither odd nor even



(C)  , 1  2

(D) (– 1, 1) if x is irrational if x is rational

(B) even (D) odd as well as even

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.37

5  12  π Number of value of x satisfying the equation sin–1   + sin–1   = is x x 2 (A) 0 (B) 1 (C) 2 (D) more than 2

Q.38

Consider a real valued function f(x) such that

 a+b  is satisfied are f (a) + f (b) = f   1 + ab  (A) a ∈ (–∞, 1); b ∈ R (C) a ∈ (–1, 1) ; b ∈ [–1, 1] Q.39

(

Q.43

−1

Q.45

−1

(

(C) 3 + 10

)

(

(D) 10 + 3

)

Quest 1 2

(C) 2π

(D) none of these

(B) ±

2

(C) ± 4 2

(D) ± 2

Which of the following is true for a real valued function y = f (x) , defined on [ – a , a]? (A) f (x) can be expressed as a sum or a difference of two even functions (B) f (x) can be expressed as a sum or a difference of two odd functions (C) f (x) can be expressed as a sum or a difference of an odd and an even function (D) f (x) can never be expressed as a sum or a difference of an odd and an even function cos  2 tan −1  1   equals  7   (B) sin(3cot–14)

(C) cos(3cot–14)

(D) cos(4cot–13)

Let f(x) = sin [a ] x (where [ ] denotes the greatest integer function) . If f is periodic with fundamental period π, then a belongs to : (A) [2, 3) (B) {4, 5} (C) [4, 5] (D) [4, 5) The range of the function, f(x) = cot–1 log 0.5 (x 4 − 2x 2 + 3) is: (A) (0, π)

Q.46

)

(B) π 2

(A) sin (4cot–13) Q.44

(

(B) 10 + 3

2 2  4  The real values of x satisfying tan–1  x +  – tan–1   – tan–1  x −  = 0 are x x  x  (A) ±

Q.42

)

The period of the function cos 2 x + cos 2x is : (A) π

Q.41

(B) a ∈ (– ∞, 1); b ∈ (–1, ∞) (D) a ∈ (–1, 1); b ∈ (–1, 1)

1  −1 The value of tan  cot (3)  equals 2  (A) 3 + 10

Q.40

1 − e f (x ) = x. The values of 'a' and 'b' for which 1 + e f (x)

 3π  (B)  0,  4 

 3π  (C)  , π  4

 π 3π  (D)  ,  2 4 

Which of the following is the solution set of the equation sin–1x = cos–1x + sin–1(3x – 2)?

1  (A)  , 1 2 

1  (B)  , 1 2 

1  (C)  , 1 3 

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

1  (D)  , 1 3 

Q.47

Which of the following functions are not homogeneous ? (A) x + y cos

Q.48

y x

xy x + y2

(C)

 1 (B) − ,  3

1 3 

(D)

1  (C)  , 1 3 

The function f : R → R, defined as f(x) =

x  y y  x ln   + ln   y  x x  y

1  (D)  , 1 2 

x 2 − 6 x + 10 is : 3x − 3 − x 2

(A) injective but not surjective (C) injective as well as surjective Q.50

x + y cos x y sin x + y

Which of the following is the solution set of the equation 3cos–1x = cos–1(4x3 – 3x)? (A) [–1, 1]

Q.49

(B)

(B) surjective but not injective (D) neither injective nor surjective

The solution of the equation 2cos–1x = sin–1 (2x 1 − x 2 ) (A) [–1, 0]

(B) [0, 1]

 1  ,1 (D)   2 

(C) [–1, 1]

Quest

Q.51

The period of the function f (x) = sin(x + 3 – [x + 3 ] ), where [ ] denotes the greatest integer function is (B) 2π (C) 1 (D) 3 (A) 2π + 3

Q.52

If tan–1x + tan–1 2x + tan–13x = π, then (A) x = 0 (B) x = 1

Q.53

If f(x + ay, x − ay) = axy then f(x, y) is equal to : x2 − y2 (A) 4

Q.54

Q.55

(D) x ∈ φ

(C) x = –1

(B)

x2 + y2 4

(C) 4 xy

(D) none

x 1  π 3 − 3x 2  = holds good is The set of values of x for which the equation cos–1x + cos–1  + 2 2  3 1   1 (A) [0, 1] (B) 0,  (C)  ,1 (D) {–1, 0, 1} 2   2 The range of the function y =

8 is 9 − x2

8  (A) (– ∞, ∞) – {± 3} (B)  , ∞  9 

 8 (C)  0,   9

8  (D) (– ∞, 0)∪  , ∞  9 

   tan 2 x  cot 2 x  + log 1   is Q.56 The domain of definition of the function f (x) = log 1  2 cos ec 2 x + 5  3 sec 2 x + 5  2 2 (A) R – {nπ, n ∈ I} (C) R – {nπ, (2n + 1)

Q.57

(B) R – {(2n + 1)

