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INFOMATHS
WORK SHEET(IIT-JNU QUESTIONS)
SETS AND RELATIONS
1.
Let S = {x ℚ | x2 {1, 20, 21}. Then the number of elements in
the set S is
IIT-2012
(A) 1
(B) 2
(C) 4
(D) 6
2.
The set S ={(x, y) ℝ2 | xℚ y ℤ}is
IIT-2012
(A) (ℝ\ℚ) (ℝ\ℤ)
(B) (ℝ ℝ) (ℚ\ ℤ)
(C) (ℝ\ℚ) ℝ
(d) ℝ (ℝ\ℤ)
3.
The set (Q×Q) \ (N×N) equals
15.
16.
IIT-2011
(A) (Q\ N) × (Q\ N)
(B) [(Q\ N)×Q] U [Q×(Q\ N)]
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
JNU – 2001
(a) (A C) (B C)
(b) (A C) (B C)
(c) (A B) (A C)
(d) (A B) (A C)
Let L be an equivalence relation on a set S on n elements.
Consider the set Sa = {x : x a and aLx}. Further, let D S be
such that no two elements in D are related under L. The number of
elements in S a is at most
(IIT – 2009)
aD
n(n 1)
(a)
(b) n – 1 (c) 2n
2
(C) [(N×Q) \ (Q×N)] U[(Q×N) \ (N×Q)]
4.
JNU – 2003
(a) 1400
(b) 6000
(c) 3300
(d) 4000
The expressions A (B C) is the same as
(D) (Q×N) \ (N×Q)
In the set of integers, a relation R is defined as aRb, if and only if
b = |a|. Relation R is
(JNU : MCA - 2009)
(a) reflexive
(b) irreflexive
(c) symmetric
(d) antisymmetric
Out of 120 students, 80 students have taken mathematics, 60
students have taken physics, 40 students have taken chemistry, 30
students have taken both physics and mathematics, 20 students
have taken both chemistry and mathematics and 15 students have
taken both physics and chemistry. If every student has taken at
least one course, then how many students have taken all the three
courses?
(MCA : IIT – 2008)
(a) 5
(b) 25
(c) 15
(d) 10
The degree of the Cartesian product of two relations P and Q is
given by
(MCA : JNU - 2008)
(a) |P| * |Q|
(b) |P| + |Q|
(c) max (|P|, |Q|)
(d) None of the above
In a cricket match, five batsmen A, B, C, D and E scored an
average of 36 runs. D scored 5 more than E ; E scored 8 fewer
than A ; B scored as many as D and E combined ; and B and C
scored 107 between them. How many runs did E score?
(MCA : JNU - 2008)
(a) 20
(b) 45
(c) 28
(d) 62
A relation R is said to be partial order if
(JNU : MCA - 2007)
(a) R is reflexive, symmetric and transitive
(b) R is reflexive, asymmetric and transitive
(c) R is reflexive, antisymmetric and transitive
(d) R is reflexive, antisymmetric but not transitive
Let P be the set of all planes in R3. The relation being normal in P
is
(IIT : MCA – 2006)
(a) symmetric and transitive
(b) symmetric and reflexive
(c) symmetric but not transitive
(d) transitive but not reflexive
For sets P, Q, R which of the following is NOT correct?
(IIT : MCA – 2006)
(a) (P Q) R = (P R) (Q R)
(b) (P\Q)\R = (P\R)\(Q\R)
(c) If P Q = P R, then Q = R
(d) P (Q R) = (P Q) (P R)
A survey shows that 63% of Indians like banana whereas 76% like
apples. If x% of Indians like both banana and apples, then
JNU – 2005
(a) x = 39
(b) x = 63
(c) 39 ≤ x ≤ 63
(d) None of these
If sets A and B are defined as
(1) A = {(x, y) : y = ex, x R}
(2) B = {(x, y) : y = x, x R}
then
JNU – 2005
(a) B A
(b) A B
(c) A B =
(d) A B = A
In a committee of 47 persons 13 take tea but not coffee and 28
take tea. The number of persons taking coffee but not tea is
JNU – 2004
(a) 6
(b) 19
(c) 32
(d) 34
In a town of 10,000 families, it was found 40% buy newspaper A,
20% buy newspaper B and 10% buy newspaper C. Five percent
(5%) of the families buy A and B, 3% buy B and C, 4% buy A and
C. If 2% buy all the three newspapers, then the number of families
which buy none of the newspapers A, B and C is
17.
18.
What is the possible number of binary relations on a set S having
n elements which are symmetric and antisymmetric?
(MCA JNU – 2009)
(a) 0
(b) 1
(c) n2 (d) 2n
Let An = {1, 2, 3, …., n} and An c = N – An where n 1 is a
natural number and N is the set of all natural numbers. Which one
of the following sets is a finite set?
(MCA : IIT – 2008)
n
(a)
20.
21.
22.
23.
24.
25.
1
c
An
i 1
19.
(d) n2
n
(b)
c
Ai
i 1
(c)
c
Ai
i n
(d)
c
Ai
i n
Let S be a nonempty symmetric and transitive binary relation on a
nonempty set A. Consider any pair (a, b) S. Since S is
symmetric, (b, a) S. Further since S is transitive, (a, a) S.
Which one of the following statements is true?
(MCA : IIT – 2008)
(a) S is a reflexive relation since (a, a) S
(b) S is a reflexive relation since the reasoning holds for any
pair of elements in S.
(c) S is a reflexive relation because the above reasoning is true
only for the specific pair (a, a) S that has been considered.
(d) S need not be reflexive because there may be other elements
in A which are not related to any element in A.
Let P be a set having n > 10 elements. The number of subsets of P
having odd number of elements is
(IIT : MCA paper – 2006)
(a) 2n – 1 – 1
(b) 2n – 1
(c) 2n – 1 + 1
(d) 2n – 1
The number of non-empty even subsets (even set is the set having
even number of elements) of set having n elements is
(IIT : MCA paper 2005)
(a) 2n
(b) 2n – 1 + 1
(c) 2n – 1 – 1
(d) 2n – 1
If X Z = Y Z for the non-empty sets X, Y and Z, where
represents the symmetric difference, then
(IIT : MCA paper 2005)
(a) X = Y
(b) X 1 Y
(c) X is a proper subset of Y
(d) Y is a proper subset of X
The enrolments of the third year MCA student of a college in
three elective papers, namely, AA (Advanced Algorithmic), AOS
(Advanced Operating System), and CAN (Advanced Computer
Network) are as follows:
30 students have taken both AA and AOS. 20 students have taken
both AOS and CAN 30 students have taken both CAN and AA. 50
students have taken AA. 60 students have taken ACN and 70
students have taken AOS. 5 students have taken all the three
subjects. If each students in the class has taken at least one of
AOS, AA, and ACN, then the total number of students in the class
is
(IIT : MCA paper 2005)
(a) 75 (b) 95 (c) 100
(d) 105
Let A and B be non-empty subsets of real line R. Which of the
following statements would be equivalent to sup(A) inf (B)?
JNU Paper – 2007
(a) For every a in A there exists a b in B such that a b
(b) there exists a in A and b in B such that a B
(c) For every a in A and every b in B, we have a b
(d) there exists a in A such that a b for all b in B
Let A and B be any two arbitrary sets. If P(X) and denote the set
of all subsets of a set X and the empty set respectively, then which
one of the following is not true?
IIT – 2008
(a) P(A B) ≠ P(A) P(B)
(b) P(A B) = P(A) P(B)
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
(c) {} P(A)
(d) P(A)
1
(a) , 2
(b) 1, 2
2
THEORY OF EQUATIONS
1
1
(c) ,1
(d) 1,
1. The only value of x satisfying the equation
2
2
x
x4
3
2
14.
Let
f(x)
=
2x
–
x
+
2x
–
5.
,
where
x
R
is
6
2
11
x4
x
Consider the following statements about the roots of
(MCA : JNU - 2009)
f(x) = 0
(a) 4/35 (b) –4/35 (c) 16/3 (d) –16/3
P. At least one root is positive
2. The value of a for which the quadratic equation
Q. At least one root is negative
2
2
2
3x + 2(1 + a )x + (a – 3a + 2) = 0 possesses roots of
R. There is a root between x = 1 and x = 2
opposite sign lies in
Which one of the following is TRUE?
(MCA : JNU – 2008)
(IIT : MCA – 2007)
(a) (-, 1) (b) (-, 0) (c) (1, 2) (d) (1.5, 2)
(a)
P,
Q
and
R
are
valid
statements
2
2
3. If x is real and k = (x – x + 1) / (x + x + 1), then
(b) P and Q are valid statements but R is NOT a
(MCA : JNU – 2008)
valid statement
(a) (1/3) k 3
(b) k 5
(c) P and R are valid statements but Q is NOT a
(c) k 0
(d) None of the above
valid statement
4. The quadratic equation whose roots are reciprocal of
(d)
P is a valid statement but Q and R NOT valid
the roots of the equation x3 – 3x + 2 = 0 is
statements
(JNU : MCA – 2007)
15. The number of solutions of
2
2
(a) 3x – 2x + 1 = 0
(b) 2x – x – 1 = 0
65
(c) x2 – 3x + 2 = 0
(d) None of these
10 2 / x 251 / x (50 1 / x ) is
3/4
1/4
-1/4
8
5. The roots of the equation 6x = 7x – 2x are
(JNU : MCA – 2007)
(JNU : MCA – 2007)
(a) zero (b) four (c) two (d) infinite
(a) 4/9 and 1/9
(b) 9/4 and 1/4
16. The number of irrational solutions of the equation
(c) 4/9 and 1/4
(d) None of these
6. If sin and cos are the roots of the equation ax2 –
x 2 x 2 11 x 2 x 2 11 4 is
bx + c = 0, then a, b and c satisfy the relation
(JNU : MCA – 2007)
(JNU : MCA – 2006)
(a) 0
(b) 2
(c) 4
(d) indefinite
(a) a2 + b2 + 2ac = 0
(b) a2 – b2 + 2ac = 0
2x 1
(c) a2 + c2 + 2ab = 0
(d) a2 – b2 – 2ac = 0
0 is
17. The set of real x such that
2 x 3 3x 2 x
7. If x 2 2 2 .... , then x =
(JNU : MCA – 2006)
(JNU : MCA – 2006)
(a) (- , - 1)
(b) (-, 0)
(a) 4
(b) 2
(c) 3.14 (d) None of these
(c) (-, )
(d) None of above
8. The number of real solutions of sin (ex) = 5x + 5-x is
18. The equation 3x – 1 + 5x – 1 = 34 has
(JNU : MCA solved – 2006)
(JNU : MCA – 2006)
(a) infinite (b) 5
(c) 0
(d) None of above
(a)
no
solution
9. Solution of the equation esinx – e-sin x = 4 is
(b) one solution
JNU – 2004
(c) two solutions
(d) more than two solutions
4
17
4
17
(a) sin 1 ln
(b) sin 1 ln
19. S is defined as S
2
2
1
1
1
= x 1 x x x x Find the
4
17
4
17
1
1
2
3
4
5
(c) sin
(d) sin
2
2
value of x for which S is minimum
10. If and are the root of 4x2 + 3x + 7 = 0, then the
JNU – 2006
(a) 1/2
(b) 1/3
(c) 2/3
(d) 78/80
value of (1/) + (1/) is
JNU – 2002 20. The number of solutions of the equation 5x – 5-x =
(a) -3/4 (b) -3/7 (c) 3/7
(d) 4/7
log10 25, (x R) is
11. If the roots of x2 – bx + c = 0 are two consecutive
JNU – 2005
integers, then b2 – 4c is
(a) 0
(b) 1
(c) 2
(d) infinitely many
JNU – 2002 21. If every pair from among the equations x2 + px + qr
(a) 0
(b) 1
(c) 2
(d) None of these
= 0, x2 + qx + rp = 0 has a common root, then the
12. If 1, a1, a2, …., an-1 are the roots of unity, then (1 –
sum of the three common root is
a1) (1 – a2) ...... (1 – an-1) equals ……
JNU – 2005
JNU – 2001
(a) 2(p + q + r)
(b) p + q + r
(a) n
(b) n2
(c) –(p + q + r)
(d) pqr
(c) n – 1
(d) None of these
22. If a and b ( 0) are the roots of the quadratic
13. If a2 + b2 + c2 = 1, then ab + bc + ca lies in the
equation x2 + ax + b = 0, then the least value of x2 +
interval
ax + b (x R) is
JNU – 1999
JNU – 2005
(a) 9/4
(b) -9/4 (c) -1/4 (d) 1/4
2
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
m–1
23. The number of real roots of the equation ex-1 + x – 2
= 0 is
JNU – 2005
(a) 1
(b) 2
(c) 3
(d) 4
2x
A
Bx C
24. If 3
, then
x 1 x 1 x2 x 1
JNU – 2005
(a) A = B = C
(b) A = B C
(c) A B = C
(d) A B C
25. The solution set of the equation log2 (3 – x) + log2 (1
– x) = 3 is
JNU – 2004
(a) {-1} (b) {5} (c) {-1, 5} (d)
26. The real solution of the following simultaneous
equations is
xy +3y2 – x + 4y – 7 = 0
2xy + y2 – 2x – 2y + 1 = 0
JNU – 2004
(a) x = 0, y = 1
(b) x = 1, y = 0
(c) x = - 2, y = 3
(d) x = 2, y = - 3
27. The equation x + ex = 0 has
JNU – 2002
(a) no real root
(b) two real roots
(c) one real negative root (d) one real positive root
28. If 1, 2, ……, n are the roots of equation xn – nax
– b = 0 and (1 - 2) (1 - 3) ….(1 - n) = A, the
value of A - n1n-1 is
JNU – 2000
(a) – na (b) na
(c) n2a
(d) 2na
29. The solution set of the inequality ||x| - 1 | < 1 – x is is
JNU - 2006
(a) (1, 1) (b) (0, ∞) (c) (2, ∞) (d) None of these
30. The solution set of the inequality 4-x+0.5 – 7.2-x – 4 <
0 (x R) is
JNU - 2006
(a) (-∞, ∞) (b) (-2, ∞) (c) (2, ∞) (d) (2, 3.5)
8.
