past 40 years iit entrance mathematics problems

7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems

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Complex Numbers Entrance Questions

Q1. The number of complex numbersz such that |z1| = |z + 1| = |zi| equals AIEEE2010

(a) 0 (b) 1 (c) 2 (d)

Q2. If 4zz

= 2, then the maximum value of |z| is equal to AIEEE2009

(a) 3 + 1 (b) 5 +1 (c) 2 (d) 2 + 2

Q3. The conjugate of a complex number is1

1i. Then the complex number is AIEEE2008

(a)1

1i (b)

1

1i(c)

1

1i(d)

1

1i

Q4. If |z + 4| 3, then the maximum value of |z + 1| is AIEEE

2007(a) 4 (b) 10 (c) 6 (d) 0

Q5. If |z| = 1 andz 1, then all the values of21

z

zlie on AIEEE2007

(a) a line not passing through the origin (b) |z| = 2

(c) the x-axis (d) the y-axis

Q6. The value of10

1

2 2sin cos

11 11k

k ki is AIEEE2006

(a) 1 (b) 1 (c) i (d) i

Q7. Ifw = + i , where 0 andz 1, satisfies the condition that

1

w wis purely real, then the

set of values of is IIT JEE2006

(a) |z| = 1,z = 2 (b) |z| = 1 andz 1

(c) z =z (d) None of these

Q8. Ifw =

3

z

iz

and |w| = 1, thenz lies on AIEEE

2005

(a) a circle (b) an ellipse (c) a parabola (d) a straight line

Q9. The locus ofz which lies in shaded region is represented by IIT JEE2005


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(a) z : |z + 1| > 2, | (z + 1) | 2, | (z1) | a, then the equation (xa)(xb)1 = 0 has IIT JEE2000

(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, )

Inequalities & Logarithms

Q1. For all x, x2 + 2ax + (103a) > 0, then the interval in which a lies is IIT JEE2004

(a) a 5 (d) 2 < a < 5

Q2. If 1, log31x3 2 , log3(

*1) are in AP, then x is equal to AIEEE2002


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(a) log34 (b) 1log34 (c) 1log43 (d) log43

Q3. The set of all real numbers x for which x2|x + 2| + x > 0 is IIT JEE2002

(a) ( ,2) (2, ) (b) ( , 2 ) ( 2 , )

(c) ( ,1) (1, ) (d) ( 2 , )

Sequences & Series

Q1. A person is to count 4500 currency notes. Let an denotes the number of notes he counts in the nth

minute. Ifa1 = a2= .= a10 = 150 and a10, a11are in AP with common difference 2, then the

time taken by him to count all notes, is AIEEE2010

(a) 24 min (b) 24 min (c) 125 min (d) 135 min

Q2. The sum of the infinity of the series 1 + 2

3+

2

6

3+

2

10

3+

4

14

3+ . is AIEEE2009

(a) 3 (b) 4 (c) 6 (d) 2

Q3. The first terms of a geometric progression add upto 12. The sum of the third and the fourth terms

is 48. If the terms of the geometric progression are alternately positive and negative, then first

term is AIEEE2008

(a) 4 (b) 4 (c) 12 (d) 12

Q4. In a geometric progression consisting of positive term, each term equals to the next two terms.

Then, the common ratio of this progression equals AIEEE2007

(a)1

2(1 5 ) (b)

15

2(c) 5 (d) 1 5 1

2

Q5. Let a1, a2, , be terms of an AP. If1 2

1 2

.....

p

q

a a a

a a a=

2

2

p

q,p q, then 6

21

a

aequals

AIEEE2007

(a)7

2

(b)2

7

(c)11

41

(d)41

11

Q6. If x =0

n

n

a , y =0

n

n

b , z =0

n

n

c where a, b, c are in AP and |a| < 1, |b| < 1, |c| < 1, then x, y, z

are in AIEEE2005

(a) AP (b) GP (c) HP (d) AGP


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(c) Statement I is true; Statement II is false

(d) Statement I is false; Statement II is true

Q3. LetA be a 2 2 matrix with non-zero entries and letA2

=Iis 2 2 identity matrix.

Define Tr(A) = sum of diagonal elements ofA and |A| = determinant of matrixA. AIEEE2010

Statement-I Tr(A) = 0.

Statement-II |A| = 1.

Q4. LetA be 2 2 matrix. AIEEE2009

Statement-I adj(adjA) =A

Statement-II |adjA| =A

Q5. LetA be a 2 2 matrix with real entries. LetIbe the 2 2 identity matrix. Denote by tr(A), the

sum of diagonal entries ofA. Assume thatA2 =I. AIEEE2008

Statement-I IfA IandA I, then det(A) =1.

Statement-II IfA IandA I, then tr(A) 0.

Q6. LetA be a square matrix all of whose entries are integers. Then, which one of the following is

true? AIEEE2008

(a) If det(A) = 1, thenA1 need not exist

(b) If det(A) = 1, thenA1

exists but all its entries are not necessarily integers

(c) If det(A) 1, thenA1 exists and all its entries are non-integers

(d) If det(A) = 1, thenA1

exists and all its entries are integers

Q7. LetA =5 5

0 5

0 0 5

If det(A2) = 25, then | | is AIEEE2007

(a) 1 (b)1

5 (c) 5 (d) 5

2

Q8. IfA andB are 3 3 matrices such thatA2B2 = (AB) (A +B), then AIEEE2006

(a) eitherA orB is zero matrix (b) either A orB is unit matrix(c) A =B (d) AB =BA

Q9. LetA =1 2

3 4andB =

0

0

a

b, a, b, ,Nthen AIEEE2006

(a) there exists exactly oneB such thatAB =BA

(b) there exists infinitely manBs such thatAB =BA


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(c) there cannot exist anyB such thatAB =BA

(d) there exist more than but finite number ofBs such thatAB =BA

Q10. The system of equations

ax + y + z = 1

x + y + z = 1

x + y + z = 1

has no solution if is AIEEE2005

(a) 2 or 1 (b) 2 (c) 1 (d) 1

Q11. IfP =3 1

2 2

1 3

2 2

,A =1 1

0 1and Q = PAP

T, then P

TQ

2005P is equal to AIEEE2005

(a) 1 20050 1

(b) 4 + 2005 3 6015

2005 4 2005 3

(c)2 3 11

4 1 2 3(d)

