Jee 2014 Booklet5 Hwt Differential Calculus 2
Short Description
Jee 2014 Booklet5 Hwt Differential Calculus 2...
Description
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [1]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One correct option. 1.
The graph of y x 1 1 is : 2
Y
X
O
2.
3.
*5.
2 Given f x tan x2 16 (A) Range of f (x) is A
(C)
X
(D)
(B)
[0, 1]
(B)
(C)
1, 1
1 0, 3
(D)
x3 meets the curve again at the point Q. The coordinates of Q are : 10 x
1 1, 3
(C)
2, 2
2, 1
(D)
and A R 0, 1
Maximum value of f (x) =1
(B)
Range of f (x) is A’
(D)
Maximum value of
f x
1
1
The coordinates of a point on the curve x3 y 3 6 xy at which the tangent is parallel to the x-axis are : (A)
7.
1 1, 3
O
1 x2 1 x2 sec 1 the set of points at which the tangent is parallel to the x-axis is : For the curve defined as y cos ec 1 2x 1 x2 (A) [0, 1] (B) (0, 1) (C) None of these 1, 0 1, (D)
(C) 6.
1 3 , 1
X
O
1 is : 2 cos x
The tangent at the point (5, 5) on the curve y 2 (A)
4.
(B)
The range of the function f x (A)
X
O
(A)
Y
Y
Y
24 / 3 , 25 / 3
(B)
24 / 3 , 25 / 3
(C)
25 / 3 , 24 / 3
(D)
None of these
If and are the lengths of perpendiculars from the origin to the tangent and normal to the curve x 2 / 3 y 2 / 3 52 / 3 respectively then 4 2 2 is : (A)
8.
625
(B)
125
(C)
25
(D)
252/3
The curve y ax3 bx 2 cx 5 touches x-axis at A 2, 0 . The curve intersects the y-axis at a point B where its slope equals 3. The value of ‘a’ is : (A)
2
VMC/Differential Calculus-2
(B)
2
(C)
69
1 2
(D)
1 2
HWT/Mathematics
Vidyamandir Classes 9.
*10.
The function f x x 4 42 x 2 80 x 32
3
is :
(A)
Monotonically increasing in 4, 1 5,
(B)
Monotonically increasing in , 4 1, 5
(C)
Monotonically increasing in 4 , 5
(D)
None of these
If f x max x 2 4, | x 2 |, | x 4 | then : (A)
f (x) is continuous for all x R
(B)
f (x) is differentiable except at x
(C)
f (x) has a critical point at x = 2
(D)
f (x) has no maximum
VMC/Differential Calculus-2
70
1 33 2
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [2]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One correct option. *1.
Given that f (x) is a linear function. Then the curves :
(B)
y
1
(C)
y
(D)
y
(A)
2.
(D)
24
Let
7.
14 10 x x 2 x 3 Let f x 2 3 log10 p 4
8.
None
*9.
Side of a regular hexagon increases at the rate of 3 cm/hour. At the instant when the side is 120 3 cm , find the rate of increase in the radius of the inscribed circle of hexagon :
(C) 5.
3 cm/hr 2
(B)
3 3 cm / hr
(D)
3 3 cm / hr 2
*10.
If the only point of inflection of the function f x x a
m
x b n , m, n N
x = a, then : (A) m, n are even (B) m is odd, n is even (C) m is even, n is odd (D)
VMC/Differential Calculus-2
, x 1
.
and m n is at
(D
14 , 2 2, 14
The set of values of P for which the function x x f x P 2 5P 6 cos 4 sin 4 P 3 x K 4 4 has no critical point is : (A) (0, 4) (B) , 0
(C)
find the rate of increase in the area of the hexagon : (A) 3040 cm2/hr (B) 3140 cm2/hr 2 (C) 3240 cm /hr (D) 3340 cm2/hr
3 cm/hr
14 , 14
(C)
Side of a regular hexagon increases at the rate of
(A)
, x 1
3 cm/hour. At the instant when the side is 120 3 cm ,
4.
with f x a0 a1 x 2 a2 x 4 . . . an x 2n a1 , a2 , . . . an 0 . Then f (x) has : (A) only one maxima (B) only one minima (C) no extrema (D) None of these
6.