9.
10.
then
(a)
11.
12.
13.
14.
15.
2.
3.
x
4.
5.
x 2 x3 x 4 x5
.....
2 3 4 5
converges to
(5 2 ) x (4 5 ) x 8 2 5 0 is
(a) 2
6.
JNU-2010
(b) 6
For 0 < a < /2, if
(c) 8
7.
JNU-2010
x cos
n 0
n 0
n 0
8.
9.
(JNU–2009)
10.
(d) 4
2n
a,
11.
y sin 2n a, z cos 2n a sin 2n a, then (JNU–2008)
(a) xyz = xz + y
(c) xyz = xy + z
(b) x + y + z + xyz = 0
(d) xy2 + x2y = z
n 1
7.
Let f(1) = 1 and f(n) =
2 f (r ). Then
r 1
(c)
ab
ab
(d)
ab
ab
If pth, qth and rth terms of a GP are x, y, z respectively, then x q-r
yr-p zp-q is equal to
JNU– 2005
(a) 0
(b) 1
(c) – 1
(d) None of these
If the product of n positive integers is unity, then their sum is
JNU– 2005
(a) a positive number
(b) divisible by n
(c) equal to n + 1 / n
(d) never less than n
If p, q, r be three positive numbers, then the value of (p + q) (q +
r) (r + p) is
JNU– 2004
(a) < 4 pqr
(b) < 8 pqr
(c) > 8 pqr
(d) > 4 pqr but < 8 pqr
The value of x for which log32, log3 (2x – 5) and log3 (2x – 7/2) are
in arithmetic progression, is
JNU– 2004
(a) 2
(b) 3
(c) 5
(d) 7
If
(a)
(a) ex (b) sin x
(c) ln x
(d) ln (1 + x)
Let S1 = {2}, S2 = {4, 6}, S3 = {8, 10, 12}, S4 = {14, 16, 18, 20}
and so on. The sum of elements of S10 is
(IIT – 2009)
(a) 990
(b) 1000
(c) 1010
(d) 1020
The harmonic mean of the roots of the equation
2
2ab
ab
3 5 7 .... nterms
7 , the value of n is
5 8 11 .... 10 terms
(b) 42
(c) 37
(d) 35
BINOMIAL
1.
Consider the identity (1 + x + x2)25 = a0 + a1x + a2x2 + … + a50x50.,
We find 2 (a0 + a2 + a4 + ….) equals
JNU-2010
(a) 325
(b) 325 + 1 (c) 326
(d) 326 – 1
2.
The positive integer just greater than (1 + 0.0001)10000 is
(JNU - 2009)
(a) 4
(b) 5
(c) 2
(d) 3
3.
The number of 0s at the end of 95! is
IIT – 2008
(a) 19
(b) 20
(c) 21
(d) 22
4.
The number of distinct terms in the expansion of
(x1 + x2 + x3 + … .+ xn)3 is
JNU– 2007
(a) n+1C3
(b) n+2C3
(c) n+3C3
(d) n+4C3
5.
If A and B are coefficients of xn in the expansions of (1 + x)2n and
(1 + x)2n-1 respectively, the A/B is equal to
JNU– 2007
(a) 1
(b) 2
(c) 1/2
(d) 1/n
6.
The sum of infinite series 1 + 3x + 6x2 + 10x3 + …, x < 1 is
JNU– 2007
1 1
1
1 2 .... is less than 2
,
2 2
1000
is
(a) 5
(b) 7
(c) 8
(d) 10
For – 1 x 1, the infinite power series
(b)
JNU – 2005
JNU– 2004
Suppose the numbers a, b, c are in AP and |a|, |b|, |c| < 1. If x = 1 +
a + a2 + ….. , y = 1 + b + b2 + … , z = 1 + c + c2 + …. then
x, y, z are in
JNU-2010
(a) AP
(b) GP
(c) HP
(d) None of these
The greatest value of the positive integer n so that the sum to n
terms of the series
A1 A2
is equal to
G1G2
ab
2ab
(a) 49
SEQUENCE & SERIES
1.
(a) 3
–1
(b) 3m – 1
(c) 3m – 1
(d) None of the above
In an arithmetic progression, the first term is 2, the last term is 29
and the sum is 155. The difference is
(JNU–2007)
(a) 3
(b) 5
(c) 4
(d) 2
If the sum of m terms to the sum of n terms in an AP is m2 to n2,
then the mth term to the nth terms is
(JNU–2007)
(a) m – 1 : n – 1
(b) 2m + 1 : 2n + 1
(c) 2m – 1 : 2n – 1
(d) None of these
If A1, A2 be two AMs and G1, G2 be two GMs between a and B
The term independent of x in the expansion of (2x2 – 1/x)12 is
JNU– 2007
(a) 7910
(b) 7920
(c) 7930
(d) 7900
The remainder obtained on dividing 21680 by 1763 is
IIT– 2006
(a) 1
(b) 3
(c) 13
(d) 31
The inequality n! > 2n-1 is true for
JNU– 2006
(a) all n N (b) n > 2
(c) n > 1
(d) n N
The coefficient of x99 in the expansion of (x – 1) (x –
2)…………….. (x – 100) is equal to
JNU– 2005
(a) 5050
(b) 5000
(c) -5050
(d) -5000
If n is even and nC0 < nC1 < nC2 < … < nCr > nCr+1 > nCr+2 > … >
n
Cn, then r is equal to
JNU– 2005
(a)
m
f (n) is equal to
12.
n1
(JNU–2008)
3
1
1
1
1
(b)
(c)
(d)
3
1 3x
1 x3
1 x2
1 x
n
2
(b)
n 1
2
(c)
n2
2
(d)
n2
2
The coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 +
……. + (1 + x)30
JNU– 2005
(a) 51C5
(b) 9C5
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
(c) C6 – C6
(d) C5 + C6
For n N, 32n + 2 – 8n – 9 is divisible by
31
13.
14.
21
30
10
1
10 10
(c)
1 2
16.
17.
x
3.
JNU– 2004
(a) 81 (b) 72 (c) 64 (d) 49
The coefficient of x2 in the trinomial expansion of (1 + x + x2)10
JNU– 2003
(b)
(c)
4.
If in the expansion of (x + y)n the coefficients of 4th and 13th terms
are equal, then n is
JNU– 2002
(a) 15 (b) 17 (c) 9 (d) Cannot be determined
n-1
C3 + n-1C4 > nC3 if n > …….
JNU– 2002
(a) 5
(b) 6
(c) 7
(d) 8
The value of
n
2n
C0
2n3
Cn n C1
2n1
Cn n C2
Cn ..... (1) n n Cn n Cn
2 n 2
18.
(c) (-1)n
(b) 1
Cn n C3
is
(d) 2n
r r (r 1)...( r k 1)
, when k is nonk (k 1)...(1)
k
r
7 .2
negative and 0, when k is negative. Thus
k
2
We define
equals to
19.
(a) 0
(b) 29.52
(c) 1.52
(d)
For n 5, the expression
1 + 2x + 3x2 + 4x3 + … + nxn – 1, x 1, is equal to
JNU – 2008
IIT– 2007
nx n (1 x) x n 1
(a)
(1 x) 2
nx n ( x 1) x n 1
(c)
20.
(b)
(d)
(1 x) 2
(1 x) 2
9.
nx n
10.
JNU– 2007
21.
3 2
(b)
9 3
The coefficient of
(c)
5 7
11.
(d) None of these
X 1 X 22 X 33 X 44 X 55 in the expansion of (X1 +
X2 + X3 – X4 + X5) is
(a) 1!2!3!4!5
15
JNU– 2004
12.
(b) 15.14.13.12.11
15!
15!
(d)
1!2!3!4!5!
15
k
m n
m n
A
, where A is
k
i 0 l k i
n
mn
n
(a)
(b) 2
(c) 2
(d) 1
k
(c)
22.
23.
13.
JNU– 2004
14.
The value of 4{nC1 + 4. nC2 + 42. nC3 + …. + 4n-1} is
JNU– 2002
(a) 0
(b) 5n + 1
(c) 5n
(d) 5n - 1
EXPONENTIAL & LOGARITHMIC SERIES
1.
The sum of the series
1
12 22 12 22 32 12 22 32 42
2!
3!
4!
15.
16.
is
JNU – 2005
17
e
(b)
6
(a) 3e
2.
n0
loge x
n!
13
e
(c)
6
17.
19
e
(d)
6
18.
n
is equal to
n 1
(b)
n
1
n
n 1
log e a
(d)
1
n 1
log a e
n
1
n
n
log a e
JNU – 2002
The coefficient of x2 in the expansion of e3x + 4 is
(a) 9e2/2
(b) 9e4/2
(c) 3e4/2
(d) 3e2/2
n
(a)
3
nx n ( x 1) x n 1
(1 x) 2
5 5 7 5 7 9
The sum of the series 1 . . . .... is equal to
3 3 6 3 6 8
(a)
1
PERMUTATIONS & COMBINATIONS
1.
Let S be a set with 10 elements. The number of subsets of S
having odd number of elements is
IIT-2011
(A) 256
(B) 512
(C) 752
(D) 1024
2.
The number of subsets of {1, 2, ... , 10} which are disjoint from
{3,7,8} is
IIT-2011
(A) 128
(B) 1021
(C) 1016
(D) 7
3.
How many numbers from 1 to 1000 are not divisible by 2, 3, and
5?
JNU-2010
(a) 266
(b) 500
(c) 333
(d) None of these
4.
In a singles tennis tournament that has 125 entrants, a player is
eliminated whenever she loses a match. How many matches are
played in the entire tournaments?
JNU-2010
(a) 62
(b) 63
(c) 124
(d) 246
5.
How many four-digit numbers have only even digits?
JNU-2010
(a) 96
(b) 128
(c) 500
(d) 625
6.
The number of rectangles that one can find on a chessboard is
JNU-2010
(a) 1082
(b) 1296
(c) 1128
(d) 1632
7.
Given a 10 10 matrix. Each element of the matrix is a Boolean
variable. How many different matrices can be formed? JNU-2010
(a) 2100
(b) 1002
(c) 210
(d) 102
8.
Given an array of n elements. Each element can take three values
– 1, 0, 1. How many different arrays can be formed? JNU-2010
(JNU - 2009)
(a) 0
(a) loge
(b) x
(c) logxe
(d) None of these
The coefficient of xn in the expansion of loga(1 + x) is
JNU – 2005
(a)
10
2
10
(d)
3
(a)
15.