2005 2 31

4 2 + 3 2005

Q12. IfA =0 0 1

0 1 0

1 0 0

, then AIEEE2004

(a) A is zero matrix (b) A = (1)I (c) A1 does not exist (d) A2 =I

Q13. IfA =2

2and detA

3 = 125, then us equal to IIT JEE2004

(a) 1 (b) 2 (c) 3 (d) 5

Q14. IfA =a b

b aandB

2= , then AIEEE2003

(a) = a2

+ b2, = ab (b) = a

2+ b

2, = 2ab

(c) = a2 + b

2, = a2b2 (d) = 2ab, = a2 + b2

Q15. IfA =0

0 1andB =

1 0

5 1, thenA

2 =B for IIT JEE2003

(a) = 4 (b) =1 (c) = 1 (d) no


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Determinants

Q1. Let a, b, c be such that (b + c) 0. If1 1

1 1

1 1

a a a

b b b

c c c

+

n+2 1

1 1 1

1 1 1

(1) (1) (1)n n

a b c

a b c

a b c

= 0 then the

value of n is AIEEE

2009

(a) zero (b) any even integer (c) any odd integer (d) any integer

Q2. 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 AIEEE2008

(a) 1 (b) 2 (c) 1 (d) 0

Q3. IfD =

1 1 1

1 1 1

1 1 1

x

y

for xy 0, thenD is divisible by AIEEE2007

(a) both x and y (b) x but not y (c) y but not x (d) neither x nor y

Q4. Ifa2

+ b2

+ c2

=2 andf(x) =

2 2 2

2 2 2

2 2 2

1 x (1 )x (1 )x

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

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

a b c

a b c

a b c

, thenf(x) is a polynomial of degree

AIEEE2005

(a) 0 (b) 1 (c) 2 (d) 3

Q5. Ifa1, a2, a3,.. are in GP, then =1 2

3 4 5

6 7 8

log log log

log log log

log log log

n n n

n n n

n n n

a a a

a a a

a a a

is equal to AIEEE

2004

(a) 0 (b) 1 (c) 2 (d) 4

Q6. Given 2xy + 2z = 2, x2y + z =4, x + y + z = 4, then the value of such that the given

system of equation has no solution, is IIT JEE2004

(a) 3 (b) 1 (c) 0 (d) 3

Q7. Of 1, ,2

are the cube roots of unity, then =

2

2

2

1

11

n n

n n

n n

is equal to AIEEE2003

(a) 0 (b) 1 (c) (d)2

Q8. If ( 1) is a cubic roots of unity, then

2 2

2

1 1+i+

1 1 1

i 1 1

i

i

equals AIEEE2002


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(a) 0 (b) 1 (c) i (d)

Q9. If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the

value ofa is IIT JEE2002

(a) 0 (b) 1 (c) 1 (d) no real values

Binomial Theorem & Its Applications

Q1. Statement-I

0

1n

r

r nCr = (n + 2)n1

Statement-II

0

1n

r

r nCr xr = (1 + x)n + nx(1 + x)n

1 AIEEE2008

(a) StatementI is true, StatementII is true;

StatementII is a correct explanation for StatementI

(b) StatementI is true, StatementII is true;StatementII is not a correct explanation for StatementI

(c) StatementI is true; StatementII is false

(d) StatementI is false; StatementII is true

Q2. In the expansion of (ab)n

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

a

b is equal to

AIEEE2007, IIT JEE2001

(a) 5

6

n(b)

4

5

n (c)

5

4n(d)

6

5n

Q3. If the expansion, in powers of x of the function1

1 x 1 xa bis a0 + a1x + a2x

2+ , then an,

is AIEEE2006

(a)

n n

a bb a

(b)

1 1

n n

a bb a

(c)

1 1

n n

b ab a

(d)

n n

b ab a

Q4. If the coefficients of x7 in

11

2 1xx

ab

and x7 in

11

2

1x

xa

bare equal, then

AIEEE2005

(a) a + b = 1 (b) ab = 1 (c) ab =1 (d) ab = 1


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Q5.30

0

30

10

30

1

30

11+

30

2

30

12.+

30

20

30

30is equal to IIT JEE2005

(a)30

11(b)

60

10(c)

30

10 (d)

65

55

Q6. The coefficient of xn in the expansion of (1 + x)(1x)n is AIEEE2004

(a) n1 (b) (1)n (1n) (c) (1)n1(n1)2 (d) (1)n

1x

Q7. The coefficients of the middle term in the binomial expansion in powers of x of (1 + x)4 and of

(1 x)6 is the same, if AIEEE2004

(a) 5

3 (b)

3

5 (c)

3

10 (d)

10

3

Q8. Ifn1

Cr = (k23)n Cr+1, then kbelongs to IIT JEE2004

(a) ( ,2] (b) [2, ) (c) 3, 3 (d) ( 3 , 2]

Q9. The number of integral terms in the expansion of ( 3 + 51/8)256 is AIEE2003

(a) 32 (b) 33 (c) 34 (d) 35

Q10. The coefficient of x24

in (1 + x2)

12(1 + x

12)(1 + x

24) is IIT JEE2003

(a)12

6 (b)

12

6+ 1 (c)

12

6+ 2 (d)

12

6+ 3

Q11. Let Tn denote the number of triangles which can be formed by using the vertices of regular

polygon ofn sides. AIEEE2002

If Tn+1Tn = 21, then n is equal to

(a) 5 (b) 7 (c) 6 (d) 4

Q12. The sumi

10

m

i

20

mi, when

p

q= 0, ifp < q is maximum for m is equal to

IIT JEE2002

(a) 5 (b) 10 (c) 15 (d) 20

Q13. For 2 r n,n

r+ 2

1

n

r+

2

n

ris equal to IIT JEE2000

(a)1

1

n

r (b) 2

1

1

n

r (c) 2

2

n

r (d)

2

n

r


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Mathematical Induction

Q1. The remainder left out when 82n

(62)2n+1 is divided by 9 is AIEEE2009

(a) 0 (b) 2 (c) 7 (d) 8

Q2. Statement-I For every natural number n 2.

1

1+

1

2+ +

1

n> n .

Statement-II For every natural number n 2.

1n n < n + 1. AIEEE2008

(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

(c) Statement-I is true; Statement-II is false(d) Statement-I is false; Statement-II is true

Q3. IfA =1 0

1 1andI=

1 0

0 1, then which one of the following holds for all n 1, by the

principle of mathematical induction ? AIEEE2005

(a) An = 2n1A + (n1)I (b) An = nA + (n1)I

(c) An = 2n1A(n1)I (d) An = nA(n1)I

Q4. Let S(k) = 1 + 3 + 5 + + (2k1) = 3 + k2. Then which of the following is true ?

AIEEE2004

(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


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Permutations & Combinations

Q1. There are two urns. UrnA 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 AIEEE2010

(a) 3 (b) 36 (c) 66 (d) 108

Q2. Let S1 =10

1j

j (j1)10Cj, S2 =10

1j

j 10Cj and S3 =10

1j

j 2 10Cj

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 is10

C5.