π , n ∈ I} 2

The solution set of the equation sin–1 (A) [–1, 1] – {0}

π , n ∈ I} 2

(D) none

1 − x2

(B) (0, 1] U {–1}

 1 − x2   x  – sin–1x  

+ cos–1x = cot–1 

(C) [–1, 0) U {1}

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) [–1, 1]

Q.58

Given the graphs of the two functions, y = f(x) & y = g(x). In the adjacent figure from point A on the graph of the function y = f(x) corresponding to the given value of the independent variable (say x0), a straight line is drawn parallel to the X-axis to intersect the bisector of the first and the third quadrants at point B . From the point B a straight line parallel to the Y-axis is drawn to intersect the graph of the function y = g(x) at C. Again a straight line is drawn from the point C parallel to the X-axis, to intersect the line NN ′ at D . If the straight line NN ′ is parallel to Y-axis, then the co-ordinates of the point D are (A) f(x0), g(f(x0)) (B) x0, g(x0) (C) x0, g(f(x0)) (D) f(x0), f(g (x0))

Q.59

The value of sin–1(sin(2cot–1( 2 – 1))) is (A) –

π 4

(B)

π 4

(C)

3π 4

(D)

7π 4

Q.60

The function f : [2, ∞) → Y defined by f(x) = x2 − 4x + 5 is both one-one and onto if : (A) Y = R (B) Y = [1, ∞) (C) Y = [4, ∞) (D) [5, ∞)

Q.61

If f(x) = cosec–1(cosecx) and cosec(cosec–1x) are equal functions then maximum range of values of x is

Q.62

Quest

 π   π (A) − ,−1 ∪ 1,   2   2

 π   π (B) − ,0 ∪ 0 ,   2   2

(C) (− ∞ , − 1]∪ [1, ∞ )

(D) [− 1, 0) ∪ [0 ,1)

If 2 f(x2) + 3 f(1/x2) = x2 − 1 (x ≠ 0) then f(x2) is : 1 − x4 (A) 5 x2

Q.63

3

Range of the function f (x) = (A) [0 , 1)

Q.65

Q.66

π 12

is

(C) – 2

(D) –

1 (B) 0,   2

 1 (C) 0,   2

Number of solution(s) of the equation cos–1(1 – x) – 2cos–1x = (B) 2

3

{x} where {x} denotes the fractional part function is 1 + {x}  1 (D)  0,   2

Range of the function sgn [ ln (x2 – x + 1) ] is (A) {–1, 0, 1} (B) {–1, 0} (C) – {1}

(A) 3 Q.67

(B) 2

2 x 4 + x2 − 3 (D) − 5 x2

5 x2 (C) 1 − x4

Sum of the roots of the equation, arc cot x – arc cot (x + 2) = (A)

Q.64

1 − x2 (B) 5x

(D) {–1, 1} π is 2

(C) 1

(D) 0

Let f (x) and g (x) be functions which take integers as arguments. Let f (x + y) = f (x) + g (y) + 8 for all integer x and y. Let f (x) = x for all negative integers x, and let g (8) = 17. The value of f (0) is (A) 17 (B) 9 (C) 25 (D) – 17

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.68

T

h e r e

(A) Q.69

e x

i s t s

a

p

o

s i t i v

e

π 10

r e a l

n

(B)

u

m

b

e r

x

s a t i s f y i n g

π 5

(C)

2π 5

(D)

4π 5

x 2 + 2 x −3

 1  2

 

1 2

 

(B) [1, 3] 3 2

 

 1 1  2 2

(C)  , 1 ∪  , ∞

1 2

 

3 2

 

(D)  − ,  ∪  , 1 ∪  , ∞

 1  7π 2π  − sin  is equal to  cos cos–1  5 5   2 (A)

Q.71

s ( t a n

 x2    2  is  

cos–1 

The domain of the function, f(x) = (x + 0.5)log0.5 + x 4 x 2 − 4 x −3 is : (A)  − , ∞

Q.70

c o

–1x) = x. The value of

23π 20

(B)

13π 20

(C)

33π 20

(D)

17π 20

Quest

Let f (x) be a function with two properties (a) for any two real number x and y, f (x + y) = x + f (y) and (b) f (0) = 2. The value of f (100), is (A) 2 (B) 98 (C) 102

(D) 100

Q.72

Let f be a function such that f (3) = 1 and f (3x) = x + f (3x – 3) for all x. Then the value of f (300) is (A) 5050 (B) 4950 (C) 5151 (D) none

Q.73

If f (x) is an invertible function, and g (x) = 2 f (x) + 5, then the value of g–1(x), is (A) 2 f –1(x) – 5

Q.74 Q.75

(B)

1 2 f −1 ( x ) + 5

(C)

1 −1 f (x ) + 5 2

If f (2x + 1) = 4x2 + 14x, then the sum of the roots of f (x) = 0, is (A) 9/4 (B) 5 (C) – 9/4