20
JNU – 2005
4
(b) n
3
n
(c) 3
(d)
n
1
3
A student is allowed to select at the most n books from a
collection of (2n + 1) books. If the total number of ways in which
he can select a book is 63, the value of n is
JNU – 2007
(a) 1
(b) 7
(c) 5
(d) 3
The number of times the digit 3 will be written when listing the
integers from 1 to 1000 is
JNU– 2005
(a) 269
(b) 300
(c) 271
(d) 302
There are two bags each containing n balls. A boy has to select an
equal number of balls from both the bags. The number of ways in
which boy can choose at least one ball from each bag is
JNU– 2005
(a) 2nCn
(b) (nCn)2
(c) 2nC1
(d) 2nCn – 1
If the letters of the word ‘REGULATION’ be arranged at random,
the probability that there will be exactly 4 letters between R and E
is
JNU– 2005
(a) 1/10
(b) 1/9
(c) 1/5
(d) 1/2
Twenty-five members of a new club meet each day lunch at a
round able. They decide to sit such that every member has
different neighbours at each lunch. How may days can this
arrangement last?
JNU– 2003
(a) 25 days (b) 12 days (c) 18 days (d) 13 days
There are 20 guests at a party. Two of them do not get along well
with each other. In how many ways can they be seated in a row so
that these two persons do not sit next to each other? JNU– 2003
(a) 20!
(b) 20! – 2(19!)
(c) 19!
(d) None of the above
A polygon has 90 diagonals then it has
JNU– 1999
(a) 10 sides (b) 15 sides (c) 20 sides (d) 25 sides
The number of zeroes that 1000! ends up with, when expanded out
is
JNU– 1999
(a) 237
(b) 249
(c) 261
(d) 280
If in a party everybody shakes hand with everybody else and these
are 36 handshakes in all, how many persons are there in the party?
JNU– 1999
(a) 9
(b) 12
(c) 18
(d) 72
A polygon has 44 diagonals then the number of its sides are
JNU– 1998
(a) 7
(b) 9
(c) 11
(d) None of these
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
19.
20.
21.
22.
Each student in a class takes at least one elective out of the three
available electives. Each one of the electives is taken by 100
students. The number of students who have taken by two electives
is either 50 or 51, and the number of students who have taken all
three electives is 34. The total number of students in the class is
bounded by
(IIT - 2009)
(a) 150 and 153
(b) 180 and 183
(c) 181 and 184
(d) 179 and 182
There are sixteen 2 by 2 matrices whose entries are 1s and 0s. Of
these, how many are invertible?
(JNU - 2009)
(a) 6
(b) 8
(c) 10
(d) None of these
In MCA, JNU Entrance Examination, a student scores 4 marks for
every correct answer and loses 1 marks for every wrong answer. If
the attempts 75 questions and secures 125 marks, the number of
questions he attempts correctly is
(JNU - 2009)
(a) 35
(b) 40
(c) 42
(d) 46
The digits 1, 2, 3, 4 are to be put in a 4 4 matrix in such a way
that each digit appears only once in each row, each column and
main diagonals. If first two rows are [1, 3, 4, 2] and [4, 2, 1, 3]
then the last two columns are
IIT – 2008
1
3
4
2
(a) and
2
4
3
1
1
2
2
1
(c) and
3
4
4
3
23.
24.
6.
List I
7.
4
2
1
3
(b) and
3
1
2
4
4
2
1
3
(d) and
2
4
3
1
2.
5.
(C)
5
12
(D)
255
256
(B)
127
128
(C)
63
64
(D)
1
27
9
1
(c)
27
27
(a)
11
15
P ( Ac | B )
4.
P( Ac | B c )
Then the correct match is
IIT-2010
(a) (1, Q), (2, S), (3, R), (4, P) (b) (1, S), (2, R), (3, P), (4, Q)
(c) (1, Q), (2, S), (3, P), (4, R) (d) (1, S), (2, R), (3, Q), (4, P)
Eight couples are participating in a game. Four persons are chosen
randomly. The probability that at least one couple will be among
the chosen person is
IIT-2010
5
13
1
26
(c)
25
26
(d)
2
5
9.
16
11.
13.
14.
(d) 1
15
4
5
(c)
4
4
(d)
5
1
45
1
2
(b)
1
4
(c)
1
3
1
6
(d)
4
25
(b)
5
26
(c)
5
25
(d)
1 1
.
5 26
If X is uniformly distributed over (0, 10), the probability that 1 <
X < 6 is
JNU – 2007
(a) 3/10
(b) 1/10
(c) 5/10
(d) None of these
The choice of throwing 12 in a single throw with three dice is
JNU – 2007
(a) 12/216 (b) 21/216 (c) 15/216 (d) 25/216
Two teams A and B play a series of four matches. If the
probability that team A wins a match is 2/3, then the probability
that team A wins three matches, loses one and the third win occurs
in the fourth match is
IIT – 2006
(a)
15.
(b)
Two letters are chosen one after another without replacement from
the English alphabet. What is the probability that the second letter
chosen is a vowel?
IIT – 2007
(a)
12.
5
An element of S is
Consider the experiment of throwing two fair dice. What is the
probability that the sum of the numbers obtained in these dice is
even?
IIT – 2008
(a)
31
32
(b)
a b
S
: a, b, cd {0,1} .
c d
4
4
9
2
1
, P ( B ) , then P(A B) is
5
3
3
13
2
(b)
(c)
(d)
5
15
15
3.
chosen randomly. Then the probability that the chosen matrix is
an invertible matrix is
IIT-2010
(a) 3/8
(b) ½
(c) 5/8
(d) 3/4
If a student is likely to choose any of the four choices with equal
probability in a multiple choice examination with five questions
then the probability that the student answer at least four questions
correctly is
(IIT - 2009)
10.
Two coins are available, one unbiased and the other two-headed.
Choose a coin at random and toss it; assume that the uniased coin
is chosen with probability 3/4. Given that the result is head, find
the probability that the two-headed coin was chosen. JNU-2010
(a) 1/5
(b) 2/5
(c) 3/8
(d) 3/16
If A and B are two independent events and
P( A | B)
P( A | B c )
Let
3
(b)
2.
7
20
13
Q.
20
3
R.
5
2
S.
5
P.
8.
There are 27 students in a college debate team. Find the
probability that at least 3 of them have their birthday in the same
month.
JNU-2010
(a)
4.
1
2
P( A | B)
(a)
An unbiased coin is tossed eight times. The probability of
obtaining at least one head and at least one tail is
IIT-2011
(A)
3.
(B)
List II
1.
(a)
PROBABILITY
1.
Three unbiased dice of different colours are rolled. The
probability that the same number appears on at least two of the
three dice is
IIT-2011
5
36
3
4
13
P ( A) , P ( B ) and P( A B)
.Consider
5
5
25
the following lists:
Given 18 one rupee coins of which one is counterfeit and weighs
less than any of the others. Given a two pan balance, the minimum
number of weighings required to identify the counterfeit is
IIT–2008
(a) 4
(b) 5
(c) 3
(d) 6
There are 7 sets of numbers S1, S2, ….., S7 such that S1 = {a}, S2 =
{b, c, d}, S3 = {c, d, e}, S4 = {f, g}, S5 = {h}, S6 = {i, j, k} and S7
= {j, k, l}. If a – l represents numbers in ascending order, and we
need to create a new list M such that it contains exactly one
element from each set S1 in strict sequence and the elements are in
strict ascending order, how many different such lists can be
created?
HCU-2011
(a) 162
(b) 32
(c) 36
(d) 72
(A)
Let
8
27
(b)
16
27
(c)
Let A and B be events with
P A B
8
81
(d)
P A
32
81
2
1
, P B and
3
2
1
. Match Lists I and II and select the correct
3
answer:
IIT-2010
IIT– 2006
List I
1.
5
List II
P A Bc
P.
2
3
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
16.
17.
18.
2.
P A Bc
3.
P Ac B c
4.
P Ac B c
1
3
5
R.
6
1
S.
6
Q.
(a)
(a) (1, P), (2, Q), (3, R), (4, S) (b) (1, R), (2, Q), (3, S), (4, P)
(c) (1, Q), (2, R), (3, S), (4, P) (d) (1, Q), (2, R), (3, P), (4, S)
For the events A and B to be independent, the probability that
both occur is 1/6 and probability that neither of them occur is 1/3.
Then the probability of occurrence of A is
IIT – 2005
(a) 1/2 or 1/3
(b) 1/4
(c) 1/2 only
(d) 1/3 only
For n independent events A1, A2, …, An, let
P(A) = 1 /(i + 1),
i = 1, 2 …,n. Then, the probability that none of the events will
occur is
IIT– 2005
(a) n/(n + 1)
(b) n – 1 /(n + 1)
(c) 1/(n + 1)
(d) 1/n
Let A, B and C be independent and mutually exclusive events
with
probability of
occurrences
20.
21.
22.
x
:
0
k
2k
2k
3k
2k2
25.
26.
Let
31.
32.
33.
34.
35.
36.
(a) 0.1 (b) -0.1
(c) – 1
(d) 1
The probability that a non-leap year should have 53 Sunday is
JNU – 2002
(a) 53/365 (b) 52/365 (c) 6/7 (d) 1/7
A speaks the truth in 70% cases and B in 80% cases. In what
percentage of cases are they likely to contradict each other while
narrating the same incident?
JNU – 2001
(a) 42 (b) 30 (c) 38 (d) 34
If P(A) = 0.59, P(B) = 0.30, P(A B) = 0.21, Then P(A' B') is
…………………
JNU – 2001
(a) 0.42
(b) 0.32
(c) 0.34
(d) 0.72
37.
38.
1
,
6
it was the first man?
JNU – 1999
(a) 3/31
(b) 6/31
(c) 7/30
(d) None of these
From a pack of 52 cards two are drawn at random. What is the
probability that one is a king and the other a queen?
JNU – 1998
40.
6
2
25
(b)
1
11
(c)
2
21
(d)
1
5
5
11
(b)
6
11
(c)
7
11
(d)
8
11
Subway trains on a certain line run every half hour between
midnight and six in the morning. Find the probability that a person
entering the station at a random time during this period will have
to wait at least twenty minutes.
JNU – 2008
(a) 1/2
(b) 2/3
(c) 1/3
(d) 1/6
A Cow is tied with a pole by a 100 meter long rope. What is the
probability that at some point of time the cow is at least 60 meters
away from the pole?
IIT – 2007
(a)
39.
1
1
and
. If only one hits the target, what is the probability that
3
4
27.
1 a
P
, where a, b, c are chosen randomly from the
b c
The incidence of occupational disease in an industry is such that
the workers have 20% chance of suffering from it. The probability
that out of 6 workers 4 or more will catch the disease is
(JNU - 2009)
(a) 2/3
(b) 40/3125 (c) 53/3125 (d) 50/3125
The UPSC has a list of 150 persons. Out of these 50 are women
and 100 are men. 125 of them know Hindi and remaining do not
know Hindi. 90 of them are teachers and remaining are not
teachers. What is the probability of selecting a Hindi – knowing
woman teacher as examiner?
(JNU - 2009)
(a) 1/6
(b) 3/5
(c) 2/9
(d) 5/6
Two dice are rolled until the sum of the numbers appearing on
these dice is either 7 or 8. What is the probability that the sum is
7?
IIT – 2008
(a)
7k2 +k
The probability that three men hit a target are respectively
(d) None of these
29.
30.
JNU – 2004
24.
10
687
Three persons play a game by tossing a fair coin each
independently. The game ends in a trial if all of them get the same
outcome in that trial, otherwise they continue to the next trial.
What is the probability that the game ends in an even number of
trials?
(IIT - 2009)
(a) 2/7
(b) 3/7
(c) 1/2
(d) 4/7
(a)
The value of k is
23.
(c)
(IIT - 2009)
1 3 p 1 4 p 1 p
,
,
,
2
3
6
K2
8
663
set {1, 2, 3, 4, 5}. The probability that P is singular is
IIT – 2005
(a) [-1/4, 5/6]
(b) [-1/4, 1/3]
(c) [-1/4, 1/2]
(d) [-1/2, 1/3]
The probability of getting a defective floppy in three boxes A, B
and C are 1/3, 1/6 and 3/4, respectively. A box is selected
randomly and a floppy is drawn from it. The probability that the
floppy is defective and is drawn from box A is
IIT – 2005
(a) 4/15
(b) 12/15
(c) 2/15
(d) 3/5
In the cigarette smoking population 70% are men and 30% are
women. 10% of these men and 20% of these women smoke
“WILLS’. The probability that a person smoking ‘WILLS’ will be
a men is
JNU – 2009
(a) 6/13
(b) 7/13
(c) 3/13
(d) 10/13
If A B = and B C = , then P(A B C) =
JNU – 2004
(a) P(A) + P(B) + P(C)
(b) P(A) P(B)P(C)
(c) P(A) P(B) + P(B) P(C) + P(C) P(A)
(d) P(A B) + P(B C)
A random variate has the following distribution:
:
0 1
2
3
4
5
6
7
p(x)
(b)
28.
respectively, then p lies in
19.
