PAST 40 YEARS IIT Entrance Mathematics Problems

Complex Numbers Entrance Questions Q1.

The number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals (a)

Q2.

If z –

0

3 +1

2

(d) AIEEE–2009

(c)

2

(d)

1 . Then the complex number is i –1

1 i –1

(c)

1

(d)

i 1

3, then the maximum value of |z + 1| is

4

(b)

If |z| = 1 and z

10

(c)

1, then all the values of

6

(d)

z lie on 1 – z2 |z| =

(c)

the x-axis

(d)

the y-axis

2k 11

i cos

(a)

1

(b)

If w =

+ i , where

0 and z

2k 11

is

–1

(c)

–i

(b)

|z| = 1 and z

(c)

z= z

(d)

None of these

(a) Q9.

i z– 3

(d)

i

1

and |w| = 1, then z lies on

a circle

0

IIT JEE–2006

|z| = 1, z = 2

z

i 1

1, satisfies the condition that w – w is purely real, then the 1–

(a)

If w =

1

AIEEE–2006

set of values of is

Q8.

–

2

(b)

sin

AIEEE–2008

AIEEE–2007

a line not passing through the origin

The value of

2

2+

AIEEE–2007

(a)

k 1

Q7.

–

(b)

10

Q6.

5 +1

(b)

1 i –1

If |z + 4| (a)

Q5.

(c)

The conjugate of a complex number is (a)

Q4.

1

4 = 2, then the maximum value of |z| is equal to z

(a) Q3.

(b)

AIEEE–2010

(b)

an ellipse

AIEEE–2005

(c)

a parabola

The locus of z which lies in shaded region is represented by

(d)

a straight line IIT JEE–2005

Q10.

Q11.

(a)

z : |z + 1| > 2, | (z + 1) | <

(c)

z : |z + 1| < 2, | (z + 1) | <

2

z : |z – 1| < 2, | (z – 1) | <

(a)

the real axis

(b)

an ellipse

(c)

a circle

(d)

imaginary axis

If a, b, c are integers not all equal and 2

2

is a cube root of unity (

IIT JEE–2004 (b)

1

(b)

n

1 n

1 z 1

If (a)

2

z 1

(b)

–128

AIEEE–2003

2

(c)

z 1 – (c)

2

(d)

1, then real part of

2

1 2

n

(c)

is an imaginary cube root of unity, then (1 + 128

is equal to

1

1

–1

(b)

(d)

2n

2n

1

If z is a complex number such that |z| = 1, z

(a)

3 2

(c)

are the cube roots of unity, then

0

2

1), then minimum value of |a

| is equal to

0

If 1, ,

(a)

Q15.

(d)

4

AIEEE–2004

2n

Q14.

z : |z – 1| > 2, | (z – 1) | <

2

(a)

Q13.

(b)

If |z – 1| = |z| + 1, then z lies on 2

+b +c

Q12.

4

2

z –1 is z 1

IIT JEE–2003

(d)

2 7

) equals 128

2

0 AIEEE–2002

(d)

–128

2

For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 – 3 – 4i| = 5, the minimum value of |z1 – z2| is equal to (a)

0

IIT JEE–2002 (b)

2

(c)

7

(d)

17

Quadratic Equations Entrance Questions Q1.

If

are the roots of the equation x2 – x + 1 = 0, then

and

2009

+

2009

is equal to AIEEE–2010

(a) Q2.

–2

–1

(b)

(c)

1

(d)

If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2

Q3.

2

AIEEE–2009

(a)

greater than 4ab

(b)

less than 4ab

(c)

greater than – 4ab

(d)

less than – 4ab

The quadratic equations x2 – 6x + a = 0, x2 – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ration 4 : 3. Then, the common root is AIEEE–2008 (a)

Q4.

2

(b)

1

(c)

4

(d)

If the difference between the roots of the equation x2 + ax + 1 = 0 is less than possible values of a is

Q5.

(a)

(–3, 3)

(b)

(–3, )

(c)

Let ,

be the roots of the equation x2 – px + r = 0 and

(3, )

2

(d)

(– , –3)

, 2 be the roots of the equation x2 – IIT JEE–2007

(a)

2 (p – q)(2q – p) 9

(b)

2 (q – p)(2p – q) 9

(c)

2 (q – 2p)(2q – p) 9

(d)

2 (2p – q)(2q – p) 9

All the values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0 are greater than –2 but less than 4 lie in the interval (a)

Q7.

5 , then the set of AIEEE–2007

qx + r = 0. Then, the value of r is

Q6.

3

m>3

(b)

AIEEE–2006

–1 < m < 3

(c)

1 a, then the equation (x – a)(x – b) – 1 = 0 has (a)

both roots in (a, b)

(b)

both roots in (– , a)

(c)

both roots in (b, )

(d)

one root in (– , a) and other in (b, )

IIT JEE–2000

Inequalities & Logarithms Q1.

For all x, x2 + 2ax + (10 – 3a) > 0, then the interval in which a lies is (a)

Q2.

a < –5

If 1, log3

31–x

(b)

–5 < a < 2

(c)

a>5

2 , log3( * –1) are in AP, then x is equal to

IIT JEE–2004 (d)

2 10 AIEEE–2002

2

(c)

3 2

(d)

4

Matrices Q1.

Consider the system of linear equations x1 + 2x2 + x3 = 3 2x1 + 3x2 + x3 = 3 3x1 + 5x2 + 2x3 = 1 The system has

Q2.

AIEEE–2010

(a)

Infinite number of solutions

(b)

Exactly 3 solutions

(c)

A unique solution

(d)

No solution

The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is AIEEE–2010 (a)

less than 4

(b)

5

(c)

6

(d)

at least 7

Directions (Q. No. 36 to 38) : For the following questions choose the correct answer from the codes (a), (b), (c), (d) defined as follows : (a)

Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I

(b)

Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I

Q3.

(c)

Statement I is true; Statement II is false

(d)

Statement I is false; Statement II is true

Let A be a 2 × 2 matrix with non-zero entries and let A2 = I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A. AIEEE–2010 Statement-I

Tr(A) = 0.

Statement-II Q4.

Q5.

|A| = 1.

Let A be 2 × 2 matrix.

AIEEE–2009

Statement-I

adj(adj A) = A

Statement-II

|adj A| = A

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2 = I.

Q6.

Statement-I If A

I and A

–I, then det(A) = –1.

Statement-II If A

I and A

–I, then tr(A)

AIEEE–2008

0.

Let A be a square matrix all of whose entries are integers. Then, which one of the following is true?

AIEEE–2008

(a)

If det(A) =

1, then A–1 need not exist

(b)

If det(A) =

1, then A–1 exists but all its entries are not necessarily integers

(c)

If det(A)

1, then A–1 exists and all its entries are non-integers

(d)

If det(A) =

1, then A–1 exists and all its entries are integers

5

Q7.

0

(a) Q8.

Q9.

5

Let A = 0

5 0

If det(A2) = 25, then | | is

AIEEE–2007

5

1

(b)

1 5

(c)

5

If A and B are 3 × 3 matrices such that A2 – B2 = (A – B) (A + B), then

(d)

52 AIEEE–2006

(a)

either A or B is zero matrix

(b)

either A or B is unit matrix

(c)

A=B

(d)

AB = BA

Let A =

1 2 3 4

and B =

a 0 , a, b, , N then 0 b

(a)

there exists exactly one B such that AB = BA

(b)

there exists infinitely man B’s such that AB = BA

AIEEE–2006

Q10.

(c)

there cannot exist any B such that AB = BA

(d)

there exist more than but finite number of B’s such that AB = BA

The system of equations ax + y + z =

–1

x+ y+z=

–1

x+y+ z=

–1

has no solution if

Q11.

(a)

–2 or 1

If P =

3 2 1 – 2

(a)

(c)

1 2 3 2

0

1

Q13.

0

1

1

0

1

(c)

1

(d)

and Q = PAPT, then PTQ2005 P is equal to

4 + 2005 3

(b)

2005

3

1

(d)

2– 3

–1 AIEEE–2005

6015 4 – 2005 3

1 2005 4 2+ 3

2– 3 2005

0 , then 0

–1 –1 0

AIEEE–2004

A is zero matrix (b) 2

If A =

–2

–1

If A = 0 (a)

,A=

2005

1 2 4 –1

AIEEE–2005 (b)

1

0

Q12.

is

A = (–1) I

and det A3 = 125, then

A–1 does not exist

(c)

us equal to

(d)

A2 = I

IIT JEE–2004

2

(a) Q14.

Q15.