−1  x − 5   (D) f   2 

(D) – 5

If y = f (x) is a one-one function and (5, 1) is a point on its graph, which one of the following statements is correct? (A) (1, 5) is a point on the graph of the inverse function y = f –1(x) (B) f (5) = f (1) (C) the graph of the inverse function y = f –1(x) will be symmetric about the y-axis

(

)

(D) f f −1 (5) = 1 Q.76

Domain of definition of the function f (x) = (A) (– ∞, 0] (C) (– ∞, –1) ∪ [0, 4)

Q.77

3x − 4 x is x 2 − 3x − 4 (B) [0, ∞) (D) (– ∞, 1) ∪ (1, 4)

Suppose f and g are both linear functions, with f (x) = – 2x + 1 and f ( g ( x ) ) = x. The sum of the slope and the y-intercept of g, is (A) – 2 (B) – 1 (C) 0 (D) 1 Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.78

 1 (A)  0,   3 Q.79

x +4 −3 is x −5

The range of the function f (x) =

 1   1 1 (B)  0,  ∪  ,   6   6 3

(C) (– ∞, 0) ∪ (0, ∞) (D) (0, ∞)

If f (x, y) = (max(x, y) )min( x , y ) and g (x, y) = max(x, y) – min(x, y), then

   3 f  g  − 1, − , g (−4, − 1.75)  equals 2    (A) – 0.5 (B) 0.5 (C) 1 Q.80

(D) 1.5

The domain and range of the function f(x) = cosec–1 log 3− 4 sec x 2 are respectively 1− 2 sec x

 π π (A) R ;  − ,   2 2

 π (B) R+ ;  0,   2

π π   π (C)  2nπ − ,2nπ +  − {2nπ};  0,  2 2   2

π π   π π (D)  2nπ − ,2nπ +  − {2nπ};  − ,  − {0} 2 2   2 2

Quest

More than one alternatives are correct. Q.81

The values of x in [–2π, 2π], for which the graph of the function y = y=–

1 + sin x – secx and 1 − sin x

1 − sin x + secx, coincide are 1 + sin x

3π   3π   (A) −2π, −  ∪  , 2 π 2 2  

π   π 3π   3π (B)  − , −  ∪  ,   2 2  2 2 

 π π (C)  − ,   2 2

3π   π (D) [–2π, 2π] – ± , ±  2  2

Q.82

sin-1(sin3) + sin-1 (sin4) + sin-1(sin5) when simplified reduces to (A) an irrational number (B) a rational number (C) an even prime (D) a negative integer

Q.83

The graphs of which of the following pairs differ . (A) y =

sin x 1 + tan 2 x

+

cos x

; y = sin 2x

1 + cot 2 x

(B) y = tan x cot x ; y = sin x cosec x sec x + cos ecx

(C) y = cos x + sin x ; y =

sec x cos ecx

(D) none of these Q.84

1



 1











If f(x) = cos  π 2  x + sin  − π 2  x , [x] denoting the greatest integer function, then 2 2 (A) f (0) = 1

 π  3

(B) f   =

1 3 +1

 π  2

(C) f   = 0

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) f(π) = 0

Q.85

1 2

 π   10 

 7π   5

(A) cos  − Q.86

 14 π       is : 5  

 

The value of cos  cos −1  cos  −  (B) sin 

The functions which are aperiodic are : (A) y = [x + 1] (B) y = sin x2 where [x] denotes greatest integer function

 3π    5

 2π (C) cos    5

(D) − cos 

(C) y = sin2 x

(D) y = sin−1 x

Q.87

tan−1 x , tan−1 y , tan−1 z are in A.P. and x , y , z are also in A.P. (y ≠ 0 , 1 , − 1) then (A) x , y , z are in G.P. (B) x , y , z are in H.P. (C) x = y = z (D) (x − y)2 + (y − z)2 + (z − x)2 = 0

Q.88

Which of the following function(s) is/are periodic with period π . (A) f(x) = sin x (B) f(x) = [x + π] (C) f(x) = cos (sin x) (where [ . ] denotes the greatest integer function)

(D) f(x) = cos2x

Q.89

For the equation 2x = tan(2tan–1a) + 2tan(tan–1a + tan–1a3), which of the following is invalid? (A) a2x + 2a = x (B) a2 + 2ax + 1 = 0 (C) a ≠ 0 (D) a ≠ –1, 1

Q.90

Which of the functions defined below are one-one function(s) ? (A) f(x) = (x + 1) , ( x ≥ − 1) (B) g(x) = x + (1/x) ( x > 0) 2 (C) h(x) = x + 4x − 5, (x > 0) (D) f(x) = e −x, ( x ≥ 0)

Q.91

If cos–1x + cos–1y + cos–1z = π, then (A) x2 + y2 + z2 + 2xyz = 1 (B) 2(sin–1x + sin–1y + sin–1z) = cos–1x + cos–1y + cos–1z (C) xy + yz + zx = x + y + z – 1