Then f (x) attains the absolute minimum value at x = 1if p takes values in the interval : (A) (4, 14) (B) 14 , 2 2 , 14
A balloon is in the form of a right circular cone surmounted by a hemisphere having the radius equal to half the heights of the cone. Air leaks through a small hole ; but the balloon keeps its shape. What is the rate of change of volume with respect to the total height (H) of the balloon when H = 18 cm. (A) (B) 48 (C)
3.
x are orthogonal f x and y f x are orthogonal f x and y f 1 x are orthogonal f x and y f 1 x are orthogonal
y f x and y f
1
0,
(D)
Let g x log 1 3x 2 x 2 3x
0, 3 3, 4 5x2 . Then g x is 2
increasing on 1 (A) (B) (0, 1) 0, 2 (C) (0, 2) (D) (0, 3) 2 2 ax , x0 x 2 e a Let f x where a > 0, the x3 2 x 2 , x 0 a a2 interval in which f x is increasing : (A) (C)
1 0, 2 (0, 2)
(B)
(0, 1)
(D)
(0, 3)
m, n are odd
71
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [3]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.
The points on the curve y x3 2 x 2 x at which the tangent line is parallel to the line y 3 x 2 is(are) : (A)
2.
4.
7.
8.
2 14 , 3 27
(D)
2, 2 , 2, 14
(B)
1, 1 , 3, 4
(C)
(2, 14), (2, 2)
(D)
None of these
None of these
(B)
f (x) is decreasing in , 0 2
(C)
f (x) is increasing in 0 , 2
(D)
f (x) is decreasing in , 2
Let f x x 2
2/3
2 x 1 . Then critical points of f (x) are : (D)
x = 1, 2
x=1
(B)
x=2
(C)
x = 0, 2
(D)
None
Equation of normal to the curve x y x y where it cuts x-axis, is : (A) (B) (C) x+y=0 x y 1 0 x y 1 0
(D)
None of these
Maximum area of a rectangle of perimeter 176 cm, is : (A) 1936 cm2 (B) 1854 cm2
(D)
None of these
If the tangent at (1, 1) on y 2 x 2 x meets the curve at P, then P is : 2
(4, 4)
(B)
1, 2
(C)
9 3 4 , 8
(C)
2110 cm2
(B) (D)
decreasing for x < 0 only decreasing for all real x
1 log x 2 1 . Then f (x) is : 2 increasing for x > 0 only increasing for all real x
Let f x x
If the tangent at P 4m 2 , 8m3 to the curve x3 y 2 is also a normal, them m 2 (A)
10.
2, 2 ,
f (x) is increasing in , 2 2
(A) (C) 9.
(C)
(A)
(A) 6.
14 2 3 , 27
Let f (x) = cos (cos x). Then which one is not correct ?
(A) 5.
(B)
The co-ordinates of the point on the curve y x 2 3 x 4 the tangent at which passes through the origin are : (A)
3.
2, 2
2
(B)
1/9
(C)
2/9
(D)
None of these
If at each point of the curve y x3 ax 2 x 2 , the tangent is inclined at an acute angle with the positive direction of x-axis, then : (A) (B) a>0 a 3 or a 3 (C)
3a 3
VMC/Differential Calculus-2
(D)
72
None of these
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
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TEST CODE : DC-2 [4]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.
2.
3.
Let f : a, b R be a function such that for c a, b , f c f c f c f iv c f v c 0 , then : (A) f has local extremum at x = c (B) f has neither local maximum nor local minimum at x = c (C) f is necessarily a constant function (D) It is difficult to say whether (A) or (B) The largest value of 2 x3 3 x 2 12 x 10 for 2 x 4 occurs at x = (A) (B) (C) 2 2 1
2
(B) n
9/2
(D)
None
(B)
2
(C)
4
(D)
n0
(D)
None of these
(D)
(2, 4)
a sin x b cos x is monotonically decreasing in its domain if : c sin x d cos x
ad bc 0
(B)
ab cd 0
(C)
ad bc
The complete set of values of x in which f x 2 log e x 2 x 2 4 x 1 increases, is : (A)
7.
(C)
n
1
The function f x (A)
6.