If A =

1

(b)

a

b

b

a

and B2 =

2

(c)

3

, then

= a2 + b2,

= ab

(b)

= a2 + b2,

(c)

= a2 + b2,

= a2 – b2

(d)

= 2ab,

(a)

0 0

1 =4

and B =

1 5 (b)

= 2ab

= a2 + b2

0 , then A2 = B for 1 = –1

5

AIEEE–2003

(a)

If A =

(d)

IIT JEE–2003 (c)

=1

(d)

no

Determinants Q1.

a 1 a –1

a

Let a, b, c be such that (b + c)

a 1

0. If –b b 1 b –1 + a –1 c –1

c

c 1

b 1

c –1

b –1

c 1

(–1) n+2 a

(–1)n 1 b

= 0 then the

(–1)n c

value of ‘n’ is (a) Q2.

zero

AIEEE–2009 (b)

any even integer

(c)

any odd integer

(d)

any integer

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then, a2 + b2 + c2 + 2abc is equal to (a)

Q3.

Q4.

1

(b)

2

–1

(c)

1 1 1 If D = 1 1 x 1 for xy 1 1 1 y

0, then D is divisible by

(a)

x but not y

(c)

1 a2 x

(1 c 2 )x

both x and y

(b)

(1 b2 )x

If a + b + c = –2 and f(x) = (1+a 2 )x 1+b2 x 2

2

2

AIEEE–2008

(d)

0

AIEEE–2007

y but not x

(1 + c 2 )x

(d)

neither x nor y

, then f(x) is a polynomial of degree

(1+a 2 )x (1+b2 )x 1+c 2 x

AIEEE–2005 (a) Q5.

(b)

1

If a1, a2, a3,….. are in GP, then

(a) Q6.

0

0

(b)

(c) log an

= log a n log an

2

log an

(d)

1

log an

2

3

log an

4

log an

5

6

log an

7

log an

8

1

(c)

is equal to

2

AIEEE–2004

(d)

Given 2x – y + 2z = 2, x – 2y + z = –4, x + y + z = 4, then the value of

Q7.

3

(b)

Of 1, ,

2

IIT JEE–2004

1

are the cube roots of unity, then

(c)

0

1

n

=

n

0

(b)

1

If (

1+i+

1) is a cubic roots of unity, then 1 – i

1

is equal to

–i

–1

(d) 2

2

2 2

–i

AIEEE–2003

n

1

–1

–3

2n

(c) 1

Q8.

(d)

2n

2n

(a)

4

such that the given

system of equation has no solution, is (a)

3

–1 equals –1

AIEEE–2002

(a) Q9.

0

(b)

1

(c)

i

(d)

If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the value of a is (a)

IIT JEE–2002

0

(b)

–1

(c)

1

(d)

no real values

Binomial Theorem & Its Applications Q1.

Statement-I n

r 1

n

Cr = (n + 2)n–1

r 0

Statement-II n

r 1 nCr xr = (1 + x)n + nx(1 + x)n–1

AIEEE–2008

r 0

(a)

Statement–I is true, Statement–II is true; Statement–II is a correct explanation for Statement–I

(b)

Statement–I is true, Statement–II is true; Statement–II is not a correct explanation for Statement–I

Q2.

(c)

Statement–I is true; Statement–II is false

(d)

Statement–I is false; Statement–II is true

In the expansion of (a – b)n, n

5, the sum of 5th and 6th term is zero, then

a is equal to b

AIEEE–2007, IIT JEE–2001 (a)

Q3.

n–5 6

(b)

n–4 5

(c)

If the expansion, in powers of x of the function

5 n–4

(d)

1

is a0 + a1x + a2x2 + …, then an,

1 – ax 1 – bx

AIEEE –2006

is (a)

Q4.

6 n–5

a n – bn b–a

(b)

a n 1 – bn b–a

If the coefficients of x7 in ax 2

1 bx

1

(c)

bn 1 – a n b–a

11

and x–7 in ax –

1 bx 2

1

(d)

bn – a n b–a

11

are equal, then AIEEE–2005

(a)

a+b=1

(b)

a–b=1

(c)

ab = –1

(d)

ab = 1

Q5.

30 0

30 30 – 10 1 30 11

(a) Q6.

Q8.

n–1

x)6 is the same, if

(a)

–

30 10

(c)

IIT JEE–2005

(–1)n (1 – n)

(c)

65 55

(d)

AIEEE–2004

(–1)n–1(n–1)2

(d)

(–1)n–1x x)4 and of

5 3

AIEEE–2004 (b)

3 5

(c)

–

3 10

(d)

If n–1Cr = (k2 – 3)n Cr+1, then k belongs to (– , –2]

(b)

[2, )

32

(b)

– 3, 3

(c)

33

(d)

(c)

34

12

(b)

6

6

+1

( 3 , 2] AIEE–2003

(d)

The coefficient of x24 in (1 + x2)12 (1 + x12)(1 + x24) is

12

10 3 IIT JEE–2004

The number of integral terms in the expansion of ( 3 + 51/8)256 is

(a) Q11.

(b)

(1 –

(a) Q10.

60 10

(b)

30 is equal to 30

The coefficients of the middle term in the binomial expansion in powers of x of (1 +

(a) Q9.

30 30 –….+ 12 20

The coefficient of xn in the expansion of (1 + x)(1 – x)n is (a)

Q7.

30 30 + 11 2

35 IIT JEE–2003

12

(c)

6

+2

12

(d)

6

+3

Let Tn denote the number of triangles which can be formed by using the vertices of regular polygon of n sides.

AIEEE–2002

If Tn+1 – Tn = 21, then n is equal to (a)

5

(b) m

Q12.

The sum i

10

20

i

m–i

7

, when

(c)

p q

6

(d)

4

= 0, if p < q is maximum for m is equal to IIT JEE–2002

(a) Q13.

For 2

(a)

5

(b) r

n,

n 1 r –1

n r

+2

10

(c)

15

(d)

n n + is equal to r–2 r –1 (b)

2

n 1 r 1

(c)

20 IIT JEE–2000

2

n 2 r

(d)

n 2 r

Mathematical Induction Q1.

The remainder left out when 82n – (62)2n+1 is divided by 9 is (a)

Q2.

0

(b)

2

(c)

Statement-I For every natural number n

1 1 + + 1 2

7

AIEEE–2009 (d)

8

2. +

Statement-II For every natural number n

1 > n

n.

2.

n n 1 < n + 1. (a)

AIEEE–2008

Statement-I is true, Statement-II is true; Statement-II is a correct explanation for Statement-I

(b)

Statement-I is true, Statement-II is true; Statement-II is not a correct explanation for Statement-I

Q3.

(c)

Statement-I is true; Statement-II is false

(d)

Statement-I is false; Statement-II is true

If A =

1 0 1 1

and I =

1 0 0 1

, then which one of the following holds for all n

1, by the

principle of mathematical induction ?

Q4.

AIEEE–2005

(a)

An = 2n–1 A + (n – 1)I

(b)

An = nA + (n – 1)I

(c)

An = 2n–1 A – (n – 1)I

(d)

An = nA – (n – 1)I

Let S(k) = 1 + 3 + 5 + + (2k – 1) = 3 + k2. Then which of the following is true ? AIEEE–2004 (a)

S(1) is correct

(b)

S(k)

S(k + 1)

(c)

S(k)

S(k+ 1)

(d)

Principle of mathematical induction can be used to prove the formula

Permutations & Combinations Q1.

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done, is (a)

3

(b) 10

Q2.

AIEEE–2010 36 10

j 1

66

(d)

108

10

j (j – 1)10Cj, S2 =

Let S1 =

(c)

j 10Cj and S3 = j 1

j 2 10Cj j 1

Statement-I S3 = 55 × 29. Statement-II S1 = 90 × 28 and S2 = 10 × 28. Q3.

In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement-I The number of different ways the child can buy the six ice-creams is 10C5. Statement-II The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row. (a)

AIEEE–2008

Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I

(b)

Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I

Q4.

(c)

Statement-I is true; Statement-II is false

(d)

Statement-I is false; Statement-II is true

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent ? (a)

Q5.

6

7 C4

C4

(b)

AIEEE–2008 6

8 C4

7

C4

(c)

8

6 7 C4

(d)

6 8 7C4

The set S = {1, 2, 3,….., 12} is to be partitioned into three sets A, B, C of equal size. Thus, A C = S, A (a)

Q6.

8

B=B

12 /3 (4 )3

C=A (b)

C = . The number of ways to partition S is 12 /3 (3 )4

(c)

12 /(4 )3

(d)

B

AIEEE–2007 12 (3 )4

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is (a)

360

IIT JEE–2007 (b)

192

(c)

96

(d)

48

Q7.

At an election, a voter may vote for any number of candidates not greater than the number to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote, is (a)

Q8.