Quest

1  1  1  (D)  x +  +  y +  +  z +  > 6  y  x  z Q.92

Which of the following homogeneous functions are of degree zero ? (A)

Q.93

x y y x ln + ln y x x y

(B)

x (x − y ) y (x + y )

(C)

xy x + y2

(D) x sin

2

 x sin α   x − cos α   – tan–1   The value of tan–1   1 − x cos α   sin α  is, for α ∈

y y − y cos x x

 π  0,  ; x ∈ R+ , is  2

(B) independent of α

(A) independent of x π (C) –α 2

(D) none of these

Q.94

D ≡ [− 1, 1] is the domain of the following functions, state which of them has the inverse. (A) f(x) = x2 (B) g(x) = x3 (C) h(x) = sin 2x (D) k(x)= sin (πx/2)

Q.95

Which of the following function(s) have no domain? (A) f(x) = logx – 1(2 – [x] – [x]2) where [x] denotes the greatest integer function. (B) g(x) = cos–1(2–{x}) where {x} denotes the fractional part function. (C) h(x) = ln ln(cosx) (D) f(x) =

1

sec

-1

(sgn (e )) −x

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.94

Q.93 A,C

A,B,C

Q.89

Q.88 A,C,D

A,B,C,D

Q.84

Q.83 A,B,C

B, D

Q.82

Q.79

Q.78 B

C

Q.77

C

Q.74

Q.73 D

A

Q.72

C

Q.71

Q.69

Q.68 C

A

Q.67

C

Q.66

Q.64

Q.63 C

D

Q.62

A

Q.61

Q.59

Q.58 C

C

Q.57

C

Q.56

Q.54

Q.53 A

B

Q.52

C

Q.51

Q.49

Q.48 D

B,C

Q.47

A

Q.46

Q.44

Q.43 A

C

Q.42

B

Q.41

Q.39

Q.38 D

B

Q.37

A

Q.36

Q.34

Q.33 C

D

Q.32

D

Q.31

Q.29

Q.28 A

A

Q.27

A

Q.26

Q.24

Q.23 A

A

Q.22

C

Q.21

Q.19

Q.18 C

C

Q.17

B

Q.16

Q.14

Q.13 A

A

Q.12

A

Q.11

Q.9

C

Q.8

A

Q.7

C

Q.6

Q.4

A

Q.3

C

Q.2

B

Q.1

B

Q.60

D

Q.55

D

Q.50

C

Q.45

D

Q.40

B

Q.35

C

A

Q.30

B

D

Q.25

D

C

Q.20

A

B

Q.15

C

D

Q.10

D

B

Q.5

D

Quest Q.92

A,B

Q.91

Q.87

A,B,D

Q.86

A,C

Q.81

Q.76

A

D

D

C

B

C D D D A,B,C

C

Q.80

A

Q.75

D

Q.70

A

Q.65

Q.85 B,C,D

Q.95 A,B,C,D

B,D

Q.90 A,C,D

B,C

ANSWER KEY

TARGET IIT JEE

MATHEMATICS

METHOD OF DIFFERENTIATION

Question bank on Method of differentiation There are 72 questions in this question bank. Select the correct alternative : (Only one is correct)

Q.1

If g is the inverse of f & f ′ (x) = (A) 1 + [g(x)]5

Q.2

tan−1

If y =

(B)

1 1 + [g(x)]5

(C) −

1 1 + [g(x)]5

(D) none


n e2  d 2y x  −1 3 + 2
n x then  + tan = 
n ex 2  dx 2 1 − 6
n x  

(A) 2

Q.3

1 then g ′ (x) = 1+ x5

(B) 1

(D) − 1

(C) 0

dy  3x + 4  =  & f ′ (x) = tan x2 then dx  5x + 6 

If y = f 

2

Q.4

Quest (B) − 2 tan 

 3 tan x 2 + 4   tan x2 (C) f   5 tan x 2 + 6 

(D) none

If y = sin−1  x 1 − x + 

(C) sin−1 x

(D) none of these

dy  2x − 1 =  & f ′ (x) = sin x then 2 dx  x + 1

(C)

1 + x − x2

(1 + x ) 2

2

1 − x + x2

(

1+ x

)

2 2

5 210

(B)

 2x − 1   x2 + 1

(D) none

2

sin 

(B)

1 + a2 a 10

If sin (xy) + cos (xy) = 0 then (A)

y x

( ) sin  2x − 1  x + 1 (1 + x )

2 1 + x − x2

 2x − 1   x2 + 1

sin 

Let g is the inverse function of f & f ′ (x) =

(A)

Q.7

dy 1 = + p, then p = dx 2 x (1 − x)

If y = f 

(A)

Q.6

x 1 − x 2  &

(B) sin−1 x

(A) 0

Q.5

 3x + 4  1 .  2  5x + 6  (5x + 6)