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 6As and 4Bs in a row. AIEEE2008

(a) Statement-I is true; Statement-II is true;


(b) Statement-I is true; Statement-II is true;


(c) Statement-I is true; Statement-II is false

(d) Statement-I is false; Statement-II is true

Q4. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in

which no two S are adjacent ? AIEEE2008

(a) 76C4

8C4 (b) 8

6C4

7C4 (c) 6 7

8C4 (d) 6 8

7C4

Q5. The set S = {1, 2, 3,.., 12} is to be partitioned into three setsA,B, Cof equal size. Thus,A B

C= S,A B =B C=A C= . The number of ways to partition S is AIEEE2007

(a) 12 /3 (4 )3

(b) 12 /3 (3 )4

(c) 12 /(4 )3

(d) 12 (3 )4

Q6. 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 IIT JEE2007

(a) 360 (b) 192 (c) 96 (d) 48


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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 AIEEE2006

(a) 6210 (b) 385 (c) 1110 (d) 5070

Q8. 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 AIEEE2005

(a) 600 (b) 601 (c) 602 (d) 603

Q9. The number of ways of distributing 8 identical balls in 3 distinct boxes so that no box is empty, is

AIEEE2004

(a) 5 (b)8

3(c) 38 (d) 21

Q10. 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 AIEEE2003

(a) 140 (b) 196 (c) 280 (d) 346

Q11. 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 AIEEE2003

(a) 6 5 (b) 30 (c) 5 4 (d) 5 7

Q12. The number of arrangements of the letters of the word BANANA, which the twoNs do not

appear adjacently is IIT JEE2002

(a) 20 (b) 40 (c) 60 (d) 80

Sets, Relations & FunctionsQ1. 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

AIEEE2010

(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


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Q2. IfA,B, and Care three sets such thatA B =A CandA B =A C, then AIEEE2009

(a) A = C (b) B = C (c) A B = (d) A =B

Q3. For real x, letf(x) = x3

+ 5x +1, then AIEEE2009

(a) fis one-one but not onto R

(b) fis onto R but not one-one

(c) fis one-one and onto R

(d) fis neither one-one nor onto R

Q4. Letf(x) = (x + 1)21, x 1 AIEEE2009

Statement-I The set {x :f(x) =f1(x)} = {0,1}

Statement-II fis 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) : xy is an integer}

Which one of the following is true? AIEEE2008

(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

Q6. Letf: N Y be a function defined asf(x) = 4x +3 for some x N}. Show thatfis invertible

and its inverse is AIEEE

2008

(a) g(y) = 3

4

y(b) g(y) =

3 4

3

y(c) g(y) = 4+

3

4

y (d) g(y) =

3

4

y

Q7. The largest interval lying in ,2 2

for which the functionf(x) =2 x4 + cos1

x1

2+

log(cos x) is defined, is AIEEE2007

(a) [0, ] (b) ,2 2

(c) ,4 2

(d) 0,2

Q8. Let Wdenotes the words in the English dictionary. Define the relation R by R = {(x, y) W W

: the words x and y have at least one letter in common}. Then, R is AIEEE2006

(a) reflexive, symmetric and not transitive

(b) reflexive, symmetric and transitive

(c) reflexive, not symmetric and transitive


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(d) not reflexive, symmetric and transitive

Q9. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the setA =

{3, 6, 9, 12}. The relation is AIEEE2005

(a) an equivalence relation

(b) reflexive and symmetric

(c) reflexive and transitive

(d) only reflexive

Q10. Let F: (1, 1) B be a function defined byf(x) = tan12

2x

1 x, thenfis both one-one and

onto whenB is in the interval AIEEE2005

(a) ,2 2

(b) ,2 2

(c) 0,2

(d) 0,2

Q11. f(x) =x, if x is rational

0, if x is irrationaland

g(x) =0, if x is rational

x, if x is irrational. Then ,fg is IIT JEE2005

(a) one-one and into

(b) neither one-one nor onto

(c) many-one and onto

(d) one-one and onto

Q12. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a 8 relation on the setA = {1, 2, 3, 4}. The relation

R is AIEEE2004

(a) reflexive (b) transitive (c) not symmetric (d) a function

Q13. Iff(x) = sin x + cos x, g(x) = x21, then g{f(x)} is invertible in the domain IIT JEE2004

(a) 0,2

(b) ,4 4

(c) ,2 2

(d) [0, ]

Q14. A functionffrom the set of natural numbers to integers defined by

f(n) =

1,

2

,2

nn

nn

is AIEEE2003

(a) one-one but not onto


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(b) onto but not one-one

(c) one-one and onto both

(d) neither one-one nor onto

Q15. Domain of definition of the functionf(x) = 1sin (2x)

6

for real valued x, is IIT JEE2003

(a)1 1

,4 2

(b)1 1

,2 2

(c)1 1

,2 9

(d)1 1

,4 4

Q16. The domain of definition of the functionf(x) =2

10

5 x xlog

4is AIEEE2002

(a) [1, 4] (b) [1, 0] (c) [0, 5] (d) [5, 0]

Q17. Supposef(x) = (x + 1)2

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

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

2002

(a) x 1, x 0 (b)2

1

x + 1

, x >1

(c) x + 1 , x 1 (d) x 1, x 0

Q18. Letf(x) =x

x + 1, x 1. Then, for what value of isf[f(x)] = x ? IIT JEE2001

(a) 2 (b) 2 (c) 1 (d) 1

Q19. The domain of definition off(x) = 22

log x 3

x 3x 2is IIT JEE2001

(a)1, 2

r(b) (2, ) (c)

1, 2, 3

R(d)

3,

1, 2

Q20. Letf( ) = sin (sin + sin3 ). Then,f( ) IIT JEE2000

(a) 0 only when 0 (b) 0 for all real

(c) 0 for all real (d) 0 only when 0


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Limits, Continuity & Differentiability

Q1. Iff: (1, 1) R be a differentiable function withf(0) =1 andf (0) = 1. Let g(x) = [f(2f(x) +

2)]2. Then g(0) is equal to AIEEE2010

(a) 4 (b) 4 (c) 0 (d) 2

Q2. Letf: R R be a positive increasing function withx

(3x)lim

(x)

f

f= 1. Then,

x

(2x)lim

(x)

f

fis

equal to AIEEE2010

(a) 1 (b)2

3(c)

3

2 (d) 3

Q3. Letf: R R be continuous function defined byf(x) =x x

1

2e e AIEEE2010

Statement-I f(c) =1

3, for some c R.