AIEEE–2006 6210

(b)

385

(c)

1110

(d)

If the letters of the word SACHIN are arranged in all possible ways and these words are written in dictionary order, then the word SACHIN appears at serial number (a)

Q9.

5070

600

(b)

601

(c)

602

AIEEE–2005 (d)

603

The number of ways of distributing 8 identical balls in 3 distinct boxes so that no box is empty, is AIEEE–2004 (a)

Q10.

5

(b)

8 3

(c)

38

(d)

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is (a)

Q11.

140

(b)

196

(c)

280

AIEEE–2003 (d)

346

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is (a)

Q12.

21

AIEEE–2003

65

(b)

30

(c)

54

(d)

57

The number of arrangements of the letters of the word BANANA, which the two N’s do not appear adjacently is (a)

20

IIT JEE–2002 (b)

40

(c)

60

(d)

80

Sets, Relations & Functions Q1.

Consider the following relations R = {(x, y)| x, y are real numbers and x = wy for some rational number w};

S=

m p , n q

m, n, p and q are integers such that n, q

0 and qm = pm. Then AIEEE–2010

(a)

R is an equivalence relation but S is not an equivalence relation

(b)

Neither R nor S is an equivalence relation

(c)

S is an equivalence relation but R is not an equivalence relation

(d)

R and S both are equivalence relations

Q2.

If A, B, and C are three sets such that A (a)

Q3.

Q4.

A=C

(b)

B=A

B=C

C and A (c)

A

B=A

C, then

B=

(d)

AIEEE–2009 A=B

For real x, let f(x) = x3 + 5x +1, then (a)

f is one-one but not onto R

(b)

f is onto R but not one-one

(c)

f is one-one and onto R

(d)

f is neither one-one nor onto R

Let f(x) = (x + 1)2 – 1, x

AIEEE–2009

–1

AIEEE–2009

Statement-I The set {x : f(x) = f –1(x)} = {0, –1} Statement-II f is a bijection. Q5.

Let R be the real line. Consider the following subsets of the plane R × R S = {(x, y): y = x + 1 and 0 < x < 2} T = {(x, y) : x – y is an integer} Which one of the following is true?

Q6.

AIEEE–2008

(a)

T is an equivalence relation on R but S is not

(b)

Neither S nor T is an equivalence relation on R

(c)

Both S and T are equivalence relations on R

(d)

S is an equivalence relation on R but T is not

Let f : N

Y be a function defined as f(x) = 4x +3 for some x

N}. Show that f is invertible

and its inverse is (a)

Q7.

g(y) =

y–3 4

AIEEE–2008 (b)

g(y) =

The largest interval lying in –

3y 4 3

,

2 2

(c)

g(y) = 4+

y 3 4

(d)

2

–x for which the function f(x) = 4 + cos–1

log(cos x) is defined, is (a) Q8.

[0, ]

g(y) =

y 3 4

x –1 + 2 AIEEE–2007

(b)

–

, 2 2

(c)

–

, 4 2

(d)

0,

2

Let W denotes the words in the English dictionary. Define the relation R by R = {(x, y) : the words x and y have at least one letter in common}. Then, R is (a)

reflexive, symmetric and not transitive

(b)

reflexive, symmetric and transitive

(c)

reflexive, not symmetric and transitive

W×W

AIEEE–2006

(d) Q9.

not reflexive, symmetric and transitive

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is

Q10.

AIEEE–2005

(a)

an equivalence relation

(b)

reflexive and symmetric

(c)

reflexive and transitive

(d)

only reflexive

2x , then f is both one-one and 1 – x2

B be a function defined by f(x) = tan–1

Let F : (–1, 1)

onto when B is in the interval

–

(a)

Q11.

–

(b)

x, if x is rational

f(x) =

0, if x is irrational

g(x) =

Q12.

, 2 2

AIEEE–2005

, 2 2

(c)

0,

(d)

2

0, if x is rational x, if x is irrational

. Then , f – g is

(a)

one-one and into

(b)

neither one-one nor onto

(c)

many-one and onto

(d)

one-one and onto

IIT JEE–2005

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a 8 relation on the set A = {1, 2, 3, 4}. The relation

(a)

AIEEE–2004 reflexive

(b)

transitive

(c)

not symmetric

(d)

If f(x) = sin x + cos x, g(x) = x2 – 1, then g{f(x)} is invertible in the domain (a)

Q14.

2

and

R is

Q13.

0,

0,

(b)

2

–

4

,

4

(c)

–

2

,

2

(d)

a function IIT JEE–2004 [0, ]

A function f from the set of natural numbers to integers defined by n –1 , 2 f(n) = n – , 2

(a)

n

is

n

one-one but not onto

AIEEE–2003

Q15.

(b)

onto but not one-one

(c)

one-one and onto both

(d)

neither one-one nor onto

Domain of definition of the function f(x) =

1 1 – , 4 2

(a)

Q16.

1 1 – , 2 2

(b)

[1, 4]

(b)

6

[1, 0]

for real valued x, is

1 1 – , 2 9

(c)

The domain of definition of the function f(x) = (a)

Q17.

sin –1 (2x)

5x – x 2 4

log10 (c)

(d)

is

[0, 5]

AIEEE–2002 (d)

f(x) w.r.t. the line y = x, then g(x) equals –

Q18.

x – 1, x 0

1

(b)

(a) Q19.

x , x –1. Then, for what value of x+1

2

(b)

–

2

r –1, –2

(b)

2

, x > –1

x – 1, x 0

(d)

The domain of definition of f(x) =

(a)

Q20.

x + 1 , x –1

Let f(x) =

is f [f(x)] = x ?

(c)

1

0 only when

(c)

0 for all real

0

IIT JEE–2001 (d)

log 2 x 3 is x 2 3x 2

(–2, )

–1 IIT JEE–2001

(c)

R –1, –2, –3

(d)

Let f( ) = sin (sin + sin3 ). Then, f( ) (a)

[5, 0]

IIT JEE–2002 x+1

(c)

1 1 – , 4 4

–1. If g(x) is the function whose graph is reflection of the graph of

Suppose f(x) = (x + 1)2 for x

(a)

IIT JEE–2003

–3, –1, –2 IIT JEE–2000

(b)

0 for all real

(d)

0 only when

0

Limits, Continuity & Differentiability Q1.

R be a differentiable function with f(0) = –1 and f ’(0) = 1. Let g(x) = [f(2f(x) +

If f : (–1, 1)

2)]2. Then g’(0) is equal to (a) Q2.

4

Let f : R

AIEEE–2010 –4

(b)

(c)

0

R be a positive increasing function with lim x

–2

(d)

f (3x) f (2x) = 1. Then, lim is x f (x) f (x)

equal to (a)

Q3.

AIEEE–2010

1

Let f : R

(b)

(c)

1 , for some c 3

Statement-II 0 < f(x)

1 2 2

(d)

e

x

1 2e – x

3

AIEEE–2010

R.

, for all x

R.

1

Let f(x) = (x –1)sin x –1 , if x 1 0

, if x 1

Then which one of the following is true?

Q5.

3 2

R be continuous function defined by f(x) =

Statement-I f(c) =

Q4.

2 3

(a)

f is differentiable at x = 1 but not at x = 0

(b)

f is neither differentiable at x = 0 nor at x = 1

(c)

f is differentiable at x = 0 and at x = 1

(d)

f is differentiable at x = 0 but not at x = 1

Let f : R

AIEEE–2008

R be function defined by f(x) = {x + 1, |x| + 1}. Then, which of the following is true? AIEEE–2007

(a)

f(x)

1 for all x

R

(b)

f(x) is not differentiable at x = 1

(c)

f(x) is differentiable everywhere

(d)

f(x) is not differentiable

2x

Q6.

lim x

4

(a)

Q7.

Q8.

Q9.

Q10.

f (t )dt

2

equals

2

x –

IIT JEE–2007

2

8

16

f(2)

2

(b)

(– , –1)

lim

1 n2

(a)

1 tan1 2

n

2

1 n2

2 n2

(– , )

4 n2

n n2

(b)

1 2

f

(d)

4 f(2)

is differentiable, is

1 x

(b) 2

2

(c)

x

The set of points, where f(x) = (a)

f(2)

(c) 2

AIEEE–2006

(0, )

(d)

(– , 0)

1 equals to

tan 1

AIEEE–2005

1 2

(c)

1

(d)

1 2

Let f be twice differentiable function satisfying f(1) = 1, f(2) = 4, f(3) = 9, then (a)

f”(x) = 2, x

(b)

f’(x) = 5 f”(x), for some x

(c)

there exists at least one x

(d)

none of the above

Let f(x) =

(0, )

1 IIT JEE–2005

(R)

1 – tan x ,x 4x –

4

,x

(1, 3) (1, 3) such that f”(x) = 2

0,

2

. If f(x) is continuous in 0,

2

, then f

4

is

AIEEE–2004 (a)

Q11.