4
x y x y The curve 2 touches the line 2 at the point (a, b) for n = a b a b (A)
5.
4
If the curves y 2 16 x and 9 x 2 by 2 16 cut each other at right angles, then the value of b is : (A)
4.
(D)
(1, 2)
(B)
(2, 3)
(C)
5 2 , 3
Segment of the tangent to the curve xy c 2 at the point x , y' which is contained between the co-ordinate axes, is bisected at the point (A)
8.
9.
y
(B)
y , x
(C)
x y 2, 2
(D)
None
The tangent and normal to the curve y 2 sin x sin 2 x are drawn at P x . The area of the quadrilateral formed by the 3 tangent, the normal and co-ordinate axes is : (A) (B) (C) (D) None 3 3/2 3
In 1, 2 the function f x x x 1 is : (A)
10.
x ,
increasing
(B)
decreasing
(C)
constant
(D)
None of these
f x The number of solutions of the equations a g x 0 , where a 0 , g x 0 and has minimum value 1/2, is :
(A)
infinitely many
VMC/Differential Calculus-2
(B)
only one
(C)
73
two
(D)
zero
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [5]
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PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.
The curves ax 2 by 2 1 and a x 2 b y 2 1 intersect orthogonally if : (A)
2.
5.
7.
f x
1 1 1 1 a a b b
(D)
None of these
circle
(C)
parabola
a sin x cos x is increasing for all values of x, then : sin x cos x (B) a1
(B)
The tangent to the curve x a
(A) 8.
1 1 1 1 a b a b
: x 1 6 then for f (x), x = 1 is : f x 7 x : x 1 (A) a point of local maxima (C) neither a point of local minima nor maxima
(A) 6.
straight line
If the function f x (A)
4.
(B)
All the points on the curve y x sin x at which the tangent is parallel to x-axis lie on a / an : (A)
3.
1 1 1 1 a b a b
[cos 1, 1]
(B)
[sin 1, 1]
(C)
[cos 1, sin 1]
(D)
[0, 1]
f x max . 4 , 1 x 2 , x 2 1 x R . Total number of points, where f (x) is not differentiable, is equal to :
(A)
2
VMC/Differential Calculus-2
(B)
4
(C)
74
6
(D)
None
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [6]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One correct option. 1.
f x x
(A) (C) 2.
1/2
0 1 t3 dt
1
2b/3
(D)
None
(D)
None
at the point where x = 1 is : (C)
1/4
(B) (D)
a minimum at x = 0 f (x) is not derivable at x = 0
(0, 1)
(B)
(1, 3)
(C)
(1, 0)
(D)
None
(B)
3
(C)
5
(D)
4
x : x 1 sin The function f x , where [.] denotes the greatest integer function, is : 2 2 x 3 x : x 1 Continuous and differentiable at x = 1 Discontinuous at x = 1
(B) (D)
Continuous but not differentiable at x = 1 None of these
The curve C1 : y 1 cos x, x 0, and C2 : y 3 2 | x | a will touch each other if a =
3 2 3
(B)
3 2 2 3
(C)
1 2 3
(D)
None
one point
(D)
no point
1/e2
(D)
2/e2
The straight line y 3 x 1 touches the curve y x 4 2 x 2 3 x at : (A)
10.
(B)
x
(C)
f x sin x cos x , x 0, 2 , where [.] denotes the greatest integer function. Total number of points where f (x) is not
(A) 9.
8b/27
2
If f x x5 5 x 4 5 x3 10 has local max. and min. at x = p and x = q, then (p, q) =
(A) (C) 8.
(B)
a maximum at x = 0 neither of two at x = 0
differentiable is equal to : (A) 2
7.
Differentiable at x = 1 None of these
f x 3 x x 2 x 4 xe x x then :
(A) 6.
8a/27
The slope of the tangent to the curve y
(A) (C) 5.
(B) (D)
3
(A) *4.
Continuous but not differentiable at x = 1 Discontinuous at x = 1
If the relation between subnormal SN and subtangent ST at any point P on the curve by 2 x a is p SN q ST , then p/q = (A)
3.
x , where [.] and {.} denote the greatest integer function and fractional part respectively, then f (x) is :
two points
(B)
four points
(C)
Let f x x 2 e 2x (x > 0). Then f (x) has maximum value equal to : (A)
1/e
VMC/Differential Calculus-2
(B)
1/2e
(C)
75
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [7]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One correct option. 1.