Statement-II 0


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Q6.

x2

2

2x 2

4

( )

lim

x 16

f t dt

equals IIT JEE2007

(a)

8

f(2) (b)

2

f(2) (c)

2

f

1

2

(d) 4f(2)

Q7. The set of points, wheref(x) =x

1 xis differentiable, is AIEEE2006

(a) ( ,1) (b) ( , ) (c) (0, ) (d) ( , 0) (0, )

Q8. limn

2 2 2

2 2 2 2 2

1 1 2 41

n

n n n n nequals to AIEEE2005

(a)1

2tan1 (b) tan 1 (c)

1

2 1 (d)

1

2 1

Q9. Letfbe twice differentiable function satisfyingf(1) = 1,f(2) = 4,f(3) = 9, then IIT JEE2005

(a) f(x) = 2, x (R)

(b) f(x) = 5f(x), for some x (1, 3)

(c) there exists at least one x (1, 3) such thatf(x) = 2

(d) none of the above

Q10. Letf(x) =1 tan x

4x

, x

4

, x 0,

2

. Iff(x) is continuous in 0,

2

, thenf

4

is

AIEEE2004

(a) 1 (b) 1/2 (c) 1/2 (d) 1

Q11. If

2x

2xlim 1

x x

a b= e

2, then the values ofa and b are AIEEE2004

(a) a R, b R (b) a = 1, b R (c) a R, b = 2 (d) a = 1, b = 2

Q12. Iff(x) =

1 1

x xxe , x 0, then (x) is

0 , x 0

f AIEEE

2003

(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


20/45

Q13. If2x 0

nx tan x sin nxlim

x

a n= 0, where n is non-zero real number, then a is equal to

IIT JEE2003

(a) 0 (b)1n

n

(c) n (d) n +1

n

Q14.1x

1 2 3lim

p p p p

p

n

nis equal to AIEEE2002

(a)1

1p(b)

1

1p(c)

1

p

1

1p(d)

1

2p

Q15. Letf: R R be such thatf(1) = 3 andf(1) = 6. Then,

1/ x

x 0

(1 x)lim

(1)

f

fequals

IIT JEE

2002(a) 1 (b) e

1/2(c) e

2(d) e

3

Q16. The left hand derivative off(x) = [x] sin( x) at x = k, kan integer is IIT JEE2001

(a) (1)k(k1) (b) (1)

k1 (k1) (c) (1)kk (d) (1)

k1k

Q17. Letf: R R be any function. Define g : R R by g(x) = |f(x)| for all x. then, g is

IIT JEE2000

(a) onto iff is onto

(b) one-one iff is one-one

(c) continuous iff is continuous

(d) differentiable iff is differentiable

Differentiation

Q1. Let y be an implicit of x defined by x2x2xx cot y1 = 0. Then, y(1) equals AIEEE2009

(a) 1 (b) 1 (c) log 2 (d) log 2

Q2. Letf(x) = x|x| and g(x) = sin x AIEEE2009

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

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

Q3.

2

2

x

y

d

dis equal to IIT JEE2007


21/45

(a)

12

2

d y

dx(b)

12

2

d y

dx

3dy

dx9

(c)

22

2

d y dy

dx dx(d)

32

2

d y dy

dx dx

Q4. If xm yn = (x + y)m + n, thendy

dxis AIEEE2006

(a)x y

xy (b) xy (c)

x

y (d)

y

x

Q5. Iff (x) =f(x), wheref(x) is a continuous double differentiable function and g(x) =f(x). If

F(x) =

2

2

xf +

2

2

xg and F(5) = 5, thenf(10) is IIT JEE2006

(a) 0 (b) 5 (c) 10 (d) 25

Q6. If y is a function of x and log(x + y) = 2xy, then the value of y(0) is equal to IIT JEE2004

(a) 1 (b) 1 (c) 2 (d) 0

Q7. If y = (x +21 x )n, then (1 + x2)

2

2x

d y

d+ x

x

dy

dis AIEEE2002

(a) n2y (b) n2y (c) y (d) 2x2y

Application of Derivatives

Q1. The equation of the tangent to the curve y = x +2

4

x, that is parallel to the x-axis, is

AIEEE2010

(a) y = 0 (b) y = 1 (c) y = 2 (d) y = 3

Q2. Letf: R R be defined byf(x) = 2x, if x 1

2x 3, if x 1

k. Iff has a local minimum at x =1,

then a possible value ofk, is AIEEE

2010

(a) 1 (b) 0 (c) 1

2 (d) 1

Q3. Given, P(x) = x4 + ax

3 + bx2 + cx + dsuch that x = 0 is the only real root ofP(x) = 0. IfP(1) 0 and q > 0. Then, which

one of the following holds ? AIEEE2008

(a) The cubic has maxima at both3

pand

3

p

(b) The cubic has minima at

3

pand maxima at

3

p

(c) The cubic has minima at3

pand maxima at

3

p

(d) The cubic has minima at both3

pand

3

p

Q6. How many real solutions does the equation x7

+ 14x5

+ 16x3

+ 30x560 = 0 have ?

AIEEE2008

(a) 5 (b) 7 (c) 1 (d) 3

Q7. The total number of local maxima and local minima of the functionf(x) =3

2/ 3

2 x , 3 x 1

x , 1 x 2

is IIT JEE2008

(a) 0 (b) 1 (c) 2 (d) 3

Q8. A value ofc for which the conclusion of Mean Value theorem holds for the functionf(x) = loge x

on the interval [1, 3] is AIEEE2007

(a) 2 log3 e (b)

1

2 loge 3 (c) log3 e (d) loge 3

Q9. The functionf(x) = tan1

(sin x + cos x) is an increasing function in AIEEE2007

(a) ,4 2

(b) ,2 4

(c) 0,2

(d) ,2 2


23/45

Q10. The tangent to the curve y = ex drawn at the point (c, ec) intersects the line joining the points (c

1, ec1) and (c + 1, ec + 1) IIT JEE2007

(a) on the left of x = c (b) on the right of x = c

(c) at no paint (d) at all points

Q11. If x is real, the maximum value of2

2

3x 9x 17

3x 9x 7is AIEEE2006

(a) 41 (b) 1 (c)17

7 (d)

1

4

Q12. 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 AIEEE2005

(a)

5

6

(b)

1

54 (c)

1

18

(d)

1

36

Q13. The tangent at (1, 7) to curve x2

= y6 touches the circle x2 + y2 + 16x + 12y + c = 0 at

IIT JEE2005

(a) (6, 7) (b) (6, 7) (c) (6,7) (d) (6,7)

Q14. The normal to the curve x = a(1 + cos ), y = a sin at always passes through the fixed point

AIEEE2004

(a) (a, a) (b) (0, a) (c) (0, 0) (d) (a, 0)

Q15. Iff(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

Q16. Letf(a) = g(a) = k and their nth derivativesfn(a), g

n(a) exist and are not equal for some n.