(A) tan x3

(B) −

x10

(1 + x ) 2

(C)

2

2

. If g(2) = a then g ′ (2) is equal to

a 10 1 + a2

(D)

1 + a 10 a2

dy = dx y x

(C) −

x y

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

x

(D) y

[2]

Q.8

(A)

Q.9

dy  2x is : 2 then dx   x = −2 1+ x

If y = sin−1 2 5

2 5

(C) −

5

(D) none



1 1  is :  w.r.t. 1 − x 2 at x = 2  2x − 1

The derivative of sec−1  (A) 4

Q.10

2

(B)

2

(B) 1/4

(C) 1

(D) none

 d   3 d 2y    y . 2  equals :  dx   dx 

If y2 = P(x), is a polynomial of degree 3, then 2  (A) P ′′′ (x) + P ′ (x)

(B) P ′′ (x) . P ′′′ (x)

(C) P (x) . P ′′′ (x)

(D) a constant

Q.11

Let f(x) be a quadratic expression which is positive for all real x . If g(x) = f(x) + f ′ (x) + f ′′ (x), then for any real x, which one is correct . (A) g(x) < 0 (B) g(x) > 0 (C) g(x) = 0 (D) g(x) ≥ 0

Q.12

If xp . yq = (x + y)p + q then

Quest dy is : dx

(A) independent of p but dependent on q (C) dependent on both p & q

Q.13

g (x) . cos x1

if x ≠ 0

Let f(x) = 

if x = 0

0

(B) dependent on p but independent of q (D) independent of p & q both .

where g(x) is an even function differentiable at x = 0, passing

through the origin . Then f ′ (0) : (A) is equal to 1 (B) is equal to 0

Q.14

If y = (A)

Q.15

Lim x→0

+x

p− m

+

1 1+ x

(B)

m− n

+x

p− n

+

emn/p

1 1+ x

(C)

m− p

+x

n− p

then

(D) does not exist np dy at e m is equal to: dx

enp/m

(D) none

log sin2 x cos x x has the value equal to log 2 x cos sin 2 2

(B) 2

(C) 4

If f is differentiable in (0, 6) & f ′ (4) = 5 then Limit x→2 (A) 5

Q.17

1+ x

emnp

(A) 1 Q.16

1 n− m

(C) is equal to 2

(B) 5/4

(D) none of these

c h=

f (4) − f x 2 2−x

(C) 10

(D) 20

Lim+ xm (ln x)n where m, n ∈ N then : Let l = x→ 0

(A) l is independent of m and n (B) l is independent of m and depends on m (C) l is independent of n and dependent on m (D) l is dependent on both m and n Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[3]

Q.18

Q.19

cos x x Let f(x) = 2 sin x x 2 tan x x

1 f ′ (x) = 2x . Then Limit x→0 x 1

(A) 2

(B) − 2

(D) 1

cos x sin x cos x  π Let f(x) = cos 2x sin 2x 2 cos 2x then f ′   = 2 cos 3x sin 3x 3 cos 3x

(A) 0 Q.20

(C) − 1

(B) – 12

(C) 4

(D) 12

People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D*f(x) by the formula f 2 (x + h) − f 2 (x) where f2 (x) means [f(x)]2. If f(x) = x lnx then D*f(x) = Limit h h→ 0 D * f ( x ) x = e has the value

(A) e

Q.21

Quest (B) 2e

If f(4) = g(4) = 2 ; f ′ (4) = 9 ; g ′ (4) = 6 then Limit x→4 (A) 3 2

Q.22

If y = x + ex then

x −2

(C) 0

2

(B) 5f ′ (x)

is equal to :

(D) none

ex

(1+ e ) x

If x2y + y3 = 2 then the value of 3 4

(C) 0

(D) none

d 2x is : dy 2

(B) −

ex

(A) − Q.25

3

f (x) − g (x)

h

(A)

Q.24

(B)

(D) none

f (x + 3h) − f (x − 2h) If f(x) is a differentiable function of x then Limit = h→0

(A) f ′ (x) Q.23

(C) 4e

(B) −

3 8

3

(C) −

ex

(1+ e ) x

2

(D)

−1

(1+ e )

x 3

d 2y at the point (1, 1) is : dx 2

(C) −

5 12

(D) none

g (x) . f (a ) − g (a ) . f (x) If f(a) = 2, f ′ (a) = 1, g(a) = − 1, g ′ (a) = 2 then the value of Limit is: x→a x−a

(A) − 5

(B) 1/5

(C) 5

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) none

[4]

Q.26

If f is twice differentiable such that f ′′ (x) = − f (x), f ′ (x) = g(x)

h ′ (x) = [ f (x)] + [ g(x)] and h (0) = 2 , h (1) = 4 2

then the equation y = h(x) represents : (A) a curve of degree 2 (C) a straight line with slope 2

Q.27

The derivative of the function, f(x)=cos-1 w.r.t. 1 + x 2 at x = (A)

3 2

2

(B) a curve passing through the origin (D) a straight line with y intercept equal to − 2.