5
629
9
25
(b)
13
25
(c)
16
25
(d)
18
25
A speaks truth 3 times out of 4 and B speaks 7 times out of 10.
They both assert that a white ball has been drawn from a bag
containing 6 different color balls. Find the probability of the truth
of the assertion
JNU – 2007
(a) 21/40
(b) 35/36
(c) 39/40
(d) None of these
In four throws with a pair of dice, what is the chance of throwing
doublets at least twice?
JNU – 2007
(a) 1/144
(b) 25/144 (c) 19/144 (d) 26/144
A fair coin is tossed twice. Let A be the event that at least one tail
appears and B be the event that both head and tail appear. Then P
(A/B), the probability of A given B, is
IIT – 2006
(a) 1/4
(b) 1/2
(c) 2/3
(d) 1
A communication system consists of n components. Each of these
functions independently with probability p. The system function
correctly if and only if at least half of its components functions.
For what range of p, the probability that a five-components system
functions correctly is higher than the probability that a threecomponents system functions correctly?
IIT – 2005
(a) [0.4, 0.6] (b) [0, 0.5] (c) [0, 1]
(d) [0.5, 1]
Rahul and Sarvesh take turns in throwing two dice; the first to
throw 10 (sum of two dice) is being awarded the prize. If Rahul
gets the turn to throw the dice, their chances of winning are in the
ratio
JNU – 2009
(a) 10 : 11 (b) 11 : 30 (c) 11 : 12 (d) 12 : 11
A number is chosen from each of the two sets {1, 2, 3, 4, 5, 6, 7,
8, 9} and {1, 2, 3, 4, 5, 6, 7, 8, 9}. If p1 denotes the probability
that the sum of the two numbers be 10 and p 2 the probability that
their sum be 8, then (p1 + p2) is
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
41.
42.
43.
44.
JNU – 2002
(a) 7/729
(b) 137/729 (c) 16/81
(d) 137/81
A coin is biased so that the probability of head = 1/4. The coin is
tossed five times. The probability of obtaining two heads and
three tails with heads occurring in succession is
JNU – 2002
(a) (5 33) / 45
(b) 33 / 54
(c) 33 / 45
(d) 33 / 44
15 coupons are numbered 1, 2, …. 15. Seven coupons are selected
at random, one at a time, with replacement. The probability that
the largest number appearing on a selected coupon is 9, is
JNU – 2002
(a) (9 / 16)6 (b) (8 / 15)7 (c) (3/ 7)7
(d) None of these
Two friends arrive at the cafeteria independently, at random times
between 9 a.m. and 10 a.m., and each one stays for exactly m
minutes. The probability that either one arrives while the other is
46.
47.
31
128
(b)
49
512
(c)
27
256
(d)
12.
13.
14.
15.
15
64
16.
In a multiple choice question there are fur alternative answers of
which one or more are correct. A candidate will get marks in the
question only if he ticks all the correct answers. The candidate
decides to tick the answers at random. If he is allowed up to three
chance to answer the question, find the probability that he’ll get
marks in the question.
JNU – 1999
(a) 1/5
(b) 6/5
(c) 1/6
(d) None of these
If two events A and B are such that P(A) = 0.3, P(B) = 0.4 and
P(A B) = 0.5, then P(B | A B) equals
(a) 3/4
(b) 5/6
(c) 1/4
(d) 3/7
The probability that a number chosen at random from the primes
between 100 and 199 is odd, is
JNU-2010
(a) 0
(b) 1
(c) 1/2 (d) 0.6
x101 y 101 z 101
17.
3.
1 sin 2 cos
4
4
19.
4.
5.
6.
7.
8.
9.
10.
20.
21.
(JNU : - 2009)
250( 3 1)m
225( 2 1)m
(d) None of these
If sin x + sin2 x = 1, then the value of
cos12 x + 3cos10 x + 3 cos8 x + cos6 x – 1
is equal to
JNU– 2006
(a) 0
(b) 1
(c) – 1
(d) None of these
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied for n I, if
(JNU–2006)
(a) x = n + cos-1 (1/3)
(b) x = n - cos-1 (1/3)
(c) x = 2n + cos-1 (1/3)
(d) None of these
y
If
1 1 sin 4 A
1 sin 4 A 1
, then one of the values of y is
(b) cot A
(d) – cot A
The expression
(2 3 4)sin x 4cos x
(b)
(2 5, 2 5)
(c)
(2 5, 2 5)
(d)
lies in the interval
(2(2 5), 2(2 5))
If A lies in the second quadrant and 3tan A + 4 = 0, the value of 2
cot A – 5 cos A + sin A is equal to
JNU– 2006
(a) – 53/10 (b) 23/10
(c) 37/10
(d) 7/10
If tan( cos ) = cot ( sin ), then cos( - /4) is equal to
JNU – 2006
22.
1
(b)
2 2
1
(c)
mn
2
(b)
2
m n
m2 n2
2mn
(d)
4 3 km
(c) 6
2 2
2mn
2
m n2
mn
mn
(b)
2 / 3 km
23.
(d) 6 km
The principal value of sin-1 [sin(2/3)] is
(a) /3
(b) -2/3
(c) 2/3
(d) 5/3
24.
The value of
1
3
is
sin10 cos10
(a) 2
(b) 4
25.
If
tan A
1 cos B
(a) (A - n)/2
(c) 2(A - n)
7
(d)
An observer at an anti-aircraft post A identifies an enemy aircraft
due east of his post at an angle of elevation of 60. At the same
instant a detection post D situated 4 km south of A reports the
aircraft at an elevation of 30. The altitude at which the plane is
flying is
JNU - 2004
(a)
(1 / 3)) cos(tan1 2 2 ) is
2
2
The value of tan 1 tan 2 … tan 89 is
JNU– 2006
(a) – 1
(b) 0
(c) 1
(d) N.O.T.
If cosA + cosB = m and sinA + sinB = n, where, m, n 0, then sin
(A + B) is equal to
JNU– 2005
(c)
(JNU-2009)
tan 1, cos 1 / 2 are
(a) n + 7/4
(b) n + (-1)n 7/4
(c) 2n + 7/4
(d) 2n + (-1)n 7/4
If A > 0, B > 0 and A + B = /3, then the maximum value of tan A
tan B is
JNU – 2008
(a) 0
(b) 1/3
(c) 3
(d) None of these
If tan A = 5/6 and tan B = 1/11, then
JNU– 2008
(a) A + B = /6
(b) A + B = /4
(c) A + B = /3
(d) None of these
In a triangle, the lengths of the two larger sides are 10 and 9
respectively. If the angles are in AP, the length of the third side
can be
MCA– 2008
(a) 3 5
(b) 5 3
(c) 5 + 6 (d) None of these
In a triangle ABC, if tan (A/2) = 5/6 and tan (B/2) = 20/37, the
sides a, b and c are in
JNU – 2008
(a) AP
(b) GP
(c) HP
(d) None of these
1
(c)
(a)
(a) 3
(b) 4
(c) 5
(d) None of the above
The value of tan 100 + tan 125 + tan 100 tan 125 is
(JNU-2009)
(a) 0
(b) 1/2
(c) –1
(d) 1
The most general values for which
The value of sin (2 tan
(b)
(a)
is
(d) 3
for
250( 3 1)m
JNU :– 2006
(a) (-4, 4)
(JNU : - 2009)
(a) 0
(b) 1
(c) 2
The maximum value of =
(a)
JNU :– 2006
TRIGONOMETRY
1.
sinh ix equals
JNU-2010
(a) coshx
(b) i sin x
(c) cos x
(d) 1
2.
If sin-1 x + sin-1 y + sin-1z = 3/2, then the value of
9
times the height of the pole is
JNU – 2007
(a) 30
(b) 45
(c) 60
(d) 135
If in a triangle ABC, sin A, sin B, sin C are in AP, then
JNU – 2006
(a) the altitudes are in AP
(b) the altitude are in HP
(c) the altitudes are in GP
(d) None of these
A person walking along a straight road observes that at two points
1 km apart, the angles of elevation of a pole in front of him are
30 and 75. The height of the pole is
JNU– 2006
(a) tan A
(c) – tan (2A)
18.
x100 y 100 z 100
3
shadow of the pole is
in the cafeteria is 40% and m a b c , where a, b and c are
positive integers and c is not divisible by the square of any prime.
Find a + b + c.
JNU – 2000
(a) 87
(b) 42
(c) 90
(d) None of these
The probability that a person tossing three fair coins will get
together all heads or all tails for the second time on the 5th toss is
JNU – 1999
(a)
45.
11.
JNU– 2007
(a) 12/13
(b) 13/14
(c) 14/15
(d) None of these
The angle of the elevation of the sum when the length of the
sin B
(c)
2 2
JNU - 2004
JNU - 2004
(d)
, then B equals
2
JNU - 2004
(b) n - 4
(d) A/2 - n
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
26.
27.
28.
29.
30.
31.
(a)
(c)
a c / 2
35.
36.
37.
38.
30.
40.
41.
(d) a c 2 / 2
(b)
cos
a / 2c 2
(b)
42.
x 2n
(c)
x n 1n
2
43.
3
,
2
44.
4
4
45.
23
8
The function
(b)
25
8
(c)
x
f ( x) sin( sin x)
2
(d) None of these
has
IIT – 2008
(a) no zero
(b) only zero at x = 0
(c) zeros at x = n, n = 0, 1, 2, …
(d) zeros at x = 2n, n = 0, 1, 2, …
The equation cos 2x + a sin x = 2a – 7 possesses a solution if
JNU – 2008
(a) a < 2
(b) 2 a 6
(c) a > 6
(d) a is any integer
The expression cos2 (A – B) + cos2 B – 2 cos (A – B) cos A cos B
is
JNU – 2008
(a) dependent of A
(b) dependent of B
(c) 0 < x < 1
(d) None of these
If x is the value of tan 3A cot A, then
JNU – 2008
(a) x < 1
(b) 1/3 < x < 3
(c) 0 < x < 1
(d) None of these
The angle of elevation of a cloud from a point x meter above a
lake is A and the angle of depression of its reflection in the lake is
45. The height of the cloud is
JNU – 2008
(a) x tan (A)
(b) x tan (45)
(c) x tan (A + 45)
(d) x cot (A + 45)
Consider the equations
sin (cos x) = x
...(1)
and
cos (sin x) = - x
…(2)
for x 0. Then
(IIT– 2007)
(a) Only Equation (1) has a solution
(b) Only Equation (2) has a solution
(c) Both Equations (1) and (2) have solutions
(d) Neither Equation (1) nor Equation (2) has a solution
The sum of the series
equal to
(d) infinite
JNU :– 2007
(a) Both I and II are true
(b) Both I and II are false
(c) I is true but II is false
(d) I is false but II is true
The number of solutions of the equation tan x + sec x = 2 cos x,
lying in the interval [0, 2] is
JNU :– 2007
(a) 0
(b) 1
(c) 2
(d) 3
The complete solution of the equation
7cos2x + sin x cos x – 3 = 0 is given by
JNU :– 2007
(a) n + /2 (n I)
(b) n - /2 (n I)
(c) n + tan-1 (4/3) (n I)
(d) n + 3/4, k + tan-1 (4/3) (n, k I)
If
f x cos
2
x
cos 2
x
sin 2 0, / 2 then
x
sin 2
cos 2
46.
roots of f(x) = 0 are
(a) 1/2, - 1
(b) 1/2, - 1, 0
(c) – 1/2, 1, 0
(d) – 1/2, - 1, 0
If sin(x + 3) = 3 sin( – x), then
47.
(a) tan x = tan
(b) tan x = tan2
3
(c) tan x = tan
(d) tan x = 3 tan
tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A is equal to
JNU– 1998
27
8
x 1 / 2
cos sin
a
b
, then
a
a
b
sec 2 cos ec 2
: n 0,1,2.....
(d) None of these
Compute : (sin 15 + cos 15)6
2
(a) zero
(b) one
(c) two
Let the two statements
(I) sin 100 sin 500 sin 700 = 1/8
sin 2
: n 0,1,2......
x
Of the following, identify the correct statement
The general solution of the trigonometric equation sin x + cos x =
1 is given by
JNU– 1999
(a) x 2n : n 0,.1,2..........