Further, ifx

( ) ( ) ( ) ( ) ( ) ( )lim

( ) ( )a

f a g x f a g a f x g a

g x f x= 4, then the value ofkis equal to

AIEEE2003

(a) 4 (b) 2 (c) 1 (d) 0

Q17. Iff(x) = x2 + 2bx + 2c2 and g(x) =x22cx + b2 such that minf(x) > g(x), then the relation

between b and c is IIT JEE2003

(a) no real values ofb and c (b) 0 < c < b 2

(c) |c| < |b| 2 (d) |c| > |b| 2


24/45

Q18. The two curves x33xy2 + 2 = 0 and 3x2yy32 = 0 AIEEE2002

(a) cut at right angled (b) touch each other

(c) cut at an angle3

(d) cut at an angle4

Q19. The length of a longest interval in which the function 3 sin x4 sin3

x is increasing is

IIT JEE2002

(a)3

(b)2

(c)3

2(d)

Q20. Iff(x) = xee(1x)

, thenf(x) is IIT JEE2001

(a) increasing on1

,12

(b) decreasing on R

(c) increasing on R (d) decreasing on

1

,12

Q21. Letf(x) =xe (x1)(x2)dx. Then, f decreases in the interval IIT JEE2000

(a) ( ,2) (b) (2,1) (c) (1, 2) (d) (2, )

Indefinite Integrals

Q1. The value of 2

sin x dx

sin x 4

is AIEEE

2008

(a) xlog cos x 4

+ c (b) x + log cos x 4

+ c

(c) xlog sin x 4

+ c (d) x + log sin x 4

+ c

Q2.dx

cos x 3 sin x

equals AIEEE2007

(a)1

2log tan

x

2 12+ c (b)

1

2log tan

x

2 12+ c

(c) log tanx

2 12+ c (d) log tan

x

2 12+ c


25/45

Q3. The value of2

3 4 2

x 1 dx

x 2x 2x 1is IIT JEE2006

(a) 22 4

2 12

x x+ c (b) 2

2 4

2 12

x x+ c

(c)1

22 4

2 12

x x(d) None of the above

Q4.

2

2

log x 1

1 log xdx is equal to AIEEE2005

(a)

x

2

xe

1 x+ c (b)

2

x

1logx+ c (c)

2

log x

log x c(d)

2

x

x 1+ c

Q5. If sin xsin(x )

dx = Ax + B log sin(x ) + c, then value of (A, B) is AIEEE

2004

(a) (sin , cos ) (b) (cos , sin ) (c) (sin , cos ) (d) (cos , sin )

Q6.dx

cos x sin xis equal to AIEEE2004

(a)1

2log

xtan

2 8+ c (b)

1

2log

xcot

2+ c

(c)

1

2 logx 3

tan 2 8 + c (d)1

2 logx 3

tan 2 8 + c

Q7.n

dx

x(x 1)is equal to AIEEE2002

(a)1

nlog

n

n

x

x 1+ c (b)

1

nlog

n

n

x 1

x+ c

(c) log

n

n

x

x 1+ c (d) None of the above


26/45

Definite Integrals

Q1. Let p(x) be a function defined on R such thatx

(3x)lim

(x)

f

f=1, p(x) = p(1 x), for all x [0, 1],

p(0) = 1 and p(1) = 41. Then,1

0(x)p dx equals AIEEE

2010

(a) 41 (b) 21 (c) 41 (d) 42

Q2.0

cot x dx, [ ] denotes the greatest integer function, is equal to AIEEE2009

(a)2

(b) 1 (c) 1 (d) 2

Q3. Let =1

0

sin x

x

dx andJ=1

0

cos x

x

dx. Then, which one of the following is true ?

AIEEE2008

(a) I>2

3andJ< 2 (b) I>

2

3andJ> 2 (c) I 1, where [x] denotes the greatest integer not exceeding x is

AIEEE2006

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

(c) af([a]){f(1) +f(2) +..+f(a)} (d) af(a){f(1) +f(2) +..+f([a])}

Q6. / 2

3 2

3 / 2[(x + ) + cos (x + 3 )] dx is equal to AIEEE2006

(a)

4

32 + 2 (b) 2 (c) 4 1 (d)

4

32

Q7. The value of

2

x

cos x

1 + adx, a > 0, is AIEEE2005, IIT JEE2001

(a) 2 (b) /a (c) /2 (d) a


27/45

Q8. If1

2

sinxt f(t) dt = 1sin x, x (0, /2), thenf

1

3is IIT JEE2005

(a) 3 (b) 3 (c) 1/3 (d) None of these

Q9. If 0 xf (sin x)dx = A

/ 2

0 (sin x)f dx, then A is equal to AIEEE

2004

(a) 0 (b) (c) /4 (d) 2

Q10. Iff(x) is differentiable and2

0x

t

f(x)dx =2

5t5, thenf

4

25equals IIT JEE2004

(a) 2/5 (b) 5/2 (c) 1 (d) 5/2

Q11. The value of the integralI=1

0x (1x)n dx is AIEEE2003

(a) 11n

(b) 12n (c) 1

1n 1

2n(d) 1

1n+ 1

2n

Q12. Iff(x) =2

2

2

x 1t

xe dt, thenf(x) increases in IIT JEE2003

(a) (2, 2) (b) no value of x (c) (0, ) (d) ( , 0)

Q13.2

2

0[x ] dx is AIEEE2002

(a) 2 2 (b) 2 + 2 (c) 21 (d) 2 3 + 5

Q14. The integral1/ 2

1/2

1 x[x] log

1 xdx equals IIT JEE2002

(a) 1/2 (b) 0 (c) 1 (d) 2 log (1/2)

Q15. Iff(x) =

cosx sin x ,

2 ,

e| x | 2, then

3

2(x)f dx is equal to IIT JEE2000

(a) 0 (b) 1 (c) 2 (d) 3


28/45

Area of Curves

Q1. The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x =3

2

is AIEEE2010

(a) (4 22) sq unit (b) (4 2 + 2)sq unit

(c) (4 21) sq unit (d) (4 2 + 1)sq unit

Q2. The area of the region bounded by the parabola (y2)2 = x1, the tangent to the parabola at the

point (2, 3) and the x-axis is AIEEE2009

(a) 6 sq unit (b) 9 sq unit (c) 12 sq unit (d) 3 sq unit

Q3. The area of the plane region bounded by the curves x + 2y2 = 0 and x + 2y2 = 1 is equal to

AIEEE2008

(a)4

3sq unit (b)

5

3sq unit (c)

1

3sq unit (d)

2

3sq unit

Q4. The area enclosed between the curves y2 = x and y = | x | is AIEEE2007

(a) 2/3 sq unit (b) 1 sq unit (c) 1/6 sq unit (d) 1/3 sq unit

Q5. 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 AIEEE2005

(a) 2 : 1 : 2 (b) 1 : 1 : 1 (c) 1 : 2 : 1 (d) 1 : 2 : 3

Q6. Letf(x) be a non-negative continuous functions. Such that the area bounded by the curve y =

f(x), x-axis and the coordinates x =4

, x = >4

is sin cos 24

. Thenf

2is AIEEE2005

(a) 1 24

(b) 1 24

(c) 2 14

(d) 2 14

Q7. The area bounded by the curve y = (x + 1)2, y = (x1)2 and the line y =1

4is AIEEE2005

(a) 1/6 sq unit (b) 2/3 sq unit (c) 1/4 sq unit (d) 1/3 sq unit


29/45

Q8. The area of the region bounded by the curve y = |x2|, x = 1, x = 3 and the axis is