RS T

UV W

1 (2 cos x − 3 sin x) + sin−1 13

RS T

1 (2 cos x + 3 sin x) 13

3 is : 4 (B)

5 2

(C)

10 3

(D) 0

Q.28

Let f(x) be a polynomial in x . Then the second derivative of f(ex), is : (A) f ′′ (ex) . ex + f ′ (ex) (B) f ′′ (ex) . e2x + f ′ (ex) . e2x x 2x (C) f ′′ (e ) e (D) f ′′ (ex) . e2x + f ′ (ex) . ex

Q.29

The solution set of f ′ (x) > g ′ (x), where f(x) =

Quest

(A) x > 1

Q.30

If y = sin−1

x (A) 4 x −1

Q.31

Q.32

1 2x + 1 (5 ) & g(x) = 5x + 4x (ln 5) is : 2

(C) x ≤ 0

(B) 0 < x < 1

(D) x > 0

dy x2 − 1 x2 + 1 −1 + sec , x > 1 then is equal to : 2 2 dx x +1 x −1 x2 (B) 4 x −1

(C) 0

dy x x x x x x ...... ∞ then = dx a + b+ a + b+ a + b+ a a b (A) (B) (C) ab + 2 by ab + 2 ay ab + 2 by

(D) 1

If y =

(D)

b ab + 2 ay

Let f (x) be a polynomial function of second degree. If f (1) = f (–1) and a, b, c are in A.P., then f '(a), f '(b) and f '(c) are in (A) G.P. (B) H.P. (C) A.G.P. (D) A.P. y

Q.33

UV W

y1

y2

If y = sin mx then the value of y 3 y 4 y 5 (where subscripts of y shows the order of derivatiive) is: y6

y7

y8

(A) independent of x but dependent on m (C) dependent on both m & x

(B) dependent of x but independent of m (D) independent of m & x .

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[5]

Q.34

If x2 + y2 = R2 (R > 0) then k =

y ′′

(1 + y ′ ) 2

(A) –

1

(B) –

R2

1 R

3

where k in terms of R alone is equal to

(C)

2 R

(D) –

2 R2

Q.35

If f & g are differentiable functions such that g ′ (a) = 2 & g(a) = b and if fog is an identity function then f ′ (b) has the value equal to : (A) 2/3 (B) 1 (C) 0 (D) 1/2

Q.36

Given f(x) = −

x3 + x2 sin 1.5 a − x sin a . sin 2a − 5 arc sin (a2 − 8a + 17) then : 3

(A) f(x) is not defined at x = sin 8 (C) f ′ (x) is not defined at x = sin 8

(B) f ′ (sin 8) > 0 (D) f ′ (sin 8) < 0

Q.37

A function f, defined for all positive real numbers, satisfies the equation f(x2) = x3 for every x > 0 . Then the value of f ′ (4) = (A) 12 (B) 3 (C) 3/2 (D) cannot be determined

Q.38

Given : f(x) = 4x3 − 6x2 cos 2a + 3x sin 2a . sin 6a +
n 2 a − a 2

Quest (

(A) f(x) is not defined at x = 1/2 (C) f ′ (x) is not defined at x = 1/2

Q.39

If y = (A + Bx) emx + (m − 1)−2 ex then (A) ex

(B) emx

)

then :

(B) f ′ (1/2) < 0 (D) f ′ (1/2) > 0

d 2y dy + m2y is equal to : 2 − 2m dx dx

(C) e−mx

(D) e(1 − m) x

Q.40

Suppose f (x) = eax + ebx, where a ≠ b, and that f '' (x) – 2 f ' (x) – 15 f (x) = 0 for all x. Then the product ab is equal to (A) 25 (B) 9 (C) – 15 (D) – 9

Q.41

Let h (x) be differentiable for all x and let f (x) = (kx + ex) h(x) where k is some constant. If h (0) = 5, h ' (0) = – 2 and f ' (0) = 18 then the value of k is equal to (A) 5 (B) 4 (C) 3 (D) 2.2

Q.42

Let ef(x) = ln x . If g(x) is the inverse function of f(x) then g ′ (x) equals to : (A) ex

Q.43

(B) ex + x

(C) e ( x

+ ex )

(D) e(x + ln x)

The equation y2exy = 9e–3·x2 defines y as a differentiable function of x. The value of

dy for dx

x = – 1 and y = 3 is (A) –

15 2

(B) –

9 5

(C) 3

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) 15

[6]

Q.44

( )

Let f(x) = x x

x

and g(x) = x

(x x )

then :

(A) f ′ (1) = 1 and g ′ (1) = 2 (C) f ′ (1) = 1 and g ′ (1) = 0 Q.45

The function f(x) = ex + x, being differentiable and one to one, has a differentiable inverse f–1(x). The d –1 value of (f ) at the point f(l n2) is dx (A)