(b)
(d) None of these
The number of real solutions of the equation
(II) If
(d) None of these
2 cos 4
19 cos
JNU :– 2007
2
4
8
16
is
cos
cos
cos
15
15
15
15
0,
2
,
2
(b)
tan 1 x x 1 sin 1
JNU - 2004
(a) 1/8
(b) 3/16
(c) 1/16
(d) 3
The greatest value of sin cos is
JNU– 2002
(a) – 1
(b) 1
(c) – 1/2
(d) 1/2
Let tan = m/(m + 1) and tan = 1/ (2m + 1), then the value of
( + ) is
JNU– 2002
(a) /3
(b) /6
(c) /2
(d)
The smallest positive root of the equation tan x – x = 0 lies in
JNU– 2000
(a)
34.
ac 2
The value of
(c)
33.
(a)
(a) 5 6 (b) 5 (c) 3 3
(d) 5 6
If A = 45 and B = 75, what is the value of side b?
JNU - 2004
(a)
32.
4 cos 1
17 8 cos
cos
(c)
2 cos
In a triangle, the lengths of the two larger sides are 10 and 9
respectively. If the angles are in AP, then the length of the third
side is
JNU - 2004
48.
7 3
3
(b)
5 3
3
(c)
2 3
3
(d)
(a) 1
51.
52.
3
3
If sin ( + ) = 1 and sin ( – ) = 1/2 where , [0, /2], then
tan( 2 )
is equal to
tan(2 )
50.
(JNU– 2007)
JNU – 2007
(a) tan 2A (b) cot A
(c) sin 3A (d) None of these
In a triangle with one angle 2/3, the lengths of the sides from an
AP. If the length of the greatest side is 7 cm, the radius of the
circumcircle of the triangle is
JNU– 2006
(a)
49.
(JNU–2007)
(b) 2
(c) 3
JNU–2006
(d) 4
2sin
1 cos sin
y , then
If
is equal to
1 cos sin
1 sin
JNU – 2006
(a) 1/y
(b) y
(c) 1 – y
(d) 1 + y
The number of solutions of the equation sin 5x cos 3x = sin 6x cos
2x in the interval [0, ] is
JNU – 2006
(a) 3
(b) 4
(c) 5
(d) 6
If cos cos cos sin sin sin 0 , then
which of the following are true?
(I) cos 2 cos 2 cos 2
0
(II) cos( ) cos( ) cos( ) 0
(III) sin 2 sin 2 sin 2 0
JNU – 2006
cos cos 2 cos 3
+ ….. will be
4
42
43
(a) (I) and (II) only
(c) (III) and (I) only
(b) (II) and (III) only
(d) (I), (II) and (III)
JNU– 2007
8
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
53.
54.
If the lines 2(sinA + sinB)x – 2 sin (A – B)y = 3 and 2(cos A +
cosB)x + 2 cos (A – B)y = 5 are perpendicular, then sin 2A + sin
2B is equal to
JNU– 2006
(a) sin (A – B) – 2 sin (A + B)
(b) 2sin (A – B) – sin (A + B)
(c) sin (2(A – B)) – sin (A + B)
(d) sin (2(A – B)) – 2 sin (A + B)
If x cos + y sin = x cos + y sin = 2 (0 < < / 2), then
it is also true that
JNU :– 2006
(a)
cos cos
(b)
cos cos
(c)
cos cos
(d)
cos cos
TWO-DIMENSIONAL GEOMETRY
1.
A straight line passes through (2, - 6) and the point of intersection
of the lines 5x – 2y + 14 = 0 and 2y = 8 – 7x. Any straight line
concurrent with the given lines is (5x – 2y + 14) + (2y – 8 + 7x)
= 0. The value of is
JNU-2010
(a) 6
(b) 36
(c) 17
(d) 16
2.
Let S x2 + y2 + 2gx + 2fy + c = 0 be equation of a circle and P
ax + by + c' = 0 be the equation of a straight line. Then the
equation S + P = 0 represents
JNU-2010
(a) circle
(b) ellipse
(c) hyperbola
(d) pair of straight lines
3.
A point moves in such a manner that the sum of its distances from
fixed points (-3, 0) and (3, 0) is 6. Then the locus of the moving
point must be
JNU-2010
(a) an ellipse
(b) a parabola
(c) a line segment joining the fixed points
(d) a circle
4ax
x y2
2
4a 2 y 2
x2 y 2
4ax
x y2
2
4a 2 y 2
x y
2
4.
2
0
55.
If
sin xdx sin 2 , then the value of satisfying 0 < <
5.
2
is
56.
JNU– 2005
(a) 3/2
(b) /6
(c) 5/6
(d) /2
The general solution of sin x – 3 sin 2x + sin 3x = cos x – 3 cos 2x
+ cos 3x is
JNU– 2005
(a)
n
8
n n
(c) 1
2
8
57.
58.
59.
60.
61.
62.
63.
n
2 8
(d)
2n cos1 3 / 2
7.
In a triangle ABC, the angle A is greater than the angle B. If the
values of the angles A and B satisfy the equation 3 sin x – 4 sin3 x
– k = 0, 0 < k < 1, then the value of C is
JNU - 2004
(a) 5/6
(b) 2/3
(c) /2
(d) /3
If cos-1p + cos-1q + cos-1 r = , then p2 + q2 + …. = 1
JNU - 2004
(a) 2p2q2 + r2 + 4pqr
(b) r2 + 2pqr
(c) r2 + 2pqr – 1
(d) None of these
The smallest positive value of x (in degree) for which tan (x +
100) = tan(x + 50) tan(x) tan(x - 50) is
JNU - 2004
(a) 75
(b) 60
(c) 45
(d) 30
In a triangle ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the
circumcircle to that of the incircle is
JNU - 2004
(a) 7/16
(b) 9/16
(c) 16/9
(d) 16/7
If A + B + C = 2S, then
cos2S + cos2(S – A) + cos2 (S – B) + cos2 (S – C) = 2 + …
JNU - 2004
(a) cos A cos B cos C
(b) 2cos A cos B cos C
(c) cos (B + C) cos (B – C)
(d) N.O.T
tan
8.
9.
10.
11.
a b
c
.......
cot , then c
ab
2
cos
(c)
2/ 3
2 2 /
3
(b)
2. 3
(d)
2 2 / 3
12.
(a)
4
5
(b)
4
(c)
6
13.
The number of solutions of the equation
cos4x + sin4x = snx cosx (0 x 2)
is ………………..
equation
| x 2 ( y 1) 2 x 2 ( y 1) 2 | K
will represented a hyperbola for
(JNU-2009)
(a) K (0, 2)
(b) K (-2, 1)
(c) K (1, )
(d) K (0, )
The equation of the tangent to the conic x2 – y2 – 8x + 2y + 11 = 0
at (2, 1) is
(JNU - 2009)
(a) x + 2 = 0
(b) 2x + 1 = 0
(c) x = 2 = 0
(d) x + y + 1 = 0
The equation of the circle through (1, 1) and the points of
2
2
intersection of x + y + 13x – 3y = 0 and
2x2 + 2y2 + 4x – 7y – 25 = 0
JNU – 2008
(a) 4x2 + 4y2 – 30x – 10y – 32 = 0
(b) 4x2 + 4y2 + 30x – 13y – 25 = 0
(c) 4x2 + 4y2 + 30x – 13y + 25 = 0
(d) None of these
The line y = x + 5 does not touch
JNU – 2008
(a) the parabola y2 = 20x
2
2
(b) the ellipse 9x + 16y = 144
(c) the hyperbola 4x2 – 29y2 = 116
(d) the circle x2 + y2 = 25
The orthocenter of the triangle with vertices (0, 0), (3, 0), (0, 4) is
MCA – 2007
(a) (0, 0)
(b) (3/2, 2) (c) (1, 4/3) (d) N.O.T
If the foci of the ellipse
x2 y 2
1
144 81 25
JNU - 2004
(a) (a + b) sin c/2 tan c/2
(b) (a – b) sin c/2
(c) (a + b) sin c/2
(d) (a – b) tan c/2 cosec c/2
The value of cos (2 cos-1 x + sin-1x), for 0 cos-1 x and -/2
sin-1 x /2 at x = 1/3, is
JNU– 2002
(a)
64.
(b)
6.
The
x2 y 2
1
25 b 2
and the hyperbola
coincide, then the value of b2 is
JNU - 2007
(a) 3
(b) 16 (c) 9
(d) 12
2
Equation of a common tangent to the curve y = 8x and xy = - 1 is
JNU– 2007
(a) 3y = 9x + 2
(b) y = 2x + 1
(c) 2y = x + 8
(d) y = x + 2
If sum of the distances of a point from two perpendicular lines in a
plane is 1, then its locus is
JNU - 2006
(a) a square
(b) a circle
(c) a straight line
(d) two intersecting lines
Two circles x2 + y2 = 6 and x2 + y2 – 6x + 8 = 0 are given. Then
the equation of the circle through their points of intersection and
the point (1, 1) is
JNU - 2006
(a) x2 + y2 – 6x + 4 = 0
(b) x2 + y2 – 3x + 1 = 0
2
2
(c) x + y – 4y + 2 = 0
(d) None of these
The straight line y = 4x + c is tangent to the ellipse
x2 y 2
1 . Then C is equal to
8
4
JNU: - 2006
JNU– 2001
5
(d)
6
14.
9
(a) 4
(b) 6
(c) 1
(d) 132
The lines x – 2y – 6 = 0, 3x + y – 4 = 0 and x + 4y + 2 = 0 are
concurrent if is equal to
JNU – 2008
(a) 2
(b) – 3
(c) 4
(d) None of these
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
15.
16.
Two of the straight lines given by 3x + 3x y – 3xy + my = 0 are
at right angles if
JNU – 2008
(a) m = - 1/3
(b) m = 1/3
(c) m = - 3
(d) m = 3
The orthocentre of the triangle with vertices (0, 0), (3, 0), (0, 4) is
JNU – 2007
(a) (0, 0)
(b) (3/2, 2)
(c) (1, 4/3)
(d) None of these
2
2
17.
If the foci of the ellipse
y2
x2
1
144 81 25
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
2
2
3
34.
35.
2
y
x
1 and the hyperbola
36.
25 b 2
coincide, then the value of b2 is
JNU– 2007
(a) 3
(b) 16 (c) 9
(d) 12
The straight line y = 4x + c is tangent to the ellipse
37.
x2 y 2
1 . Then C is equal to
8
4
(a) 4
(b) 6
(c) 1
38.
JNU – 2006
(d) 132
If a line is drawn through a fixed point P(, ) to cut the circle x2
+ y2 = a2 at A and B, then PA PB is equal to
JNU – 2005
(a) 2 + 2
(b) 2 + 2 - a2
(c) 2
(d) 2 + 2 + a2
If the lines ax + 2y + 1, = 0, bx + 3y + 1 = 0, cx + 4y + 1 = 0 are
concurrent, then a, b, c are in
JNU – 2005
(a) AP
(b) GP
(c) HP
(d) None of these
The distance between the lines 4x + 3y = 11 and 8x + 6y = 15 is
JNU – 2005
(a) 7/2
(b) 4
(c) 7/10
(d) None of these
Area of the quadrilateral formed by the lines |x| + |y| = 1 is
JNU – 2005
(a) 4
(b) 2
(c) 8
(d) None of these
The parametric coordinates of any point on the parabola y2 = 4ax
can be
JNU – 2005
(a) (-at2, -2at)
(b) (-at2, 2at)
2
(c) (- a sin t, - 2a sin t)
(d) (- a sin t, 2a cos t)
The angle between the lines given by the equation
Y2 sin2 - xy sin2 + X2 (cos2 - 1) = 0 is
JNU– 2004
(a) /4
(b) /3
(c) /2
(d) 2/3
The straight lines
7x – 2y + 10 = 0
7x + 2y – 10 = 0
and
y + 2 = 0 form
JNU – 2004
(a) obtuse-angled triangle
(b) acute-angled triangle
(c) right-angled triangle
(d) isosceles triangle
The three lines : ax + by + c = 0; bx + cy + a = 0 and cx + ay + b =
0 are congruent only when
JNU– 2004
(a) a + b + c = 0
(b) a2 + b2 + c2 – ab – bc – ca = 0
(c) a3 + b3 + c3 + 3abc = 0
(d) a3 + b3 + c3 – a2b – b2c – c2a = 0
The area of the quadrilateral with vertices at (2, - 1), (4, 3), (-1, 2)
and (-3, - 2) is
JNU – 2004
(a) 30
(b) 36
(c) 15
(d) 18
The tangent of the circle x2 + y2 = 169 at the points (5, 12) and
(12, - 5)
JNU – 2004
(a) coincide
(b) are parallel
(c) are perpendicular
(d) None of the above
The medians of a triangle meet at (0, - 3). While its two vertices
are (-1, 4) and (5, 2), the third vertex is at
JNU – 2002
(a) (4, 5)
(b) (-1, 2) (c) (7, 13) (d) (-4, -15)
The areas of the triangle having the vertices (4, 6), (x, 4), (6, 2) is
10 sq. units. The value of x is
JNU – 2002
(a) 0
(b) 1
(c) 2
(d) None of these
The angle between the tangents from the point (4, 3) to the circle
x2 + y2 – 2x – 2y = 0 is
JNU – 2002
(a) /2
(b) /3
(c) /4
(d) None of these
Consider the circle x2 + y2 = 14x. The point P(6, - 7) is
JNU – 2002
(a) on the circle
(b) in the circle
(c) outside the circle
(d) None of these
The eccentricity of a rectangular hyperbola is always JNU – 2002
39.