Let f x 0 x R and g x f 2 x f 4 x . Then g(x) is increasing in : (A)
2.
5.
1,
(D)
None
(C)
9
(D)
6
(C)
R 1, 1
(D)
None of these
(B) (D)
Discontinuous None of these
(D)
2
(D)
None
(C)
8
(B)
10
2x , then f (x) is differentiable on : 1 x2
If f x sin 1 (A)
4.
, 0
(B)
Total number of solutions of the equation sin x | ln | x || is : (A)
3.
, 1
1, 1
R 1, 1
(B)
3x : 1 x 1 Let f x . Then at x = 1, f (x) is : 4 x : 1 x 4 (A) Differentiable (C) Continuous but not differentiable
dy Given : 2 x 2 xy y 2 0 . Then dx 1, 2
(A)
2
(B)
3 2
4 3
(C)
x
6.
x2 2 If lim e , then a = x ax 1
(A) *7.
(B)
2
2, 2
(B)
1, 2
. . . .(i)
and
Should intersect orthogonally is that 1 1 1 1 (A) (B) a b a b
10.
4
2
14 2 3 , 27
(C)
(D)
(3, 6)
(D)
None of these
The condition that the curves
ax 2 by 2 1
9.
(C)
Find the points on the curve y x 2 x x at which the tangent lines are parallel to the line y 3 x 2 . 3
(A) 8.
1
a x 2 b y 2 1
1 1 1 1 a b a b
. . . .(ii) 1 1 1 1 a a b b
(C)
The equations of those tangents to 4 x 2 9 y 2 36 which are perpendicular to the straight line 5 x 2 y 10 0 , are : (A)
117 5 y 3 2 x 2
(B)
(C)
2 x 5 y 10 2 18 0
(D)
2 x 5 y 10 2 18 0
None of these
The point of intersection of the tangents drawn to the curve x y 1 y at the points where it is met by the curve xy 1 y , is given by: (A) (B) (1, 1) (C) (0, 1) (D) None of these 0, 1
VMC/Differential Calculus-2
2
76
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [8]
START TIME :
END TIME :
TIME TAKEN:
PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.
a a If the sum of the squares of the intercepts on the axes cut off by the tangent to the curve x1 3 y1 3 a1 3 (with a > 0) at , 8 8
is 2, then a has the value : (A) 1 2.
2
(C)
4
(C)
Cut at an angle
(D)
8
The two curves x3 3 xy 2 2 0 and 3 x 2 y y 3 2 (A)
3.
(B)
Cut at right angles (B)
touch each other
(D) 3
Cut at an angle
4
A curve with equation of the form y ax 4 bx3 c 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 the x for which the curve has a negative gradient are : (A)
4.
*7.
(D)
1 x 1
1
(B)
2
(C)
3 2
(D)
1 2
x=0
(B)
y=0
(C)
x y 0
(D)
x y 0
3
(D)
4
a 0, b 0
(B)
a 0, b 0
(C)
b 0, a 0
(D)
a 0, b 0
p 2 , q 7
(D)
p 2, q 7
If y 4 x 5 is a tangent to the curve y 2 px3 q at (2, 3), then : p 2 , q 7
(B)
p 2 , q 7
(C)
The curve y e xy x 0 has a vertical tangent at the point : (A)
10.
x 1
If the line ax by c 0 is normal to xy = 1, then :
(A)
9.
(C)
The number of real roots of the equation e x 1 x 2 0 is : (A) 1 (B) 2 (C)
(A)
8.
x 1
The equation of the tangent to the curve x t cos t, y t sin t at the origin is : (A)
6.