AIEEE2004

(a) 4 sq unit (b) 2 sq unit (c) 3 sq unit (d) 1 sq unit

Q9. 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) 1 (b)1

3(c)

1

3(d) None of these

Q10. The area bounded by the curve y = 2xx2 and the straight line y =x is given by

AIEEE2002

(a) 9/2 sq unit (b) 43/6 sq unit (c) 35/6 sq unit (d) None of these

Q11. The area bounded by the curves y = | x |1 and y =| x | + 1 is IIT JEE2002

(a) 1 sq unit (b) 2 sq unit (c) 2 2 sq unit (d) 4 sq unit

Differential Equations

Q1. Solution of the differential equation cos xdy = y(sin xy)dx, 0 < x 0. Then y(3) is equal to IIT JEE2005

(a) 3 (b) 2 (c) 1 (d) 0

Q10. The solution of the differential equation y dx + (x + x2y)dy = 0 is AIEEE2004

(a) 1

xy= c (b)

1

xy+ log y = c (c)

1

xy+ log y = c (d) log y = cx

Q11. If y = y(x) and2 + sin x

y + 1

dy

dx=cos x, y(0) = 1, then y

2equals IIT JEE2004

(a)1

3(b)

2

3(c)

1

3 (d) 1

Q12. The solution of the differential equation (1 + y2) + (x1tan y

e )y

x

d

d= 0 is AIEEE2003

(a) (x2) = k1tan y

e (b) 2 x1tan y

e =12tan y

e + k

(c) x1tan y

e = tan1 y + k (d) x12tan x

e = tan1 x


31/45

Q13. If y(t) is a solution of (1 + t)dy

dtty = 1 and y(0) =1, then y(1) is equal to IIT JEE2003

(a) 1

2(b) e +

1

2(c) e

1

2 (d)

1

2

Trigonometric Ratios & Equations

Q1. Let cos( + ) =4

5and let sin( ) =

5

13, where 0 ,

4. Then, tan 2 is equal to

AIEEE2010

(a)25

16(b)

56

33(c)

19

22(d)

20

7

Q2. 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 AIEEE2010

(a) there is a regular polygon withR

r=

1

2

(b) there is a regular polygon withr

R=

1

2

(c) there is a regular polygon withr

R=

2

3

(d) there is a regular polygon withr

R=

3

2

Q3. LetA andB denote the statements

A : cos + cos + cos = 0

B : sin + sin + sin = 0

If cos( ) + cos( ) + cos( ) =3

2, then AIEEE2009

(a) A is true andB is false (b) A is false andB is true

(c) bothA andB are true (d) bothA andB are false

Q4. The number of values of x in [0, 3 ] such that 2sin2x + 5sin x3 = 0 is AIEEE2006

(a) 1 (b) 2 (c) 4 (d) 6

Q5. If 0 < x < and cos x + sin x =1

2, then tan x is equal to AIEEE2006


32/45

(a) 4 7

3(b)

1 7

4(c)

1 7

4(d)

4 7

3

Q6. Let 0,4

and t1 = (tan )tan , t2 = (tan )

cot , t3 = (cot )tan , t4 = (cot )

cot , then

IIT JEE

2006

(a) t1 > t2 > t3 > t4 (b) t4 > t3 > t1 > t2 (c) t3 > t1 > t2 > t4 (d) t2 > t3 > t1 > t4

Q7. In a triangle PQR, R =2

. If tan2

Pand tan

2

Qare the roots ofax

2+ bx + c = 0, a 0,

then AIEEE2005

(a) b = a + c (b) b = c (c) c = a + b (d) a = b + c

Q8. Cos( ) = 1 and cos( + ) =1

e

where , [ , ]. The number of pairs of ,

which satisfy both the equation is IIT JEE2005

(a) 0 (b) 1 (c) 2 (d) 4

Q9. Let , such that < < 3 . If sin + sin =21

65, cos + cos =

27

65, then cos

2is AIEEE2004

(a)

3

130 (b)

3

130

(c)

6

65 (d)

6

65

Q10. Given both and are acute angles sin =1

2, cos =

1

3, then the value of + belongs to

IIT JEE2004

(a) ,3 6

(b)2

,2 3

(c)2 5

,3 6

(d)5

,6

Q11. In a triangleABC, mediansAD andBEare drawn. IfAD = 4, DAB =6

and ABE=3

, then

the area of the ABCis AIEEE2003

(a)8

3sq unit (b)

16

3sq unit (c)

32

3 3(d)

64

3sq unit


33/45

Q12. sin2

=2

4xy

x + yis true, if and only if AIEEE2002

(a) xy 0 (b) x =y (c) x = y (d) x 0, y 0

Q13. If + = 2 and + = , then tan is equal to IIT JEE

2001

(a) 2(tan + tan ) (b) tan + tan (c) tan + 2tan (d) 2tan + tan

Heights & Distances

Q1. AB is a vertical pole withB at the ground level andA at the top. A man finds that the angle of

elevation of the pointA from a certain point Con the ground is 60o. He moves away from the

pole along the lineBCto a pointD such that CD = 7 m. FromD the angle of elevation of the

pointA is 45o. Then, the height of the pole is AIEEE

2008

(a)7 3

2

1

3 1m (b)

7 3

2

1

3 1m

(c)7 3

23 1 m (d)

7 3

23 1 m

Q2. A tower stands at the centre of a circular park. A andB are two points on the boundary of the

park such thatAB(= a) subtends as an angle of 60o

at the foot of the tower and the angle of

elevation of the top of the tower fromA orB is 30o

. The height of the tower is AIEEE

2007

(a)2

3

a(b) 2a 3 (c)

3

a(d) 3

Q3. 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 AIEEE2004

(a) 20 m (b) 30 m (c) 40 m (d) 60 m

Q4. The upper 3/4th portion of a vertical pole subtends an angle tan1

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 AIEEE2002

(a) 40 m (b) 60 m (c) 80 m (d) 20 m


34/45

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 IIT JEE2001

(a) 100 3 (b)200 3

3

(c)100 3

3

(d) 200 3

Q6. A pole stands vertically inside a triangular parkABC. If the angle of elevation of the top of the

pole from each corner of the park is same, then in ABCthe foot of the pole is at the

IIT JEE2000

(a) centroid (b) circumcentre (c) incentre (d) orthocenter

Inverse Trigonometric Functions

Q1. The value of cot1 15 2cos tan

3 3ec is AIEEE2008

(a)5

17(b)

6

17 (c)

3

17(d)

4

17

Q2. If 0 < x < 1, then21 x [{x cos(cot 1 x) + sin(cot1 x)}21]1/2 is equal to IIT JEE2008