40.
(a) 1
(b) 2
(c) 3
(d) 2
If two lines a1x + b1y + c1 = 0 and a2x + b2y + c = 0 cut the
coordinate axes in concyclic points, then
JNU – 2008
(a) a1a2 + b1b2 = 0
(b) a1a2 – b1b2 = 0
(c) a1b1 + a2b2 = 0
(d) a1b1 – a2b2 = 0
If the line joining the points (0, 3) and (5, -2) is a tangent to the
curve y = c/(x + 1), then the value of c is
JNU – 2008
(a) 1
(b) – 2
(c) 4
(d) none of these
Each side of an equilateral triangle subtends an angle of 60 at the
top of a tower h meter high located at the center of the triangle. If
a is the length of each side of the triangle, then
JNU – 2008
(a) 3a2 = 2h2
(b) 2a2 = 3h2
2
2
2
2
(c) a = 3h
(d) 3a = h
The lines x – 2y – 6 = 0, 3x + y – 4 = 0 and x + 4y + 2 = 0
JNU – 2008
(a) 2
(b) – 3
(c) 4
(d) None of these
If the line y = mx is one of the bisectors of the lines x2 – y2 + 4xy
= 0, then the value of m is given by
JNU – 2008
(a) m = 1
(b) m2 – m = 0
(c) m2 + m – 1 = 0
(d) None of these
Two of the straight lines given by 3x2 + 3x2y – 3xy2 + my3 = 0 are
at right angles if
JNU – 2008
(a) m = - 1/3
(b) m = 1/3
(c) m = - 3
(d) m = 3
The length of the line joining two points on the parabola y2 = x
which is bisected at (1, 2) is
JNU– 2007
51
(a)
(b)
3 51
(c)
4 51
(d)
2 51
y
e x ex
2
41.
The shortest distance of (0, 0) from the curve
42.
JNU– 2007
(a) 1/2
(b) 1
(c) 2
(d) None of these
If a, b, c are the sides of a triangle, then the value of the
expression
43.
a
c
c
is equal to
bc ca ab
JNU– 2007
(a) 1
(b) 3/2
(c) 2
(d) 5/2
A straight line is drawn through the centre O of the circle x2 + y2 =
2ax parallel to x + 2y = 0 and intersecting the circle at A and B.
The area of the AOB is
JNU– 2007
(a)
a2
(b)
5
44.
45.
a3
5
(c)
a2
(d)
a2
2
3
The area of the portion of the circle x2 + y2 – 4y = 0 lying below
the x-axis is
JNU– 2007
(a) 24π
(b) 42π
(c) 82π
(d) 0
The parametric equations
x
a
b
1/ ; y 1/
2
2
where is a parameter, represents
46.
JNU– 2007
(a) a straight line
(b) a parabola
(c) an ellipse
(d) a hyperbola
The centre of a circle passing through the point (0, 1) and
touching the curve y = x2 at (2, 4) is
JNU - 2006
(a)
(c)
47.
48.
10
16 27
,
5 10
16 53
,
5 10
(b)
16 5
,
7 10
(d) None of these
A variable chord is drawn through the origin to the circle x2 + y2 –
2ax = 0. The locus of the centre of the circle drawn on this chord
as diameter is
JNU - 2006
(a) x2 + y2 + ax = 0
(b) x2 + y2 + ay =0
(c) x2 + y2 – ax = 0
(d) x2 + y2 – ay = 0
If G is the centroid and I is the incentre of the triangle with
vertices A (-36, 7), B(20, 7) and C(0, - 8), then GI is equal to
JNU - 2006
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
(a)
49.
50.
51.
52.
53.
(c)
55.
56.
57.
58.
59.
60.
61.
62.
63.
205
(c)
3
181
(d) None of these
3
Locus of the mid-points of the chords of the circle x2 + y2 = 4
which subtends a right angle at the centre is
JNU - 2006
(a) x + y = 2
(b) x2 + y2 = 1
(c) x2 + y2 = 2
(d) x – y = 0
If the equation of one tangent to the circle with centre at (2, - 1)
from the origin is 3x + y = 0, then the equation of the other
tangent through the origin is
JNU - 2006
(a) 3x – y = 0
(b) x + 3y = 0
(c) x – 3y = 0
(d) None of these
If the line joining the points (0, 3) and (5, -2) is a tangent to the
curve y = c/(x + 1), then the value of c is
JNU – 2008
(a) 1
(b) – 2
(c) 1/2
(d) None of these
If the line y = mx is one of the bisectors of the lines x2 – y2 + 4xy
= 0, then the value of m is given by
JNU – 2008
(a) m = 1
(b) m2 – m = 0
(c) m2 + m – 1 = 0
(d) None of the above
The center of a circle passing through the point (0, 1) and
touching the curve y = x2 at (2, 4) is
JNU – 2006
(a)
54.
250
(b)
3
16 27
,
5 10
16 53
,
5 10
(b)
16 5
,
7 10
(d) None of these
If A and B are two fixed points, then the locus of a point which
moves in such a way that the angle APB is a right angle is
JNU – 2005
(a) a circle
(b) an ellipse
(c) a parabola
(d) None of these
The mid-points of the sides of a triangle are (5, 0), (5, 12) and (0,
12). The orthocenter of this triangle is
JNU– 2005
(a) (0, 0)
(b) (10, 0)
(c) (0, 24) (d) (13/3, 8)
Points A(1, 3) and C(5, 1) are opposite vertices of a rectangle
ABCD. If the slope of BD is 2, then its equation is JNU– 2005
(a) 2x – y = 4
(b) 2x + y = 4
(c) 2x + y – 7 = 0
(d) 2x + y + 7 = 0
The locus of the point of intersection of tangents to an ellipse at
two points, sum of whose eccentric angles is constant, is a / an
JNU– 2005
(a) parabola
(b) circle
(c) ellipse
(d) straight line
The locus of the point of intersection of tangents to the parabola y2
= 4(x + 1) and y2 = 8(x + 2) which are perpendicular to each other
is
JNU– 2005
(a) x + 7 = 0
(b) x – y = 4
(c) x + 3 = 0
(d) y – x = 12
The ends of the base of an isosceles triangles are at (2a, 0) and (0,
a). The equation of one side is x = 2a. The equation of the other
side is
JNU– 2005
(a) x + 2y – a = 0
(b) x + 2y = 2a
(c) 3x + 4y – 4a = 0
(d) 3x – 4y + 4a = 0
If the point (2a, a), (a, 2a) and (a, a) enclose a triangle of area 18
sq. units, the centroid of the triangle is
JNU – 2004
(a) (6, 4)
(b) (4, 6)
(c) (-8, 8)
(d) (8, 8)
Consider the three lines :
L1 + x + y = 1, L2 : x – y = - 1, L3 : 7x – y = 6
A maximum of how many circles can be drawn each touching all
these lines?
JNU – 2004
(a) Three
(b) Two
(c) One
(d) None (zero)
Given the points A(0, 4) and B(0, - 4), the equation of the locus of
the point P(x, y) such that |AP - BP| = 6 is
JNU – 2004
(a) 9x2 + 7y2 + 63 = 0
(b) 9x2 – 7y2 – 63 = 0
(c) x2 + y2 – 9 = 0
(d) x2 + y2 – 1 = 0
In a rectangular hyperbola, the asymptotes are
JNU – 2001
(a) at right angles
64.
(b) meeting the curve at two points
(c) inclined at an angle of 60 to each other
(d) parallel to the transverse and conjugate axes
x – 2y + 4 = 0 is a common tangent to y2 = 4x and
x2 y2
2 1 . Then the value of b and the other common
4
b
JNU – 2001
tangent are given by
(a)
(b)
b = 3 ; x + 2y + 4 = 0
b = 3; x + 2y + 4 = 0
(c)
(d)
b = 3 ; x + 2y – 4 = 0
None of these
FUNCTIONS
1.
Let f (x) = 2x3 + 3x2 −12x + 4 for all x R. Then
IIT-2011
(A) f is not one-one on [−1,1]
(B) f is one-one on [−1,1] but not one-one on [−2, 2]
(C) f is one-one on [0, 2] but not one-one on [−2, 0]
(D) f is one-one on [−2, 2]
1
f x , g x x 3/ 2 , h x x 2 2 x 3.
x
2.
Suppose that
3.
Computer fgh (x) at x = 2.
JNU-2010
(a) 113(b) 11-3/2
(c) 113/2
(d) None of these
What is the range of the function f that maps R toR2 by means of
the formula
4.
5.
sin t
f t
?
cot t
JNU-2010
(a) A circle (circumference only)
(b) R2
(c) The set of all points {x, y} satisfying -1 x 1 and -1 y 1
(d) A disk consisting of a circle together with all the points
enclosed by circle
If f(x) = ax + b and g(x) = cx + d, then f (g(x)) = g(f(x)) is
equivalent to
JNU – 2005
(a) f(a) = g(c)
(b) f(b) = g(b)
(c) f(a) = g(b)
(d) f(c) = g(a)
If f(x) = cos (ln x) then f(x) f(y)
1 x
f f xy has
2 y
the value
JNU – 2000
6.
(a) – 1
(b) 1/2(c) – 2
(d) None of these
Which of the following function is periodic?
JNU – 1999
(a) f(x) = x – [x] where [x] denotes the largest integer less than or
equal to the real number x
(b)
1
f x sin , x 0 f 0 0
x
(c) f(x) = x cos x
(d) None of these
7.
Suppose
n,
if n 1
T ( n) n
5T
n, if n 1
5
The value of T(125) is
(IIT - 2009)
8.
(a) 500
(b) 400
Consider the function
(c) 375
(d) 380
m
if n 1
f (m, n)
. For positive
m f (m, n 1) if n 1
integers m and n, f(m, n) is
(IIT - 2009)
n 1
11
(m i )
(a) m + n
(b)
(c) mn
(d) mn
i 0
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
9.
10.
11.
12.
A and B are two sets with cardinality m and n respectively. The
number of possible one – to – one mappings from A to B, when m
< n is
(JNU - 2009)
(a) mn
(b) mCn
(c) nPm
(d) mP2
If f(x + y) = f(x) + f(y) – xy – 1 for all x, y, and f(1) = 1, then the
number of solutions of f(n) = n n I N, is
JNU - 2008
(a) one
(b) two
(c) four
(d) None of these
The function f defined on R by
f(x) = 3x + 4x – 5x
has
IIT – 2007
(a) exactly one zero
(b) exactly two zeros
(c) exactly three zeros
(d) infinitely many zeros
Consider the following functions of a complex variable
( R ( z )) / | z |
f1 ( z ) e
0
2
13.
14.
15.
2
2
f a
f 'a
f " a
b2
b 1
a2
a 1
2.
3.
17.
2
(a) Points inside y ≤ |x|
(c) Points inside y ≤ x
18.
is
4.
x
(b) r
x2
is
x2 1
a
is
(c) p
The value of f(0,) for which
(a) 51
(d)
f x
512
2
x4 2
sin 2 x
(b) 59
JNU-2010
(d) None of these
(c) 61
n
5.
lim
r 1
x 1
is equal to
x 1
JNU – 2008
(a) n/2
(c) 1
6.
(b) n(n + 1)/2
(d) None of these
2
n
1
lim
...
is equal to
2
2
n 1 n
1 n
1 n2
(b) – 1/2
(c) 1/2
7.
lim 1 2 / x equals
8.
(a) e
(b)
(c) e2
The value of a so that the function
JNU – 2008
(d) None of these
x
x
JNU – 2005
(d) 1/e
cos ax, x 0
f x 1
x0
2 ,
Be continuous at x = 0 is
10.
(c) 1
(b) – 1
(a) 1
The value of
lim
(a) 0
(b) 1
5 x3
0
x 0
JNU – 2005
(d) 0
et dt
x
JNU – 2004
is
(c) 5
(d)
1
sin x x x 2
6
The value of lim
is
5
x 0
x
surjection, then A is equal to
JNU – 2005
19.