(B)
If the area of the triangle included between the axes and any tangent to the curve x n y a n is constant, then n is equal to : (A)
5.
x 1
(1, 1)
(B)
at no point
(C)
(0, 1)
(D)
(1, 0)
Let a, b be two distinct roots of a polynomial f (x). Then there exists at least one root lying between a and b of the polynomial : (A)
f (x)
VMC/Differential Calculus-2
(B)
f x
(C)
77
f x
(D)
None of these
HWT/Mathematics
Vidyamandir Classes DATE :
TIME : 30 Minutes
MARKS : [ ___ /10]
TEST CODE : DC-2 [9]
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PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One correct option. 1.
(A) (C) 2.
The set of all x for which log 1 x x is : (A)
*3.
*7.
9.
1, 0
(D)
None of these
(1, 2)
(B)
(2, 3)
(C)
5 2 , 3
(D)
(2, 4)
a 2 3b 15 0
(B)
a 2 3b 15 0
(C)
a 2 3b 15 0
(D)
a 0 and b 0
even
(B)
odd
(C)
increasing
(D)
decreasing
(A)
fog is an increasing function on I
(B)
fog is a decreasing function on I
(C)
fog is neither increasing nor decreasing on I
(D)
None of these
Which of the following functions are decreasing on 0 , ? 2 cos x
(B)
cos 2 x
(C)
cos 3x
(D)
(B)
decreases everywhere
tan x
The function y x3 3 x 2 6 x 17 : (A)
increases everywhere
(C)
increases for positive x and decreases for negative x
(D)
increases for negative x and decreases for positive x
The interval in which the function x3 increases less rapidly than 6 x 2 15 x 5 , is : (A)
10.
(C)
If f is an increasing function and g is a decreasing function on an interval I such that fog exists, then :
(A) 8.
1,
The function f x log e x3 x 6 1 is of the following types : (A)
6.
(B)
Let f x x3 ax 2 bx 5 sin 2 x be an increasing function on the set R. Then a and b satisfy : (A)
*5.
0,
The function f x 2 log x 2 x 2 4 x 1 increases on the interval : (A)
4.
x x and g x where 0 x 1 , then in this interval : sin x tan x Both f (x) and g (x) are increasing functions (B) Both f (x) and g (x) are decreasing functions f (x) is an increasing function (D) g (x) is an increasing function
If f x
, 1
(B)
5, 1
(C)
1, 5
(D)
5,
A condition for a function y f x to have an inverse is that it should be : (A)
defined for all x
(C)
strictly monotone and continuous in the domain (D)
VMC/Differential Calculus-2
(B)
78
continuous everywhere an even function
HWT/Mathematics
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PARENT’S SIGNATURE :
This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. x
1.
1 The maximum value of is : x
(A) 2.
3.
5.
8.
(B)
2
(C)
x 1
(A)
2
(D)
1
(B)
2 3
(B)
13
0
(B)
2
(D)
None of these
3
(D)
1 3
on [0, 1] is : (C)
8 7
On the interval [0, 1] the function x 25 1 x
1 e
3
The difference between the greatest and least values of the function f x cos x (C) 75
9 4
1 1 cos 2 x cos 3x is : 2 3 3 (D) 8
takes its maximum value at the point :
1 4
(C)
1 2
(D)
1 3
Let P x a0 a1 x 2 a2 x 4 . . . an x 2 n be a polynomial in a real variable x with 0 a0 a1 a2 . . . an . The function P(x) has : (A) neither a maximum nor a minimum
(B)
only one maximum
(C)
(D)
None of these
(C)
20
only one minimum
The maximum value of xy subject to x y 8 is : 8
(B)
16
(D)
24
(D)
2r 2
The maximum area of the rectangle that can be inscribed in a circle of radius r is : (A)
10.
1
13
(A) 9.
e1 e
is :
x2
The greatest value of f x x 1
(A) 7.
(C)
x 1
The number of critical points of f x
(A) 6.
ee
(B)
1 The value of a for which the function f x a sin x sin 3x has an extremum at x is : 3 3 (A) 1 (B) (C) 0 (D) 1
(A) 4.
e
r2
(B)
r2
(C)
r2 4
3 x 2 12 x 1 , 1 x 2 If f x , then : , 2 x3 37 x
(A)
f (x) is increasing in 1, 2
(B)
f (x) is continuous in 1, 3
(C)
f (x) is maximum at x = 2
(D)
All of these
VMC/Differential Calculus-2
79
HWT/Mathematics
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