1

(b)

a If lim 1 x x (a)

a

–

Q12.

If f(x) = xe

0

–1/2

(d)

= e2, then the values of a and b are

R 1 x

(c)

–1

2x

b x2

R, b

1/2

1 x

(b)

, ,

a = 1, b

R

x

0, then f (x) is

x

0

(c)

a

(a)

continuous as well as differentiable for all x

(b)

continuous for all x but not differentiable at x = 0

(c)

neither differentiable nor continuous at x = 0

(d)

discontinuous everywhere

AIEEE–2004 R, b = 2

(d)

a = 1, b = 2

AIEEE–2003

Q13.

a – n nx – tan x sin nx = 0, where n is non-zero real number, then a is equal to x2

If lim x

0

IIT JEE–2003 (a)

Q14.

lim

n 1 n

(b)

1p

2 p 3p np 1

x

(a)

Q15.

0

1

n

(d)

n+

is equal to

(b)

p 1

Let f : R

np

(c)

1 1– p

1 n

AIEEE–2002 (c)

1 1 – p p –1

f (1 x) R be such that f (1) = 3 and f’(1) = 6. Then, lim x 0 f (1)

1

(d)

p 2

1/ x

equals IIT JEE–2002

(a) Q16.

(b)

e

(c)

e

2

(d)

The left hand derivative of f(x) = [x] sin( x) at x = k, k an integer is (a)

Q17.

1

1/2

(–1)k(k – 1)

Let f : R

(b)

(–1)k–1 (k – 1)

(c)

R be any function. Define g : R

(–1)k k

e

3

IIT JEE–2001 (d)

(–1)k–1 k

R by g(x) = |f (x)| for all x. then, g is IIT JEE–2000

(a)

onto if f is onto

(b)

one-one if f is one-one

(c)

continuous if f is continuous

(d)

differentiable if f is differentiable

Differentiation Q1.

Let y be an implicit of x defined by x2x – 2xx cot y – 1 = 0. Then, y’(1) equals (a)

Q2.

–1

(b)

1

(c)

Let f(x) = x|x| and g(x) = sin x Statement-I

log 2

(d)

AIEEE–2009 –log 2 AIEEE–2009

gof is differentiable at x = 0 and its derivative is continuous at that point.

Statement-II gof is twice differentiable at x = 0. Q3.

d 2x is equal to dy 2

IIT JEE–2007

Q4.

(a)

d2y dx 2

(c)

d2y dx 2

x

(d)

d2y – dx 2

y

–1

–3

dy dx dy dx

9 –3

dy is dx

(b)

AIEEE–2006

xy

(c)

x y

(d)

y x

If f ”(x) = –f(x), where f(x) is a continuous double differentiable function and g(x) = f ’(x). If

(a)

x f 2

0

2

+ g

2

x 2

and F(5) = 5, then f(10) is

(b)

5

(c)

IIT JEE–2006

10

(d)

25

If y is a function of x and log(x + y) = 2xy, then the value of y’(0) is equal to (a)

Q7.

(b)

d2y dx 2

–2

xy

F(x) =

Q6.

dy dx

If xm yn = (x + y)m + n, then (a)

Q5.

–1

1

If y = (x + (a)

n2y

(b)

–1

(c)

1 x 2 )n, then (1 + x2)

d2y dy +x is 2 dx dx

–n2y

(c)

(b)

2

(d)

IIT JEE–2004 0 AIEEE–2002

–y

(d)

2x2y

Application of Derivatives Q1.

The equation of the tangent to the curve y = x +

4 , that is parallel to the x-axis, is x2 AIEEE–2010

(a) Q2.

y=0

Let f : R

(b)

y=1

R be defined by f(x) =

(c)

y=2

k – 2x, if x

–1

2x 3, if x

–1

(d)

. If f has a local minimum at x = –1,

then a possible value of k, is (a) Q3.

1

(b)

y=3

AIEEE–2010 0

(c)

–

1 2

(d)

–1

Given, P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P’(x) = 0. If P(–1) < P(1), then in the interval [–1, 1] (a)

P(–1) is the minimum and P(1) is the maximum of P

AIEEE–2009

Q4.

(b)

P(–1) is not minimum but P(1) is the maximum of P

(c)

P(–1) is the minimum and P(1) is not the maximum of P

(d)

neither P(–1) is the minimum nor P(1) is the maximum of P

The shortest distance between the line y – x = 1 and the curve x = y2 is (a)

Q5.

3 2 8

2 3 8

(b)

(c)

3 2 5

AIEEE–2009

3 4

(d)

Suppose the cubic x3 – px + q has three distinct real roots where p > 0 and q > 0. Then, which one of the following holds ?

Q6.

AIEEE–2008

(a)

The cubic has maxima at both

(b)

The cubic has minima at

(c)

The cubic has minima at –

(d)

The cubic has minima at both

p and – 3

p 3

p and maxima at – 3

p 3

p and maxima at 3

p 3

p and – 3

p 3

How many real solutions does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have ? AIEEE–2008 (a)

5

(b)

7

(c)

1

(d)

3 3

Q7.

The total number of local maxima and local minima of the function f(x) = 2 x , – 3 x x 2 / 3 , –1 x

is (a) Q8.

2

IIT JEE–2008 0

(b)

1

(c)

2

(d)

3

A value of c for which the conclusion of Mean Value theorem holds for the function f(x) = loge x on the interval [1, 3] is (a)

Q9.

–1

2 log3 e

AIEEE–2007 (b)

1 loge 3 2

(c)

log3 e

(d)

The function f(x) = tan–1 (sin x + cos x) is an increasing function in (a)

, 4 2

(b)

–

, 2 4

(c)

0,

2

loge 3 AIEEE–2007

(d)

–

, 2 2

Q10.

The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c – 1, ec – 1) and (c + 1, ec + 1)

Q11.

(a)

on the left of x = c

(b)

on the right of x = c

(c)

at no paint

(d)

at all points

If x is real, the maximum value of (a)

Q12.

IIT JEE–2007

41

(b)

3x 2 9x 17 is 3x 2 9x 7

1

(c)

AIEEE–2006

17 7

(d)

1 4

A spherical iron ball 10 cm in radius is coated with a layer ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice 15 cm, then the rate at which the thickness of ice decreases, is (a)

Q13.

AIEEE–2005

5 6

(b)

1 54

(c)

1 18

(d)

1 36

The tangent at (1, 7) to curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 at IIT JEE–2005 (a)

Q14.

(6, 7)

(b)

(–6, 7)

(c)

(6, –7)

(d)

(–6, –7)

The normal to the curve x = a(1 + cos ), y = a sin at ‘ ’ always passes through the fixed point AIEEE–2004 (a)

Q15.

Q16.

(a, a)

(b)

(0, a)

(c)

(0, 0)

(d)

(a, 0)

If f(x) = x3 + bx2 + cx + d and 0 < b2 < c, then in (– , )

IIT JEE–2004

(a)

f(x) is strictly increasing function

(b)

f(x) has a local maxima

(c)

f(x) is strictly decreasing function

(d)

f(x) is bounded

Let f(a) = g(a) = k and their nth derivatives f n(a), gn(a) exist and are not equal for some n. Further, if lim x

a

f (a) g ( x) – f (a) – g (a) f ( x) g (a ) = 4, then the value of k is equal to g ( x) f ( x) AIEEE–2003

(a) Q17.

4

(b)

2

(c)

1

(d)

0

If f(x) = x2 + 2bx + 2c2 and g(x) = –x2 – 2cx + b2 such that min f(x) > g(x), then the relation between b and c is

IIT JEE–2003

(a)

no real values of b and c

(b)

0

2 and J < 2 3

Let f (x) = f (x) + f

(a)

1 2

The value of

1

I>

2 and J > 2 3

1 , where f (x) = x (b)

a

Q5.

(b)

x 1

(c)

I<

2 and J < 2 3

(d)

I<

log t dt. Then, f (e) equals 1 t

0

(c)

1

2 and J > 2 3

AIEEE–2007

(d)

2

x f ’(x) dx, a > 1, where [x] denotes the greatest integer not exceeding x is AIEEE–2006

(a)

[a] f (a) – {f (1) + f (2) +…..+ f ([a])}

(b)

[a] f ([a]) – {f (1) + f (2) +….+ f (a)}

(c)

a f ([a]) – {f (1) + f (2) +…..+ f (a)}

(d)

a f (a) – {f (1) + f (2) +…..+ f ([a])}

– /2

Q6.