(a)2

x

1 x (b) x (c) x

21 x (d)21 x

Q3. If sin1

x

5+ cosec

1

5

4=

2, then a value of x is AIEEE2007

(a) 1 (b) 3 (c) 4 (d) 5

Q4. If cos1

xcos12

y= , then 4x

24xy cos + y2 is equal to AIEEE2005

(a) 4 (b) 2sin2 (c) 4sin2 (d) 4sin2

Q5. If sin {cot1

(1 + x)} = cos(tan1

x), then x is equal to IIT JEE2004

(a) 12

(b) 1 (c) 0 (d) 12

Q6. The equation sin1

x = 2sin1

a has a solution for AIEEE2003

(a)1

2< |a| 0, then a

AIEEE2006

(a)1

,32

(b)1

3, 2

(c)1

0,2

(d) (3, )

Q7. If non-zero numbers a, b, c are in HP, then the straight linex

a+

y

b+

1

c= 0 always passes

through a fixed point. That point is AIEEE2005

(a)1

1, 2

(b) (1,2) (c) (1,2) (d) (1, 2)

Q8. The equation of the straight line passing through the point (4, 3) and making intercepts on the

coordinate axes whose sum is1, is AIEEE2004

(a) x

2+ y

3=1, x

2+ y

1=1 (b) x

2 y

3=1, x

2+ y

1=1

(c)x

2+

y

3= 1,

x

2+

y

1= 1 (d)

x

2

y

3= 1,

x

2+

y

1= 1


38/45

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 04

with the positive direction of x-axis. The

equation of its diagonal not passing through the origin is AIEEE2003

(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

Q10. Three straight lines 2x + 11y5 = 0, 24x + 7y20 = 0 and 4x3y2 = 0 AIEEE2002

(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

Q11. 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 IIT JEE2002

(a) 1 : 2 (b) 3 : 4 (c) 2 : 1 (d) 4 : 3

Q12. Area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx, y = nx + 1 is equal to

IIT JEE2001

(a)2

m n

m n

(b)2

m n

(c)1

m n

(d)1

m n

The Circle

Q1. The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x4y = m at two distinct points, if

AIEEE2010

(a) 85 < m


39/45

(a) (3, 4) (b) (3,4) (c) (3, 4) (d) (3,4)

Q4. 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 AIEEE2007

(a) k 1/2 (b) 1/2 k 1/2 (c) k 1/2 (d) 0 < k< 1/2

Q5. LetABCD be a quadrilateral with area 18, with sideAB parallel to the side CD andAB = 2CD.

LetAD be perpendicular toAB and CD. If a circle is drawn inside the quadrilateralABCD

touching all the sides, then its radius is IIT JEE2007

(a) 3 (b) 2 (c) 3/2 (d) 1

Q6. Let Cbe the circle with centre (0, 0) and radius 3. The equation of the locus of the mid points of

the chords of the circle Cthat subtend an angle 2 /3 at its centre is AIEEE2006

(a) x2 + y2 =27

4

(b) x2 + y2 =9

4

(c) x2 + y2 =3

2

(d) x2 + y2 = 1

Q7. If the circles x2 + y23ax + dy1 = 0 intersect in two distinct point P and Q then the line 5x +

bya = 0 passes through P and Q for AIEEE2005

(a) no value ofa (b) exactly one value ofa

(c) exactly two values ofa (d) infinitely many values ofa

Q8. If a circle passes through the point (a, b) and cuts the circle x2

+ y2

= 4 orthogonally; then the

locus of its centre is AIEEE2004

(a) 2ax + 2by + (a2

+ b2

+ 4) = 0 (b) 2ax + 2by(a2 + b2 + 4) = 0

(c) 2ax2by + (a2 + b2 + 4) = 0 (d) 2ax2by(a2 + b2 + 4) = 0

Q9. If the two circles (x1)2 + (y3)2 = r2 and x2 + y28x + 2y + 8 = 0 intersect in two distinct

points, then AIEEE2003

(a) (3, 7) (b) (4, 7) (c) (2, 5) (d) (6, 9)

Q10. The greatest distance of the point P(10, 7) from the circle x2 + y24x2y20 = 0 is

AIEEE2002

(a) 10 unit (b) 15 unit (c) 5 unit (d) none of these

Q11. Let PQ andRSbe tangents at the extremities of the diameter PR of a circle of radius r. IfPSand

PQ intersect at a point X on the circumference of the circle, then 2r is equal to IIT JEE2001

(a) PQ RS (b)2

PQ RS (c)

2PQ RS

PQ RS (d)

2 2

2

PQ RS


40/45

Q12. The PQR is inscribed in the circle x2

+ y2

= 25. IfQ andR have coordinates (3, 4) and (4, 3)

respectively QPR is IIT JEE2000

(a)2

(b)3

(c)4

(d)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 AIEEE2010

(a) x = 1 (b) 2x + 1 = 0 (c) x =1 (d) 2x1 = 0

Q2. A parabola has the origin as its focus and the line x = 2 as the directrix. Then, the vertex of the

parabola is at AIEEE2008

(a) (2, 0) (b) (0, 2) (c) (1, 0) (d) (0, 1)

Q3. Consider the two curves

C1 : y2 = 4x

C2 : x2

+ y26x + 1 = 0, then IIT JEE2008

(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

Q4. 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 AIEEE2007

(a) (0, 2) (b) (2, 4) (c) (2, 0) (d) (1, 1)

Q5. The locus of the vertices of the family of parabolas y =

3 2x

3

a+

3x

2

a2a is AIEEE2006

(a) xy =35

36(b) xy =

64

105(c) xy =

105

64(d) xy =

3

4

Q6. 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 JEE2006

(a) (x + y)2 = xy2 (b) (xy)2 = x + y2

(c) (xy)2 = 4(x + y2) (d) (xy)2 = 8(x + y2)


41/45

Q7. Ifa 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas

y2

= 4ax and x2

= 4ay, then AIEEE2004

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

(c) d2

+ (2b3c)2 = 0 (d) d2 + (3b + 2c)2 = 0

Q8. The normal at the point2

1 2,2bt bt , then AIEEE2003

(a) t2 =t11

2

t(b) t2 =t1 +

1

2

t(c) t2 = t1

1

2

t(d) t2 = t1 +

1

2

t

Q9. The focal chord to y2

= 16x is tangent to (x6)2 + y2 = 2, then the possible values of the slope of

this chord are IIT JEE2003

(a) {1, 1} (b) {2, 2} (c) {2, 1/2} (d) {2,1/2}

Q10. Two common tangents to the circle x2 + y2 = 2a2 and parabola y2 = 8ax are AIEEE2002

(a) x = (y + 2a) (b) y = (x + 2a) (c) x = (y + a) (d) y = (x + a)