20.
(a) R
(b) [0, 1]
(c) (0, 1]
(d) [0, 1)
Let f, g : R R+ defined by f(x) = 2x + 3 and g(x) = x2. The value
of (g o f) (x) is
JNU – 2004
(a) 2x2 + 3
(b) 2x + 3
(c) (2x + 3)2
(d) 4x2 + 9
Let f be a one-one function with domain {a, b, c} and range {x, y,
z}. If f(a) = y, then which of the following is true?
JNU – 2002
(a) f(b) = x, f-1 (z) = a
(b) f(b) = z, f-1 (y) = c
-1
-1
(c) f(c) = z, f (x) = b
(d) f(c) = x, f (x) = b
LIMITS & CONTINUITY
1.
Given f(x) is differentiable and f ' (4) = 5, find
lim
x 2
f 4 f x
2
JNU – 2004
(a)
11.
(b) 1/120
If f(9) = 9, f'(9) = 4, then
(c) 1/20
(d) 1/2
{ f x 3}
{ x 3}
equals
JNU – 2004
12.
(a) 0.50
(b) 1
(c) 2
(a) ½ log 5
(b) 1/5 log 2 (c) 2 log 5
x 5x 1
is equal to
lim
x 0 1 cos x
(d) 4
JNU – 2004
13.
is
xr n
IIT - 2005
(b) Points inside |y| ≤ |x|
(d) Points inside |y| ≤ x
f x
2
continuous, is
9.
If the function f : R A given by
(d) Does not exist
JNU-2010
(JNU – 2006)
IIT - 2005
(a) increases
(b) decreases
(c) remains constant
(d) oscillates
The domain of the real valued function f(x, y) defined by
1
(c)
2
tan px 2 qx r
(a) 0
(a) 0
(b) 1
(c) 2
(d) 4
If an odd function increase for x > 0, then for x < 0 it
is
If a is a repeated root of px2 + px + r = 0, then
xa
if z 0
16.
(d) – 20
(c) 5
(b) 1
(a) 0
0 0
f x, y x x y
1
1 e1/ x
lim
is equal to (-16) Then b – a is equal to
2
x 0
(a) 0
2a 1 0
2
lim
(b) 0
JNU-2010
if z 0
and f2 (z) = |z|2, where Re(z) is the real part of z. Let the two
statements
(I) f1(z) is continuous at z = 0
(II) f2(z) is analytic at z = 0
Of the following, identify the correct statement
(JNU – 2007)
(a) I is true but II is false
(b) II is true but I is false
(c) Both I and II are true
(d) Both I and II are false
Let f : R R and f(x) = logex, R being the set of real numbers,
then
JNU– 2007
(a) f is onto
(b) f is one-one
(c) f is invertible
(d) None of these
Let f be a function defined on [0, 2], then the function g(x) = f(9x2
– 1) has domain
(JNU – 2007)
(a) [0, 2]
(b) [-1/3, 1/3]
(c) [-3, 3]
(d) None of these
Let f(x) = x3, x [a, b] and the value of the determinant
f b
(a)
(d) 5 log 2
If m and n are positive numbers, then the limit
mx nx
x 0
x
lim
is
equal to
x2
JNU – 2003
JNU-2010
12
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
log
(a)
14.
IIT-2011
m
m
(b) m n (c)
n
n
(d) Does not exist
(A) f '(0) = 1 and f ''(0) = 2
(B) f '(0) = 1 but f ''(0) is not defined
lim x 1 x 1
(C) f '(0) does not exist
x
JNU – 2003
(c)
(d) 0
2
n
x x ... x n
lim
...........
x 1
x 1
(a) 1
(b)
(D) f is not continuous at x = 0
3.
It is given that
f " x f x , f ' x g x
2
15.
h x f x g x
2
JNU : – 2001
16.
17.
21/ x 1
, then lim y ........
If y 1/ x
x 0
2 1
0
,x0
1
The function f ( x)
is
sin
,x0
x
JNU– 2001
4.
18.
continuous
discontinuous
differentiable such that f'(0) = 1
differentiable such that f'(0) = - 1
Let
Sn
1
2
n
3
2
2
n
3
3
2
n
3
n 1
...
. Then lim S n
n
6.
(a) – 2
(b) – 1
(c) 1
If xy = ex-y, then dy/dx is equal to
(a)
19.
(b)
(c) 0
7.
(d) 1
Which one of the following is false?
MCA : IIT – 2008
(a) A continuous function that is never zero on an interval, never
changes sign on that interval.
(b) The function f(x) = 1 when x is rational and f(x) = 0 when x is
irrational is always continuous.
(c) If the product function h(x) = f(x)g(x) is continuous at x = 0,
then f(x) or g(x) may not be continuous at x = 0
(d) A function f(x) is continuous in [0, 1] such that f(x) [0, 1].
Then there exists a point c in [0, 1] such that f(c) = c
20.
21.
The value of
x
lim sin 1 log 3
x1
3
(a) -/2
(b) /2
If
8.
9.
is equal to
JNU – 2008
(d) None of these
(c) 0
1 h t 2
e dt 1,
h0 h
lim
w.r.t.
10.
22.
The limits
(c) 1
B
Ax sin x
A
JNU – 2006
(b) (1 + log x)-2
(d) None of these
11.
If
z e xy
2
, x = t cos t, y = t sin t then
dz
dt
at
t
2
is
3
(b)
8
3
4
(c)
3
2
(d)
3
8
12.
13.
(B) f ' is bounded on R
1
a
x sin , x 0
The function f ( x)
x
0,
x0
(C) f ' has exactly three zeroes
is differentiable at x = 0 for all a in the interval
(b) 1
(c) A
(d) 0
(c) 2 In (k)
JNU– 2006
(d) None of these
lim 2k 1/ n 1 is equal to
n
n
(a) k2
(b) 2k
DERIVATIVES
x
Let
f x t 1 t 2 5t 6 dt for all x R. Then
0
IIT-2011
(A) f is continuous but not differentiable on R
(D) f is continuous and bounded on R
2.
is
equal to
JNU – 2005
(a) a constant
(b) a function of x
(c) a function of y
(d) a function of both x and y
If f is twice differentiable function such that f”(x) = - f(x) and f'(x)
= g(x). Let h(x) = [f(x)]2 + [g(x)]2. Given that h(5) = 11, then h
(10) is
JNU – 2004
(a) 22
(b) 11
(c) 16
(d) 0
It is given that
f" (x) = - f(x)
f'(x) = g(x)
and
h(x) = (f(x))2 + (g(x))2
If h(4) = 0 then h(8) is equal to
JNU – 1998
(a) 0
(b) 2
(c) 5
(d) None of the above
(a)
JNU - 2007
1.
d3y
dx3
Consider the function f(x) = min {x + 1, |x + 1|}. Then f(x) is
MCA : IIT - 2008
(a) always continuous and differentiable.
(b) always continuous but not differentiable at all points
(c) always continuous but not differentiable at x = - 1
(d) not always continuous.
The function f defined by f (x) x [1 + 1/3 sin (log2)], x 0, f (0) =
0 ([ ] represents the greatest integer function) is
JNU – 2007
(a) continuous and differentiable at origin
(b) not continuous but differentiable
(c) continuous but not differentiable
(d) not continuous and not differentiable
(a) B
23.
y3
(IIT - 2009)
(d) 2
where x and 0 < A < 1 is
1 x 2
is
cos 1
2
1 x
If y2 = ax2 + bx + c, where a, b, c are constants, then
IIT - 2007
(b) 0
is
(d) 2
Then the value of is
(a) – 1
2
h
3
then
JNU – 2005
(a) (1 + log x)-1
(c) log x. (1 – log x)-1
IIT – 2008
2
3
2x
sin 1
2
1 x
Derivative of
is
1
3
1
h 8,
2
JNU-2010
(a) 2
(b) 2
(c) 4
(d) 8
If f is twice differentiable function such that f"(x) = - f(x), f'(x) =
g(x) and h(x) = [f(x)]2 + [g(x)]2, also if h(5) = 11, then h(10) is
equal to
JNU – 2007
(a) 22
(b) 121
(c) 16
(d) None of these
5.
2
n3
. If
equal to
(JNU : - 2009)
(a)
(b)
(c)
(d)
2
and
Let
x x2
f x
x2
if
x 0,
if
x 0.
(b) (-1, )
(d) (1, )
(a) (-, 1]
(c) (1, )
14.
Which one of the following is TRUE?
13
The function
f x
n3
sin nx
n 1
4
IIT – 2006
is
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
15.
16.
JNU – 2006
(a) continuous at x = 0 and differentiable in (0, 2)
(b) discontinuous at x = 0 and non-differentiable in (0, 2)
(c) continuous at x = 0 and non-differentiable in (0, 2)
(d) discontinuous at x = 0 and differentiable in (0,2)
There exists a functions f(x) satisfying f(0) = 1, f(0) = - 1, f(x) > 0
for all x, and
JNU – 2006
(a) f" (x) > 0 for all x
(b) – 1 < f" (x) < 0 for all x
(c) – 2 < f" (x) < - 1 for all x
(d) f" (x) < - for all x
For a real number y, let [y] denote the greatest less than or equal
to y. Function f(x) is given by
17.
18.
19.
tan( [ x ])
1 [ x]2
(a) Then the function f(x) is discontinuous at some x
(b) Then the function f (x) is continuous at all x, but the derivative
f’(x) does not exist for some x
(c) Then for the function f (x), f"(x) exists for all x (d) Then for
the function f(x), f' (x) for all x but the second derivative f" (x)
does not exist for some x
Suppose that f is continuous and differentiable on [a, b].
If f' (x) 0 on [a, c) and f' (x) 0 on (c, b], a < c < b then on [a, b]
IIT-2008
(a) f(x) is never less than f(c)
(b) f(x) is always less than f(c)
(c) f(x) is always less than f(a)
(d) f(x) is always greater than f(b)
If f(x) = a | sin x | + b e(x) + c | x |3 and if f(x) is differentiable at x
= 0, then
JNU – 2005
(a) a = b = c = 0
(b) a = 0, b = 0, c R
(c) b = c = 0, a R
(d) c = 0, a = 0, b R
Let f(x) is a function differentiable at x = c, then
equals
(a) f'(c)
(b) f"(c)
If
lim f x
x c
y f t sin{k x t }dt , then
0
d2y
k2 y
dx 2
21.
(b) y
x y y x c , then
If
(a) 2/c
22.
If
(b) -2/c
2
y x 1 log
2
(c) 2/c
x 1 x
2
(a)
xy 1 x 2 1
(c)
xy 1 /
x
2
2
JNU – 2005
(d) None of these
Let
x 1
f x
2
x 1
3
5
a ,b
4
2
3
5
(c) a , b
4
4
(a)
5.
6.
7.
xy 1 x 2 1
(d)
xy
x
2
1
8.
9.
10.
11.
12.
if x 0,
13.
if x 0.
IIT-2011
(A)
f is differentiable on R
(B)
f has neither a local maximum nor a local minimum in R
(C)
f is bounded on R
(c)
y
x3
4
x2
3
3
If
(b)
y
x3
x2
3
(d)
y
x3
4
x2
3
3
a0
a
a
a
1 2 ... n1 a n 0 , then the
n 1 n n 1
n 1
(a) at least one zero
(b) at most one zero
(c) only 3 zeros
(d) only 2 zeros
The slope of the tangent line to the curve
x = a (t – sin t), y = a(1 – cos t), t R
14.
15.
16.
17.
14
t
2
is
IIT - 2007
(a) – 1
(b) 0
(c) 1
(d)
Let f(x) = x3 – x2 + 1, 0 x 1.