–3 / 2

[(x + )3 + cos 2 (x + 3 )] dx is equal to

AIEEE–2006

4

(a)

Q7.

32

The value of (a)

2

4

+

–

2

(b)

2

(c)

4

cos 2 x dx, a > 0, is 1 + ax (b)

/a

–1

(d)

32

AIEEE–2005, IIT JEE–2001 (c)

/2

(d)

a

1

Q8.

Q9.

If

sin x

(a)

3

If

xf (sin x)dx = A

0

(a) Q10.

Q12.

2 0

(a)

2/5

Q15.

1

(b)

n 1 x2 1 x

2

(c)

1 3

is

IIT JEE–2005

1/3

(d)

None of these

f (sin x) dx, then A is equal to (c) t2

x f(x)dx =

0

1

/4

(d)

2

4 2 5 t , then f equals 25 5

–5/2

0

AIEEE–2004

(c)

1

IIT JEE–2004 (d)

5/2

x (1 – x)n dx is 1

AIEEE–2003 (c)

n 2

1 n 1

–

1 n 2

(d)

1 n 1

2

e–t dt, then f (x) increases in

(2, 2)

(b)

no value of x

2–

–1/2

(b)

2+

[x] log

1 x 1– x

–1/2

0

1 n 2

(c)

(0, )

(d)

(– , 0) AIEEE–2002

2

ecos x sin x , 2

+

IIT JEE –2003

[x 2 ] dx is

If f (x) = (a)

3

(b)

The integral (a)

(0, /2), then f

(b)

1/ 2

Q14.

0

0

If f (x) = (a)

Q13.

/2

The value of the integral I = (a)

x

(b)

If f (x) is differentiable and (a)

Q11.

t 2 f (t) dt = 1 – sin x,

,

2

(b)

0

|x|

2, then

(b)

1

2–1

(c)

(d)

– 2 – 3 +5

dx equals

IIT JEE–2002 (c)

3 –2

1

(d)

2 log (1/2)

f (x) dx is equal to (c)

2

IIT JEE–2000 (d)

3

Area of Curves Q1.

The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x = is

Q2.

AIEEE–2010

(a)

(4 2 – 2) sq unit

(b)

(4 2 + 2)sq unit

(c)

(4 2 – 1) sq unit

(d)

(4 2 + 1)sq unit

The area of the region bounded by the parabola (y – 2)2 = x – 1, the tangent to the parabola at the point (2, 3) and the x-axis is (a)

Q3.

3 2

6 sq unit

(b)

AIEEE–2009 9 sq unit

(c)

12 sq unit

(d)

3 sq unit

The area of the plane region bounded by the curves x + 2y2 = 0 and x + 2y2 = 1 is equal to AIEEE–2008 (a)

Q4.

(b)

5 sq unit 3

1 sq unit 3

(c)

2 sq unit 3

(d)

The area enclosed between the curves y2 = x and y = | x | is (a)

Q5.

4 sq unit 3

2/3 sq unit

(b)

1 sq unit

(c)

AIEEE–2007

1/6 sq unit

(d)

1/3 sq unit

The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the line x = 4, y = 4 and the coordinate axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom, then S1 : S2 : S3 is (a)

Q6.

2:1:2

(b)

AIEEE–2005 1:1:1

1:2:1

(d)

1:2:3

Let f (x) be a non-negative continuous functions. Such that the area bounded by the curve y = f (x), x-axis and the coordinates x =

2 (a)

(c)

Q7.

(c)

4

,x=

>

sin

is

4

4

cos

2

is

. Then f

AIEEE–2005

1–

4

4

– 2

(b)

2 –1

(d)

1–

4

4

2

– 2 1

The area bounded by the curve y = (x + 1)2, y = (x – 1)2 and the line y =

1 is 4

(a)

(d)

1/6 sq unit

(b)

2/3 sq unit

(c)

1/4 sq unit

AIEEE–2005 1/3 sq unit

Q8.

The area of the region bounded by the curve y = |x – 2|, x = 1, x = 3 and the axis is AIEEE –2004 (a)

Q9.

4 sq unit

(b)

2 sq unit

(c)

3 sq unit

(d)

1 sq unit

The area of the region bounded by y = ax2 and x = ay2, a > 0 is 1, then a is equal to IIT JEE–2004 (a)

Q10.

1

1 3

(b)

(c)

1 3

(d)

None of these

The area bounded by the curve y = 2x – x2 and the straight line y = –x is given by AIEEE–2002 (a)

Q11.

9/2 sq unit

(b)

43/6 sq unit

(c)

35/6 sq unit

(d)

None of these

The area bounded by the curves y = | x | – 1 and y = –| x | + 1 is (a)

1 sq unit

(b)

2 sq unit

(c)

IIT JEE–2002

2 2 sq unit

(d)

4 sq unit

Differential Equations Q1.

Q2.

Solution of the differential equation cos xdy = y(sin x – y)dx, 0 < x < (a)

sec x = (tan x + c)y

(b)

y sec x = tan x + c

(c)

y tan x = sec x + c

(d)

tan x = (sec x + c)y

2

, is

AIEEE–2010

c x The differential equation which represents the family of curves y = c1e 2 , where c1 and c2 are

arbitrary constants is (a) Q3.

2

y’ = y

AIEEE–2009 (b)

y” = y’ y

(c)

yy” = y’

(d)

yy” = (y’)2

The differential equation of the family of circles with fixed radius 5 unit and centre on the line y = 2 is

Q4.

AIEEE–2008

(a)

(x – 2) y’ = 25 – (y – 2)

(c)

(y – 2) y’2 = 25 – (y – 2)2

2

2

2

The solution of the differential equation

(b)

(x – 2) y’ = 25 – (y – 2)

(d)

(y – 2)2 y’2 = 25 – (y – 2)2

2

2

dy x + y = satisfying the condition y(1) = 1 is dx x AIEEE–2008

(a) Q5.

y = x log x + x

(b)

y = log x + x

(c)

y = x log x + x2

(d)

y = xe(x – 1)

The differential equation of all circles passing through the origin and having their centres on the x-axis is

AIEEE–2007

Q6.

Q7.

dy dx

(a)

x2 = y2 + xy

(c)

y2 = x2 + 2xy

dy dx

The differential equation

(b)

x2 = y2 + 3xy

dy dx

(d)

y2 = x2 – 2xy

dy dx

1 – y2 determines family of circles with y

dy = dx

(a)

variable radii and fixed centre at (0, 1)

(b)

variable radii and a fixed centre at (0, –1)

(c)

fixed radius 1 and variable centres along the x-axis

(d)

fixed radius 1 and variable centres along the y-axis

IIT JEE–2007

The differential equation whose solution is Ax2 + By2 = 1, where A and B are arbitrary constant, is of

Q8.

(a)

first order and second degree

(b)

first order and first degree

(c)

second order and first degree

(d)

second order and second degree

If x

(a) Q9.

x = cy y

(b)

3

log

(b)

1 =c xy

(b)

If y = y(x) and

2 + sin x y+1

(a)

Q12.

log

y = cx x

(c)

AIEEE–2005

y = cy x

x log

(d)

y log

2

(c)

–

1 3

(b)

–

1 + log y = c xy

1

(d)

dy dx

= –cos x, y(0) = 1, then y

2 3

–

(c)

(x – 2) = k e

(c)

xe

tan –1 y

tan –1 y

= tan–1 y + k

(b)

2 xe

(d)

xe

equals

1 3

0

tan –1 y

2 tan –1 x

–1

y

)

dy = 0 is dx

2 tan =e

–1

= tan–1 x

y

+k

(d)

log y = cx

IIT JEE–2004

(d)

tan The solution of the differential equation (1 + y2) + (x – e

(a)

2

= cx

AIEEE–2004

1 + log y = c xy

(c)

x y

IIT JEE–2005

The solution of the differential equation y dx + (x + x2y)dy = 0 is (a)

Q11.

dy = y(log y – log x + 1), then the solution of the equation is dx

If x dy = y(dx + y dy), y(1) = 1 and y(x) > 0. Then y(–3) is equal to (a)

Q10.

AIEEE–2006

1

AIEEE–2003

Q13.

If y(t) is a solution of (1 + t)

(a)

–

1 2

(b)

dy – ty = 1 and y(0) = –1, then y(1) is equal to dt e+

1 2

(c)

e–

1 2

(d)

IIT JEE–2003

1 2

Trigonometric Ratios & Equations Q1.

Let cos( + ) =

4 5 and let sin( – ) = , where 0 5 13

,

4

. Then, tan 2 is equal to AIEEE–2010

25 16

(a) Q2.