Q11. 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 IIT JEE2002

(a) x =a (b) x =2

a(c) x = 0 (d) x =

2

a

Q12. If the line x1 = 0 is the directrix of the parabola y2kx + 8 = 0, then one of the values ofkis

IIT JEE2000

(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 AIEEE2009

(a) x2

+ 12y2

= 16 (b) 4x2

+ 48y2

= 48 (c) 4x2

+ 64y2

= 48 (d) x2

+ 16y2

= 16

Q2. A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is1

2, then

length of semimajor axis is AIEEE2008

(a)5

3(b)

8

3(c)

2

3(d)

4

3

Q3. In an ellipse, the distance between its foci is 6 and minor-axis is 8. Then, its eccentricity is


42/45

AIEEE2006

(a) 1/2 (b) 4/5 (c) 1/ 5 (d) 3/5

Q4. If the angle between the lines joining the end points of minor-axis of an ellipse with its foci is2

,

then the eccentricity of the ellipse is AIEEE

2005

(a) 1/2 (b) 1/ 2 (c) 3/2 (d) 1/2 2

Q5. The eccentricity of an ellipse, with centre at the origin, is1

2. If one directrix is x = 4, the

equation of the ellipse is AIEEE2004

(a) 3x2 + 4y2 = 1 (b) 3x2 + 4y2 = 12 (c) 4x2 + 3y2 = 1 (d) 4x2 + 3y2 = 12

Q6. Tangent is drawn to the ellipse

2x

27+ y

2= 1 at (3 3 cos , sin ) (where, (0, /2)).

Then the value of such that the sum of intercepts on axes made by this tangent is minimum, is

IIT JEE2003

(a)3

(b)6

(c)8

(d)4

Q7. The equation of the ellipse whose foci are ( 2, 0) and eccentricity is1

2, is AIEEE2002

(a)

2 2x y

12 16= 1 (b)

2 2x y

16 12= 1 (c)

2 2x y

16 8= 1 (d) none of these

Hyperbola

Q1. Consider a branch of the hyperbola x22y22 2 x4 2 y6 = 0 with vertex at the pointA.

LetB be one of the end points of its latusrectum. IfCis the focus of the hyperbola nearest to the

pointA, then the area of the triangleABCis IIT JEE2008

(a) 213

sq unit (b) 312

sq unit

(c)2

13

sq unit (d)3

12

sq unit


43/45

Q2. For the hyperbola

2

2cos

x

2

2sin

y= 1. Which of the following remains constant when varies

? AIEEE2007/IIT JEE2003

(a) directrix (b) abscissae of vertices

(c) abscissae of foci (d) eccentricities

Q3. A hyperbola, having the transverse axis of length 2sin , is confocal with the ellipse 3x2

+ 4y2

=

12. Then, its equation is IIT JEE2007

(a) x2 cosec2 y2 sec2 = 1 (b) x2 sec2 y2 cosec2 = 1

(c) x2

sin2

y2 cos2 = 1 (d) x2 cos2 y2 sin2 = 1

Q4. If e1 is the eccentricity of the ellipse

Vector Algebra

Q3. If u

, v

, w

are non-coplanar vectors andp, q are real numbers, then the equality

3u v wp p

v w up q

2w v uq q

= 0 holds for AIEEE2009

(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)

Q4. 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 ?

AIEEE2008

(a) = 1, = 1 (b) = 2, = 2 (c) = 1, = 2 (d) = 2, = 1

Q5. The nonzero vectors a

, b

and c

are related by a

= 8, b

and c

=7 b

. Then, the angle

between a

and c

is AIEEE2008

(a) (b) 0 (c)4

(d)2

Q6. 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 IIT JEE

2008

(a)1

2cu unit (b)

1

2 2cu unit (c)

3

2cu unit (d)

1

3cu unit

Q7. Ifu and v are unit vectors and is the acute angle between them, then 2u 3v is a unit vector for

AIEEE2007


44/45

(a) exactly two values of (b) more than two values of

(c) no value of (d) exactly one value of

Q8. Let a

, b

, c

be unit vectors such that a

+b

+ c

= 0

. Which one of the following is correct ?

IIT JEE2007

(a) a

b

=b

c

= c

a

= 0

(b) a

b

=b

c

= 0

(c) a

= = = 0

(d) , , are mutually perpendicular

Q9. ABCis triangle, right angled atA. the resultant of the forces acting along AB

, AC

with

magnitudes1

ABand

1

ACrespectively is the force along AD

, whereD is the foot of the

perpendicular fromA ontoBC. The magnitude of the resultant is AIEEE

2006

(a)( )( )AB AC

AB AC(b)

1 1

AB AC(c)

1

AD(d)

2 2

2 2

+

( ) ( )

AB AC

AB AC

Q10. The distance between the line r

= 2i2j + 3k + (ij + 4k) and the plane r

(i5j + k) = 5 is

AIEEE2005

(a)10

3 3(b)

10

9 (c)

10

3(d)

3

10

Q11. Let , , are nonzero vectors such that ( a

) =1

3| | | | a

. If is acute angle

between the vectors and , then sin is equal to AIEEE2004

(a)1

3(b)

2

3(c)

2

3(d)

2 2

3

Q12. The unit vector which is orthogonal to the vector 3i + 2j + 6k and is coplanar with the vectors 2i

+j + k and ij + k, is IIT JEE2004

(a)2i 6j k

41

(b)2i3j

13

(c)2 j k

10

(d)4i + 3j 3k

34

Q13. If u

, v

, w

are three non-coplanar vectors, then ( u

+ v

w

) ( u

v

) ( v

w

) is equal to

AIEEE2003

(a) 0

(b) u

v

w

(c) u

w

v

(d) 3 v

c

a

b

b

c

a

c

a

b

b

c

c

a

a

b

c

b

c

b

c

b

c

u

w


45/45

Q14. If the vectors c

, a

= xi + yj + z k and b

=j are such that a

, c

and b

form a right handed

system, then c

is AIEEE2002

(a) zixk (b) 0

(c) yj (d) zi + zk

Q15. Let v

= 2i + jk and w

= I + 3k. If u

is a unit vector, then the maximum value of [ u

v

w

] is

IIT JEE2002

(a) 1 (b) 10 + 6 (c) 59 (d) 60

Q16. If a

, b

, c

are unit vectors, then | a

b

|2 + | b

c

|2 + | c

a

|2 does not exceed.

IIT JEE2001

(a) 4 (b) 9 (c) 8 (d) 6

Q17. If the vectors a

, b

, c

form the sidesBC, CA,AB respectively of ABC, then IIT JEE2000

(a) a

+ + = 0

(b) = =

(c) a

= = (d) + + + = 0

b

b

c

c

a

a

b

b

c

c

a

b

b

c

c

a

a

b

b

c

c

a

past 40 years iit entrance mathematics problems

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