7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
1/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
2/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
5/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
6/45
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
7/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
8/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
9/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
10/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
11/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
12/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
13/45
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;
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
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
14/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
15/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
16/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
17/45
(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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
18/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
19/45
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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)
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
7/29/2019 PAST 40 YEARS IIT Entrance Mathematics Problems
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
Top Related