Q.46

(B) f ′ (1) = 2 and g ′ (1) = 1 (D) f ′ (1) = 1 and g ′ (1) = 1

1
n2

(B)

1 3

(C)

1 4

(D) none

π log sin|x| cos 3 x If f (x) = for |x| < x≠0 3 3 x  log sin|3x| cos   2 =4 for x = 0

Quest

 π π then, the number of points of discontinuity of f in  − ,  is  3 3 (A) 0

Q.47

If y =

(A)

Q.48

(B) 3

(C) 2

(a − x) a − x − ( b − x) x − b a −x + x−b

x + (a + b )

(B)

(a − x) (x − b)

If y is a function of x then

then

(D) 4

dy wherever it is defined is equal to : dx

2 x − (a + b)

2 (a − x) (x − b)

(C) −

(a + b)

2 (a − x) (x − b)

(D)

2 x + (a + b )

2 (a − x) (x − b)

dy d2 y = 0 . If x is a function of y then the equation becomes : 2 +y dx dx 3

 dx d2 x (B)  =0 2 +y   dy dy

d2 x dx (A) =0 2 +x dy dy 2

 dx d2 x (C)  =0 2 − y   dy dy

2

 dx d2 x (D)  =0 2 − x   dy dy

Q.49

A function f (x) satisfies the condition, f (x) = f ′ (x) + f ′′ (x) + f ′′′ (x) + ...... ∞ where f (x) is a differentiable function indefinitely and dash denotes the order of derivative . If f (0) = 1, then f (x) is : (A) ex/2 (B) ex (C) e2x (D) e4x

Q.50

If y =

cos 6x + 6 cos 4 x + 15 cos 2 x + 10 cos 5x + 5 cos 3x + 10 cos x (A) 2 sinx + cosx (B) –2sinx

dy = dx (C) cos2x

, then

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D) sin2x

[7]

3

d 2 x  dy  d2y   Q.51 If + = K then the value of K is equal to dy 2  dx  dx 2 (A) 1 (B) –1 (C) 2

Q.52

(

)



(D) 0

1 2

−1 −1 If f(x) = 2 sin 1− x + sin 2 x (1− x) where x ∈  0 , 

then f ' (x) has the value equal to 2

(A)

x (1− x)

−

Q.53

 e y = f(x) =    0

Let

1 x2

2

(C) −

(B) zero

x (1− x)

(D) π

if x ≠ 0 if x = 0

Then which of the following can best represent the graph of y = f(x) ?

Quest

(A)

Q.54

(B)

 Diffrential coefficient of  x  (A) 1

Q.55

 

 . x  

Let f (x) be diffrentiable at x = h then Lim x→ h (B) 2 f(h) + hf '(h)

(D)

1 m+ n
− m  n−


 

Limit+ x→0

(A)

(B) 24 a (ax + b)2

 . x  

1 n +
m− n 
− m

 

(C) – 1

b x + hg f ( x)

w.r.t. x is

(D)

x
mn

− 2 h f ( h)

x−h (C) hf(h) + 2f '(h)

If y = at2 + 2bt + c and t = ax2 + 2bx + c, then (A) 24 a2 (at + b)

Q.57

1
+ m n−
 m− n 

(B) 0

(A) f(h) + 2hf '(h)

Q.56

(C)

is equal to (D) hf(h) – 2f '(h)

d 3y equals dx 3

(C) 24 a (at + b)2

(D) 24 a2 (ax + b)

 x x  a arc tan  has the value equal to − b arc tan a b  x x  1

a−b 3

(B) 0

(C)

(a 2 − b 2 ) 6a 2 b 2

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

a 2 − b2 3a 2 b2

[8]

Q.58

Q.59

Q.60

x Let f (x) be defined for all x > 0 & be continuous. Let f(x) satisfy f   = f ( x ) − f ( y) for all x, y & y f(e) = 1. Then : (A) f(x) is bounded

 1 (B) f   → 0 as x → 0

(C) x.f(x)→1 as x→ 0

(D) f(x) = ln x

Suppose the function f (x) – f (2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The derivative of the function f (x) – f (4x) at x = 1, has the value equal to (A) 19 (B) 9 (C) 17 (D) 14 x4 − x2 +1 dy If y = 2 and = ax + b then the value of a + b is equal to dx x + 3x + 1 (A) cot

Q.61

 x

5π 8

(B) cot

5π 12

(C) tan

5π 12

(D) tan

5π 8

Quest

Suppose that h (x) = f (x)·g(x) and F(x) = f ( g ( x ) ) , where f (2) = 3 ; g(2) = 5 ; g'(2) = 4 ; f '(2) = –2 and f '(5) = 11, then (A) F'(2) = 11 h'(2) (B) F'(2) = 22h'(2) (C) F'(2) = 44 h'(2) (D) none

Q.62

Let f (x) = x3 + 8x + 3 which one of the properties of the derivative enables you to conclude that f (x) has an inverse? (A) f ' (x) is a polynomial of even degree. (B) f ' (x) is self inverse. (C) domain of f ' (x) is the range of f ' (x). (D) f ' (x) is always positive.