56 33

(b)

(c)

19 22

(d)

20 7

For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is

Q3.

Q4.

(a)

there is a regular polygon with

1 r = 2 R

(b)

there is a regular polygon with

1 r = R 2

(c)

there is a regular polygon with

2 r = 3 R

(d)

there is a regular polygon with

3 r = 2 R

Let A and B denote the statements A : cos

+ cos

B : sin

+ sin

+ cos = 0 + sin = 0

If cos( – ) + cos( – ) + cos( – ) = –

3 , then 2

(a)

A is true and B is false

(b)

A is false and B is true

(c)

both A and B are true

(d)

both A and B are false

AIEEE–2009

The number of values of x in [0, 3 ] such that 2sin2 x + 5sin x – 3 = 0 is (a)

Q5.

AIEEE–2010

1

If 0 < x <

(b)

2

and cos x + sin x =

(c)

4

1 , then tan x is equal to 2

AIEEE–2006 (d)

6 AIEEE–2006

– 4

(a)

Q6.

7

(b)

3

0,

Let

4

1

7

(c)

4

1– 7 4

4– 7 3

(d)

and t1 = (tan )tan , t2 = (tan )cot , t3 = (cot )tan , t4 = (cot )cot , then IIT JEE–2006

(a) Q7.

t1 > t2 > t3 > t4

In a triangle PQR,

(b) R=

2

t4 > t3 > t1 > t2 . If tan

(c)

t3 > t1 > t2 > t4

(d)

t2 > t 3 > t1 > t4

Q P and tan are the roots of ax2 + bx + c = 0, a 0, 2 2

then

AIEEE–2005

(a) Q8.

b=a+c

(b)

b=c

Cos( – ) = 1 and cos( + ) =

(c)

1 where , e

c=a+b

(d)

a=b+c

[– , ]. The number of pairs of ,

which satisfy both the equation is (a) Q9.

0

Let ,

such that

– 2 (a)

Q10.

(b) <

–

IIT JEE2005

1

(c)

< 3 . If sin

+ sin

=–

2

(d)

21 , cos 65

+ cos

=–

4

27 , then cos 65

is –

AIEEE–2004

3 130

Given both

(b)

and

3 130

are acute angles sin =

(c)

6 65

(d)

–

6 65

1 1 , cos = , then the value of + belongs to 2 3 IIT JEE–2004

(a)

Q11.

,

3 6

(b)

2 2 3 ,

(c)

2 5 , 3 6

In a triangle ABC, medians AD and BE are drawn. If AD = 4,

DAB =

(d)

6

and

the area of the ABC is (a)

8 sq unit 3

5 , 6 ABE =

3

, then

AIEEE–2003 (b)

16 sq unit 3

(c)

32 3 3

(d)

64 sq unit 3

Q12.

4xy

sin2 =

x+y x–y

(a) Q13.

If (a)

+

=

2

is true, if and only if

0

2

(b)

and

AIEEE–2002

x=–y

(c)

x=y

(d)

+ = , then tan is equal to

2(tan + tan )

(b)

tan + tan

x

0, y

0

IIT JEE–2001

(c)

tan + 2tan

(d)

2tan + tan

Heights & Distances Q1.

AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60o. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45o. Then, the height of the pole is

Q2.

(a)

7 3 2

(c)

7 3 2

1 m 3 1

3 1 m

AIEEE–2008

(b)

7 3 2

1 m 3 –1

(d)

7 3 2

3 –1 m

A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB(= a) subtends as an angle of 60o at the foot of the tower and the angle of elevation of the top of the tower from A or B is 30o. The height of the tower is (a)

Q3.

2a 3

(b)

2a 3

(c)

a 3

(d)

AIEEE–2007

3

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60o and when he retires 40 m away from the tree, the angle of elevation becomes 30o. The breadth of the river is (a)

Q4.

20 m

(b)

30 m

(c)

AIEEE–2004 40 m

(d)

60 m

The upper 3/4th portion of a vertical pole subtends an angle tan–1 3/5 at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is (a)

AIEEE–2002 40 m

(b)

60 m

(c)

80 m

(d)

20 m

Q5.

A man from the top of a 100 m high tower sees a car moving towards the tower at an angle of depression of 30o. After some time, the angle of depression becomes 60o. The distance (in metres) travelled by the car during this time is (a)

Q6.

100 3

(b)

IIT JEE–2001

200 3 3

100 3 3

(c)

(d)

200 3

A pole stands vertically inside a triangular park ABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in

ABC the foot of the pole is at the IIT JEE–2000

(a)

centroid

(b)

circumcentre

(c)

incentre

(d)

orthocenter

Inverse Trigonometric Functions Q1.

The value of cot cos ec –1

(a)

Q2.

5 17

(b)

If 0 < x < 1, then

1 x Q3.

If sin–1 (a)

Q4.

6 17

(c)

3 17

(d)

4 17

+ cosec–1

1

x

x

1 x2

1 x2

(d)

5 = , then a value of x is 4 2 (b)

3

y = , then 4x2 – 4xy cos 2

4

(c)

IIT JEE–2008

(b)

(c)

AIEEE–2007 4

(d)

5

+ y2 is equal to

2sin2

(c)

–4sin2

AIEEE–2005 (d)

4sin2

If sin {cot –1(1 + x)} = cos(tan–1 x), then x is equal to (a)

Q6.

(b)

2

If cos–1 x – cos–1 (a)

Q5.

x 5

AIEEE–2008

1 x 2 [{x cos(cot –1 x) + sin(cot –1 x)}2 – 1]1/2 is equal to

x

(a)

5 2 tan –1 is 3 3

1 2

(b)

1

(c)

IIT JEE–2004 0

(d)

The equation sin–1 x = 2sin–1 a has a solution for (a)

1 1 < |a| < 2 2

(b)

all a

1 2

–

AIEEE–2003 (c)

|a|

1 2

(d)

|a|

1 2

Q7.

Q8.

tan–1

1 2 + tan–1 is equal to 4 9

(a)

3 1 cos–1 5 2

If sin–1 x –

x2 2

(b)

1 –1 3 sin 5 2

x2 x4 –1 – .... + cos x 2 – 4 2

AIEEE–2002

1 –1 3 tan 5 2

(c)

(d)

x6 – .... = , 0 < | x | < 2 4

to (a)

tan–1

1 2

2 , then x is equal IIT JEE–2001

1 2

(b)

1

–

(c)

1 2

(d)

–1

Rectangular Coordinate Systems Q1.

Three distinct points A, B and C given in the 2 – dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (–1, 0) is equal to

(a) Q2.

1 . Then, the circumcentre of the triangle ABC is at the input 3 5 ,0 4

(b)

5 ,0 2

5 ,0 3

(c)

AIEEE–2009

(d)

(0, 0)

Consider three points P = {–sin( – ), – cos } Q = {cos( – ), sin } and R = {cos( –

Q3.

+ ), sin( – )} where 0 < , ,

(a)

P lies on the line segment RQ

(b)

Q lies on the line segment PR

(c)

R lies on the line segment QP

(d)

P, Q, R are non-collinear

<

4

. Then

IIT JEE–2008

Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is 1, then the set of values which ‘k’ can take is given by AIEEE–2007 (a)

Q4.

{1, 3}

(b)

{0, 2}

(c)

{–1, 3}

(d)

{–3, –2}

Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the OPQ. The point R inside the OPQ is such that OPR, PQR, OQR are of equal area. Then R is

IIT JEE–2007

(a) Q5.

4 ,3 3

(b)

3,

2 3

(c)

4 3

(d)

(a)

1 7 – , 3 3

(b)

–1,

7 3

is

AIEEE–2005

1 7 , 3 3

(c)

(d)

(a)

2x + 3y = 9

(b)

2x – 3y = 7

(c)

3x + 2y = 5

(d) 3x – 2y = 3

The locus of centroid of triangle whose vertices are (a cos t, a sin t), (b sin t, b cos t) and (1, 0) AIEEE–2003

(a)

(3x – 1)2 + 9y2 = a2 – b2

(b)

(3x – 1)2 + 9y2 = a2 + b2

(c)

(3x + 1)2 + 9y2 = a2 + b2

(d)

(3x + 1)2 + 9y2 = a2 – b2

If the equation of the locus of a point equidistant from the points (a1, b1) and (a2, b2) is (a1 – a2)x + (b1 – b2)y + c = 0, then c is

Q9.

7 3

AIEEE–2004

where t is a variable parameter is

Q8.

1,

Let A(2 – 3) and B(–2, 1) be the vertices of a ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line

Q7.

4 3

If a vertex of a triangle is (1, 1) and the mid point of two sides of a triangle through this vertex are (–1, 2) and (3, 2), then the centroid of the

Q6.