Q.63

Which one of the following statements is NOT CORRECT ? (A) The derivative of a diffrentiable periodic function is a periodic function with the same period. (B) If f (x) and g (x) both are defined on the entire number line and are aperiodic then the function F(x) = f (x) . g (x) can not be periodic. (C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function. (D) Every function f (x) can be represented as the sum of an even and an odd function

Select the correct alternatives : (More than one are correct) Q.64

If y = tan x tan 2x tan 3x then

dy has the value equal to : dx

(A) 3 sec2 3x tan x tan 2x + sec2 x tan 2x tan 3x + 2 sec2 2x tan 3x tan x (B) 2y (cosec 2x + 2 cosec 4x + 3 cosec 6x) (C) 3 sec2 3x − 2 sec2 2x − sec2 x (D) sec2 x + 2 sec2 2x + 3 sec2 3x Q.65

dy x − x If y = e + e then equals dx e x − e− x (A) 2 x

(B)

e x − e− x 2x

(C)

1 2 x

y2 − 4

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

(D)

1 2 x

y2 + 4

[9]

Q.66

2

If y = x x then (A) 2 ln x . xx

dy = dx

2

(B) (2 ln x + 1). xx

(C) (2 ln x + 1). x x Q.67

Q.69

1 2y − 1

(B)

x x + 2y

2y (A) − x 2

1 (B) 1 − 2x

. ln ex2

1 1 + 4x

(C) 1 −

2y

(D)

y 2x + y

( (2

) − 1)

2x 1 − 2 y

(D)

2

y

x

Quest

The functions u = ex sin x ; v = ex cos x satisfy the equation :

d2u = 2v dx2

dv du −u = u2 + v2 dx dx

(B)

d 2v = − 2u dx 2

(D) none of these

x −2 x −1

Let f (x) =

x −1 −1

. x then :

(A) f ′ (10) = 1 (C) domain of f (x) is x ≥ 1

(B) f ′ (3/2) = − 1 (D) none

Two functions f & g have first & second derivatives at x = 0 & satisfy the relations, f(0) =

2 , f ′ (0) = 2 g ′ (0) = 4g (0) , g ′′ (0) = 5 f ′′ (0) = 6 f(0) = 3 then : g(0)

(A) if h(x) =

15 f (x) then h ′ (0) = 4 g(x)

1 g′ (x) (C) Limit = x→0 f ′ (x)

Q.72

(C)

dy has the value equal to : dx

(C)

Q.71

2 +1

dy = dx

If 2x + 2y = 2x + y then

(A) v

Q.70

(D) x x

Let y = x + x + x + ...... ∞ then (A)

Q.68

2 + 1

2

If y = x (
n x )

(


n (
n x )

(D) none

2

dy is equal to : dx + 2
n x
n (
n x )

, then

y
n x
n x − 1 x y (C) ((ln x)2 + 2 ln (ln x)) x
n x

(A)

(B) if k(x) = f(x) . g(x) sin x then k ′ (0) = 2

)

y (ln x)ln (ln x) (2 ln (ln x) + 1) x y
n y (D) (2 ln (ln x) + 1) x
n x

(B)

Quest Tutorials North Delhi : E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

[10]

[11] Q.1 Q.6 Q.11 Q.16 Q.21 Q.26 Q.31 Q.36 Q.41 Q.46 Q.51 Q.56 Q.61

Quest Tutorials E-16/289, Sector-8, Rohini, New Delhi. Ph. 65395439

Q.64

Q.33 D

D

Q.32

D

Q.28 D

C

Q.27

C

Q.23 B

B

Q.22

A

Q.18 B

A

Q.17

D

Q.13 B

D

Q.12

B

C

Q.8

B

Q.7

B

B

Q.3

C

Q.2

A

D e lh i :

Q.68

N o rth

Q.4 Q.9 Q.14

D

Q.35

C

Q.30

C

Q.25

C

Q.20

C

Q.15

D

C

Q.10

A

B

Q.5

D

Quest Q.70 A,B

A,B,C

Q.69

A,B,C,D

Q.66 C,D

A,C

Q.65

A,B,C

Q.63 B

D

Q.62

B

Q.58 D

D

Q.57

D

Q.53 C

B

Q.52

D

Q.48 C

B

Q.47

C

Q.43 D

C

Q.42

C

Q.38 D

B

Q.37

D

Q.19

Q.24

Q.29

Q.34

Q.39 Q.44 Q.49 Q.54 Q.59

Q.67 Q.71

C

B

D

B

A D A B A

B,D

Q.72

B

Q.60

A

Q.55

B

Q.50

B

Q.45

C

Q.40

A,C,D A,B,C

ANSWER KEY