3,

AIEEE–2003

(a)

1 2 a2 b22 – a12 – b12 2

(b)

(c)

1 2 a1 a22 b12 b22 2

(d)

a12 – a22 b12 – b22

a12 b12 – a22 b22

The incentre of the triangle with vertices A(1, 3 ), B(0, 0), C(2, 0) is (a)

1,

3 2

(b)

2 1 , 3 3

(c)

2 3 , 3 2

IIT JEE–2000 (d)

1,

1 3

Straight Lines Q1.

The line L given by has the equation (a)

23 15

x y + = 1 passes through the point (13, 32). The line K is parallel to L and 5 b

x y + = 1. Then, the distance between L and K is c 3 (b)

17

(c)

17 15

AIEEE–2010 (d)

23 17

Q2.

The lines p(p2 + 1) x – y + q = 0 and (p2 + 1)2 x + (p2 + 1)y + 2q = 0 are perpendicular to a common line for

Q3.

AIEEE–2009

(a)

exactly one value of p

(b)

exactly two values of p

(c)

more than two values of p

(d)

no values of p

The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y–intercept –4. Then, a possible value of k is (a)

Q4.

4

(b)

AIEEE–2008 1

(c)

2

(d)

Let P = (–1, 0), Q = (0, 0) and R = (3, 3 3 ) be three points. The equation of the bisector of the PQR is (a)

Q5.

–2

AIEEE / 2007, IIT JEE–2007

3 x+y=0 2

(b)

x+

3y=0

(c)

3x+y=0

(d)

x+

A straight line through the point A(3, 4) is such that in intercept between the axis is bisected at A. Its equation is (a)

Q6.

3 y=0 2

4x + 3y = 24

AIEEE–2006 (b)

3x + 4y = 25

(c)

If (a, a2) falls inside the angle made by the lines, y =

x+y=7

(d)

3x – 4y + 7 = 0

x , x > 0 and y = 3x, x > 0, then a 2 AIEEE–2006

(a)

Q7.

1 ,3 2

–3, –

(b)

1 2

(c)

0,

If non-zero numbers a, b, c are in HP, then the straight line

1 2

(d)

(3, )

x y 1 + + = 0 always passes a b c

through a fixed point. That point is (a) Q8.

1, –

1 2

(b)

(1, –2)

AIEEE–2005 (c)

(–1, –2)

(d)

(–1, 2)

The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is –1, is

AIEEE–2004

(a)

x y x y + = –1, + = –1 2 3 –2 1

(b)

x y x y – = –1, + = –1 2 3 –2 1

(c)

x y x y + = 1, + =1 2 3 2 1

(d)

x y x y – = 1, + =1 2 3 –2 1

Q9.

A square of side a lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle

0

4

with the positive direction of x-axis. The

equation of its diagonal not passing through the origin is

Q10.

Q11.

(a)

y(cos

– sin ) – x(sin

– cos ) = a

(b)

y(cos

+ sin ) – x(sin

– cos ) = a

(c)

y(cos

+ sin ) + x(sin

+ cos ) = a

(d)

y(cos

+ sin ) + x(sin

– cos ) = a

AIEEE–2003

Three straight lines 2x + 11y – 5 = 0, 24x + 7y – 20 = 0 and 4x – 3y – 2 = 0 (a)

form a triangle

(b)

are only concurrent

(c)

are concurrent with on line bisecting the angle between the other two

(d)

none of the above

A straight line through the origin meets the parallel lines 4x + 2y = 9 and 2x + y = –6 at points P and Q respectively. Then, the point O divides the segment PQ in the ration (a)

Q12.

AIEEE–2002

1:2

(b)

3:4

(c)

2:1

(d)

IIT JEE–2002 4:3

Area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, y = nx + 1 is equal to IIT JEE–2001 (a)

m n m–n

2

(b)

2

(c)

m n

1 m n

(d)

1 m–n

The Circle Q1.

The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points, if AIEEE–2010 (a)

Q2.

–85 < m < –35

(b)

–35 < m < 85

(c)

15 < m < 65

(d)

If P and Q are the points of intersection of the circles x2 + y2 + 3x + 7y + 2p – 5 = 0 and x2 + y2 + 2x + 2y – p2 = 0, then there is a circle passing through P, Q and (1, 1) and

Q3.

35 < m < 85

(a)

all values of p

(b)

all except one value of p

(c)

all except two values of p

(d)

exactly one value of p

AIEEE–2009

The point diametrically opposite to the point P(1, 0) on the circle x2 + y2 + 2x + 4y – 3 = 0 is AIEEE–2008

(a) Q4.

(3, 4)

(b)

(3, –4)

(c)

(–3, 4)

(d)

(–3, –4)

Consider a family of circles which are passing through the point (–1, 1) and are tangent to x-axis. If (h, k) is the centre of circle, then (a)

Q5.

k

1/2

(b)

–1/2

AIEEE–2007 k

1/2

(c)

k

1/2

(d)

0 < k < 1/2

Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is (a)

Q6.

3

(b)

IIT JEE–2007

2

(c)

3/2

(d)

Let C be the circle with centre (0, 0) and radius 3. The equation of the locus of the mid points of the chords of the circle C that subtend an angle 2 /3 at its centre is (a)

Q7.

x2 + y2 =

27 4

(b)

x2 + y2 =

9 4

(c)

x2 + y2 =

AIEEE–2006

3 2

(d)

AIEEE–2005

(a)

no value of a

(b)

exactly one value of a

(c)

exactly two values of a

(d)

infinitely many values of a

If a circle passes through the point (a, b) and cuts the circle x2 + y2 = 4 orthogonally; then the locus of its centre is

Q9.

AIEEE–2004

(a)

2ax + 2by + (a + b + 4) = 0

(b)

2ax + 2by – (a + b + 4) = 0

(c)

2ax – 2by + (a + b + 4) = 0

(d)

2ax – 2by – (a2 + b2 + 4) = 0

2

2

2

2

2

2

If the two circles (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct points, then (a)

Q10.

x2 + y2 = 1

If the circles x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct point P and Q then the line 5x + by – a = 0 passes through P and Q for

Q8.

1

(3, 7)

AIEEE–2003 (b)

(4, 7)

(c)

(2, 5)

(d)

(6, 9)

The greatest distance of the point P(10, 7) from the circle x2 + y2 – 4x – 2y – 20 = 0 is AIEEE–2002 (a)

Q11.

10 unit

(b)

15 unit

(c)

5 unit

(d)

none of these

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and PQ intersect at a point X on the circumference of the circle, then 2r is equal to (a)

PQ RS

(b)

PQ RS 2

(c)

2PQ RS PQ RS

(d)

IIT JEE–2001

PQ 2

RS 2 2

Q12.

The PQR is inscribed in the circle x2 + y2 = 25. If Q and R have coordinates (3, 4) and (–4, 3) respectively (a)

QPR is

IIT JEE–2000 (b)

2

(c)

3

(d)

4

6

Parabola Q1.

If two tangents drawn from a point P to the parabola y2 = 4x are at right angles, then the locus of P is (a)

Q2.

AIEEE–2010 x=1

(b)

2x + 1 = 0

(c)

x = –1

(d)

A parabola has the origin as its focus and the line x = 2 as the directrix. Then, the vertex of the parabola is at (a)

Q3.

2x – 1 = 0

(2, 0)

AIEEE–2008 (b)

(0, 2)

(c)

(1, 0)

(d)

(0, 1)

Consider the two curves C1 : y2 = 4x C2 : x2 + y2 – 6x + 1 = 0, then

Q4.

IIT JEE–2008

(a)

C1 and C2 touch each other only at one point

(b)

C1 and C2 touch each other exactly at two points

(c)

C1 and C2 intersect (but do not touch) at exactly two points

(d)

C1 and C2 neither intersect nor touch each other

The equation of the tangent to the parabola y2 = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent, is (a)

Q5.

(b)

(2, 4)

(c)

The locus of the vertices of the family of parabolas y = (a)

Q6.

(0, 2)

xy =

35 36

(b)

xy =

64 105

(c)

(–2, 0)

(d)

a3x 2 a3x + – 2a is 3 2 xy =

105 64

(d)

AIEEE–2007 (–1, 1) AIEEE–2006 xy =

3 4

The axis of a parabola is along the line y = x and the distance of its vertex from the origin is

2

and that of its focus from the origin is 2 2 . If the vertex and focus lie in the first quadrant, the equation of the parabola is

IIT JEE–2006

(a)

(x + y)2 = x – y – 2

(b)

(x – y)2 = x + y – 2

(c)

(x – y)2 = 4(x + y – 2)

(d)

(x – y)2 = 8(x + y – 2)

Q7.

If a

0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas

y2 = 4ax and x2 = 4ay, then

Q8.

AIEEE–2004

2

2

2

(a)

d + (2b + 3c) = 0

(b)

d + (3b + 2c) = 0

(c)

d + (2b – 3c) = 0

(d)

d2 + (3b + 2c)2 = 0

2

2

The normal at the point bt12 , 2bt2 , then (a)

Q9.

2

t2 = – t1 –

2 t1

(b)

t2 = –t1 +

2 t1

AIEEE–2003 (c)

t2 = t1 –

2 t1

(d)

t2 = t1 +

The focal chord to y2 = 16x is tangent to (x – 6)2 + y2 = 2, then the possible values of the slope of this chord are (a)

Q10.

IIT JEE–2003

{–1, 1}

(b)

{–2, 2}

(c)

{–2, 1/2}

(d)

{2, –1/2}

Two common tangents to the circle x2 + y2 = 2a2 and parabola y2 = 8ax are (a)

Q11.

x = (y + 2a)

(b)

y = (x + 2a)

(c)

x = (y + a)

(d)

AIEEE–2002 y = (x + a)

The locus of the mid point of the line segment joining the focus to a moving point on the parabola y2 = 4 ax is another parabola with directrix (a)

Q12.

2 t1

x = –a

(b)

x=–

a 2

IIT JEE–2002 (c)

x=0

(d)

x=

a 2

If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is IIT JEE–2000 (a)

1/8

(b)

8

(c)

4

(d)

1/4

Ellipse Q1.

The ellipse x2 + 4y2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which is turn in inscribed in another ellipse that passes through the point (5, 0). Then, the equation of the ellipse is (a)

Q2.

2

AIEEE–2009 2

x + 12y = 16

(b)

2

2

4x + 48y = 48

(c)

2

2

4x + 64y = 48

(d)

2

x + 16y2 = 16

A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is length of semimajor axis is (a)

Q3.

5 3

(b)

1 , then 2

AIEEE–2008

8 3

(c)

2 3

(d)

4 3

In an ellipse, the distance between its foci is 6 and minor-axis is 8. Then, its eccentricity is

AIEEE–2006 (a) Q4.

1/2

(b)

4/5

(c)

1/ 5

(d)

If the angle between the lines joining the end points of minor-axis of an ellipse with its foci is then the eccentricity of the ellipse is (a)

Q5.

1/2

(b)

1/ 2

3 /2

(c)

The eccentricity of an ellipse, with centre at the origin, is

(a)

2

2

3x + 4y = 1

(d)

1/2 2

1 . If one directrix is x = 4, the 2 AIEEE–2004

(b)

2

2

3x + 4y = 12

Tangent is drawn to the ellipse Then the value of

2

AIEEE–2005

equation of the ellipse is

Q6.

3/5

(c)

2

2

4x + 3y = 1

(d)

x2 + y2 = 1 at (3 3 cos , sin ) (where, 27

4x2 + 3y2 = 12 (0, /2)).

such that the sum of intercepts on axes made by this tangent is minimum, is IIT JEE–2003

(a)

Q7.

(b)

3

(c)

6

(d)

8

The equation of the ellipse whose foci are ( 2, 0) and eccentricity is

(a)

x2 12

y2 =1 16

(b)

x2 16

y2 =1 12

(c)

x2 16

y2 =1 8

1 , is 2 (d)

4 AIEEE–2002

none of these

Hyperbola Q1.

Consider a branch of the hyperbola x2 – 2y2 – 2 2 x – 4 2 y – 6 = 0 with vertex at the point A. Let B be one of the end points of its latusrectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

IIT JEE–2008

(a)

1–

2 sq unit 3

(b)

3 –1 sq unit 2

(c)

1

2 sq unit 3

(d)

3 1 sq unit 2

,

Q2.

For the hyperbola

x2 cos 2

–

y2 sin 2

= 1. Which of the following remains constant when

?

Q3.

AIEEE–2007/IIT JEE–2003

(a)

directrix

(b)

abscissae of vertices

(c)

abscissae of foci

(d)

eccentricities

A hyperbola, having the transverse axis of length 2sin , is confocal with the ellipse 3x2 + 4y2 = 12. Then, its equation is

Q4.

varies

IIT JEE–2007

(a)

x2 cosec2 – y2 sec2 = 1

(b)

x2 sec2 – y2 cosec2 = 1

(c)

x2 sin2 – y2 cos2 = 1

(d)

x2 cos2 – y2 sin2 = 1

If e1 is the eccentricity of the ellipse

Vector Algebra    Q3.

If u , v , w are non-coplanar vectors and p, q are real numbers, then the equality

         3u p v p w – p v w q u – 2w q v q u = 0 holds for

Q4.

AIEEE–2009

(a)

exactly two values of (p, q)

(b)

more than two but not all values of (p, q)

(c)

all values of (p, q)

(d)

exactly one value of (p, q)

   The vector a = i + 2j + k lies in the plane of the vector b = I + j and c = j + k and bisects   the angle between b and c . Then, which of the following gives possible values of and ? AIEEE–2008 (a)

Q5.

= 1,

=1

(a) Q6.

(b)

= 2,

=2

(c)

= 1,

=2

(d)

(b)

0

(c)

4

(d)

=1

2

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors a, b, c such that a b = b c = c a = 1/2. Then, the volume of the parallelepiped is (a)

Q7.

= 2,

       The non–zero vectors a , b and c are related by a = 8, b and c = –7 b . Then, the angle   between a and c is AIEEE–2008

1 cu unit 2

(b)

If u and v are unit vectors and

1 2 2

cu unit

(c)

3 cu unit 2

(d)

IIT JEE–2008

1 cu unit 3

is the acute angle between them, then 2u × 3v is a unit vector for AIEEE–2007

Q8.

(a)

exactly two values of

(b)

more than two values of

(c)

no value of

(d)

exactly one value of

(a) (b) (c) (d) Q9.

  









Let a , b , c be unit vectors such that a + b + c = 0 . Which one of the following is correct ?

 a  a  a  a

× × × ×

IIT JEE–2007

      b =b ×c=c ×a =0       b =b ×c = c ×a 0       b = b × c = a× c = 0      b , b × c , c × a are mutually perpendicular

 

ABC is triangle, right angled at A. the resultant of the forces acting along AB , AC with magnitudes

 1 1 and respectively is the force along AD , where D is the foot of the AB AC

perpendicular from A onto BC. The magnitude of the resultant is

AB 2 + AC 2 1 1 1 (c) (d) ( AB ) 2 ( AC ) 2 AB AC AD   The distance between the line r = 2i – 2j + 3k + (i – j + 4k) and the plane r (i – 5j + k) = 5 is (a)

Q10.

AIEEE–2006

( AB)( AC ) AB AC

(b)

AIEEE–2005 (a)

10 3 3

(b)

10 9

10 3







   Q11.

(c)

Let a , b , c are non–zero vectors such that ( a × b ) × c =





(d)

1    | b | | c | a . If is acute angle 3

between the vectors b and c , then sin is equal to (a) Q12.

1 3

(b)

2 3

(c)

3 10

AIEEE–2004

2 3

(d)

2 2 3

The unit vector which is orthogonal to the vector 3i + 2j + 6k and is coplanar with the vectors 2i + j + k and i – j + k, is

IIT JEE–2004

2i – 6 j k 2i – 3j 2j – k 4i + 3j – 3k (b) (c) (d) 10 34 41 13           If u , v , w are three non-coplanar vectors, then ( u + v – w ) ( u – v ) × ( v – w ) is equal to (a)

Q13.

(a)

 0

(b)

 u

  v ×w

(c)

 u

  w ×v

AIEEE–2003



(d)

3u

  v × w

  Q14.







system, then c is (a) Q15.

 

If the vectors c , a = xi + yj + z k and b = j are such that a , c and b form a right handed

zi – xk

(b)

 0

AIEEE–2002 (c)

yj

(d)

–zi + zk

      Let v = 2i + j – k and w = I + 3k. If u is a unit vector, then the maximum value of [ u v w ] is IIT JEE–2002

Q16.

–1

10 + 6 59 60 (c) (d)          If a , b , c are unit vectors, then | a – b |2 + | b – c |2 + | c – a |2 does not exceed. (a)

(b)

IIT JEE–2001 (a) Q17.

4

(b)

9

(c)

8

(d)

6

   If the vectors a , b , c form the sides BC, CA, AB respectively of ABC, then IIT JEE–2000              (a) a b + b c + c a = 0 (b) a × b = b × c = c × a             (c) a b = b c = c a (d) a × b + b + c + c × a = 0