Then the absolute minimum value of f(x) is
IIT - 2007
(a) 14/27
(b) 5/9
(c) 23/27
(d) 1
3
If 8x – y = 15 is a tangent at (2, 1) to the curve y = x + ax2 + b,
then (a, b) is
IIT– 2006
(a) (1, 3)
(b) (-1, 3) (c) (1, - 3) (d) (-1, - 3)
For the function y = 1 – x4, the point x = 0 is a point of IIT - 2005
(a) inflection
(b) minima
(c) maxima
(d) absolute minima
If f(x) = kx – sin x is monotonically increasing, then
JNU – 2005
(a) k > 1
(b) k > - 1 (c) k < 1
(d) k < - 1
The function f(x) = a sin x + (1/3) sin 3x has maximum value at x
= /3. The value of a is
JNU – 2005
(a) 3
(b) 1/3
(c) 2
(d) 1/2
The normal to a given curve is parallel to x-axis if
JNU – 2005
(a)
(D) f is not differentiable at x = 0 but has a local maximum at x
=0
The curve which passes through the point (2, 0) and the slope of
the tangent at any point (x, y) is x2 – 2x for all value of x, is
JNU-2010
(a) y = x3
The height of an open-cylinder of given surface and greatest
volume is equal to
(JNU-2009)
(a) two times the radius of the base
(b) half of the radius of the base
(c) radius of the base
(d) 1/9th of the radius of the base
Let f(x) = ax2 + bx + c; a, b, c R and a 0. Suppose f(x) > 0 for
all x R. Let g(x) = f(x) + f’(x) + f” (x). Then
JNU – 2008
(a) g(x) > 0 for all x R
(b) g(x) < 0 for all x R
(c) g(x) = 0 has real roots
(d) None of these
JNU – 2008
, then dy / dx is
(b)
3
5
a ,b
2
4
3
5
(d) a , b
2
2
(b)
function a0xn + a1xn-1 + a2xn-2 + … + an has in (0, 1)
Which one of the following is TRUE?
2.
(d) f has local maxima at x = 1 and x = 0
If f(x) = ax3 +bx2 + x +1 has a local maximum value 3 at x = -2 ,
then
IIT-2010
at
APPLICATION OF DERIVATIVES
1.
4.
equals
JNU – 2004
1
(b) f has local minima at x = 0 and x = 1
(c) f has a local maximum at x = 1 and a local minimum at x = 0
2
d y
dx 2
f ( x) t (t 1) dt , then
IIT-2010
(a) if has a local maximum at x = 0 and a local minimum at x = 1
(d) k2 f(x)
(c) k f(x)
if
a
equals
JNU – 2005
(a) 0
3.
JNU – 2005
(d) None of these
(c) 1/f(c)
x
20.
JNU – 2006
x2
dx
dx
dy
dy
0 (b)
1 (c)
0 (d)
1
dx
dx
dy
dy
On the interval [0, 1], the function x25 (1 – x)75 takes its maximum
value at the point
JNU – 2004
(a) 1/4
(b) 1/3
(c) 1/2
(d) 0
The ratio of the altitude of the cone of greatest volume which can
be inscribed in a given sphere, to the diameter of the sphere is
JNU – 2004
(a) 1/4
(b) 3/4
(c) 1/3
(d) 2/3
If f(x) = x3 – 2x2 + x + 6, then which one of the following is
correct?
JNU – 2004
(a) f(x) has a maximum at x = 1/3
(b) f(x) has a maximum at x = 1
(c) f(x) has a minimum at x = 1
(d) f(x) has a no maxima or minima
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
18.
19.
20.
21.
22.
2
The length of the subnormal to the parabola y = 4ax at any point
is equal to
JNU – 2004
(a) 2a
(b) 2 2a (c) 2a
(d) a / 2
The radius of a sphere is decreasing at the rate of 0.02 cm per
minute. The rate at which the weight of the sphere is varying
when the radius is 15 cm and the material weighs 0.3 kg/cc is
JNU – 2004
(a) 5 kg/ min
(b) 5.4 kg/min
(c) 6 kg/min
(d) 4.8 kg/min
The critical points of the function f(x) = (x – 2)2/3, (2x + 1) are
…………….
JNU – 2001
(a)
(b)
(c)
(d) None of these
If 2x + 5y = 3
Find maximum value of x3 y4. JNU – 2000
(a)
(b)
(c)
(d) None of these
2x + 3y = 5
and 6x + ky = 15
have an infinite number of solutions?
(a) 7
7.
24.
8.
The maximum value of
1
is equal to x
JNU – 1998
1.
If
IIT-2011
2.
3.
4.
5.
6.
(a)
1 100 500
0 1 100
0 0
1
(c)
50 100 150
0 50 100
0
0
50
(b)
1 50 100
0 1 50
0 0
1
(d)
1 50 1275
0 1
50
0 0
1
Consider the following system of equations
2x + 3y + 4z = 13
5x + 7y + 7z = 27
9x + 13y + 15z = 13
The value of for which the system has infinitely many solutions
is
b
bx
a
b
a
cx
(b) 2
0
then the total
1 2 2
A 2 1 2,
2 2 1
is
IIT – 2008
(b) A – 4I
1
( A 4I )
5
(c)
9.
(JNU - 2009)
(d) None of these
(c) 3
The multiplicative inverse of the matrix
(d)
1
( A 4I )
5
The value of the determinant
1
a2
a
cos(n 1) x cos nx cos(n 1) x is zero if.
sin(n 1) x
10.
sin nx
sin(n 1) x
JNU – 2008
(a) sin x = 0
(b) cos x = 0
(c) a = 0
(d) N.O.T
The area of triangle formed by the vertices (p, q + r), (q, p + r), (r,
p + q) is
JNU– 2007
(a) p + q + r
(b) pq + qr + rp
(c) 0
(d) N.O.T
MATRICES
1 1 1
P 0 1 1 , then P50 equals
0 0 1
c
c
given by
(a) A + 4I
x
25.
ax
If a + b + c 0 and
(a) 1
JNU – 1999
23.
(JNU : MCA - 2009)
(d) 10
(c) 9
number of different values of x is equal to
x
The set of all points where f x
is differentiable, is
1 x
(a) (-,)
(b) (0, )
(c) (-, 0) (0, )
(d) None of these
A rectangular box without a top is to have a given volume R.
What are the dimensions of the box if it is to be made using least
amount of material?
JNU – 1999
(a)
(b)
(c)
(d) None of these
Let f(x) be a continuous function in [a, b] and be differentiable in
(a, b).Suppose f(a) = f(b). Then by Rolle’s theorem, there is at
least one point a in (a, b) such that
JNU – 1998
(a)
(b)
(c)
(d) None of these
(b) 8
5 0 2
A 0 1 0
4 0 1
and I be 3 3 unit matrix, then rank of I
11.
If
12.
– A is
(a) 0
(b) 1
(c) 2
Which of the following is false?
JNU– 2007
(d) 3
JNU– 2007
(a) If A is a square matrix, then Adj A' = (Adj A)'
(b) If I is the identity matrix of order n, then Adj I = I
(c) (A*)-1 = (A-1)*
(d) If A and B are invertible, then AB = BA
13.
The determinant
1
1 i
i
1 i
i
1
i
1
1 i
equals
JNU– 2007
14.
(a) 7 + 4i
(b) 2 – 2i
(c) – 7 – 4i (d) – 2 + 2i
The inverse of the matrix
1
1
1
1
1
IIT-2011
(A) 1
(B) 2
(C) 3
(D) 4
Let P, Q, R be matrices of order 3 5, 5 7 and 7 3,
respectively. The number of scalar additions required to compute
P(QR) is
(IIT – 2009)
(a) 114
(b) 126
(c) 128
(d) 138
For which value of the following system of equations is
inconsistent?
3x + 2y + z = 10
2x + 3y + 2z = 10
x + 2y + z = 10
(IIT – 2009)
(a) 0.98
(b) 1.4
(c) 1.6
(d) 1.8
If A and B are symmetric matrices, which of these are certainly
symmetric?
(JNU : MCA - 2009)
(i) A2 – B2
(ii) (A + B) (A – B)
(iii) ABA
(iv) ABAB
(a) (i) and (iii) only
(b) (i) and (iv) only
(c) (ii) and (iii) only
(d) (ii), (iii) and (iv) only
For what value of k, will the equations
0 0 0 0
1 0 0 0
0 1 0 0 is
0 0 1 0
0 0 0 1
IIT– 2006
1 0 0 0 0
1 1 0 0 0
0
(a) 1 0 1 0
1 0 0 1 0
1 0 0 0 1
15
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
1
1
(b) 1
1
1
1
1
(c) 1
1
1
1
1
(d) 1
1
1
15.
0
0
1
0
0
0 0
0 0
0 0
1 0
0
0
0
0
1
0
1
0
0
1
0
0
0 0 0
1 0 0
0 1 0
0 0 1
0 0 0
(c)
22.
0 1
0 0
0 0
19.
a
23.
24.
(a) a symmetric matrix
(b) a skew-symmetric matrix
(c) a diagonal matrix
(d) None of these
The rank of a null matrix
25.
26.
1
cos nx cos n 1 x cos n 2 x
sin n 1 x
0 0 1
A 2 1 1 , A1
1 1 1
sin n 2 x
27.
and
1
b 2
3
Let
28.
IIT– 2005
29.
1
1
1 c
x a x b
xa
0
xb
xc
xc 0
then x is equal to
JNU– 2004
0
(a) 0
(b) a
(c) b
The value of the determinant
(d) abc
a
cos n 1 x
1
cos n 2 x
sin nx sin n 1 x sin n 2 x
is independent of
(a) n
(b) a
30.
Let
(c) x
Dr b 3 4r 2 416 1
c 7 8
r
4 3
16
is
(a) 0
(c) ab + bc + ca
31.
The value of
IIT– 2006
2
16
then the value of
1
D
k 1
k
JNU– 2001
(b) a + b + c
(d) None of these
1
1
1
a
b
c
3
3
3
a
then 8p is equal to
JNU– 2004
(d) None of these
216 1
2r
a
(b) (I – P)2 = 0
(d) P = 0
(b)
(b) a-1 b-1 c-1
(d) None of these
(a) abc
(c) – a – b – c
If a, b, c are distinct and
has
-1
(a)
1
a2
cos nx
1 0 5
P 1 2 5
1 3 1
13 4 1
15 4 3
10
0 2
1
1 b
JNU– 2004
IIT– 2006
21.
b
1
0
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) more than one but finitely many solutions
Let P be a 2 2 matrix such that P102 = 0. Then
(a) P2 = 0
(c) (I + P)2 = 0
1 a
is equal to zero, then the value of a-1 + b-1 + c-1 is
IIT– 2005
20.
is equal to zero,
0
JNU– 2005
(a) is 0
(b) is 1
(c) does not exist
(d) None of these
Let A be a square matrix of orders n n and k is a scalar, then adj
(kA) is equal to
JNU– 2005
(a) k adj A
(b) kn adj A
(c) kn-1 adj A
(d) kn+1 adj A
If a matrix A is such that 3A3 + 2A2 + 5A + I = 0, then A-1 is equal
to
JNU– 2005
(a) –(3A2 + 2A + 5)
(b) 3A2 + 2A + 5
2
(c) 3A – 2A – 5
(d) None of these
If a, b, c are all different from zero, and
is given by
(a) A2 – 2A
(b) A2 + 2A + 3I
(c) A2 – 2A – I
(d) A – 3I
The system of equations Ax = b, where A
1 1 1
3 2 1
4 3 2
b c
JNU– 2005
JNU– 2006
(a) n
(b) a
(c) x
(d) None of these
The equations 2x + 3y + 5z = 9; 7x + 3y – 2z = 8;
2x + 3y + z = have infinite number of solutions if
JNU– 2006
(a) = 5
(b) = 5
(c) = = 5
(d) None of these
For the matrix
c
(a) a, b, c are in AP
(b) a, b, c are in GP
(c) a, b, c are in HP
(d) no relation between a, b, c
The inverse of a diagonal matrix is
is independent of
18.
b
JNU– 2005
The number of values of for which the system of equations
x + ( + 3) y = 10z
( - 1) x + ( - 2)y = 5z
2x + ( + 4) y = z
has infinitely many solutions, is
IIT– 2006
(a) 1
(b) 2
(c) 3
(d) infinite
sin nx
17.
b
0
1 0
The determinant
a
if
a2
16.
The determinant
(d)
13 4 1
0 2
10
15 4 3
a b
a b b c
0
0 0
0 0
1 0
0 1
0 0
13 10 15
4
4 0
1 2 3
b
c
(a) (a + b + c) (a + b + c2)
(b) a2b2 + b2c2 + c2a2
(c) (a + b + c) (a – b) (b – c) (c – a)
(d) (a + b + c) (a + b) (b + c) (c + a)
13 15 10
0
4 4
1 3 2
16
2
is JNU– 1998
2
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
32.
Let P,M, N be n n matrices such that M and N are nonsingular.
If x is an eigenvector of P corresponding to the eigen value , then
an eigen vector of N-1 MPM-1N corresponding to the eigenvalue
is
(IIT – 2009)
(a) MN-1x
(b) M-1Nx
-1
-1
(c) NM x
(d) N Mx
1
1 2 3
33.
4 5 6 k,
If
then
7 8 9
(a) 4k + 3
34.
Matrices Let
(b) 4k – 3
2
5
6
3
7
8
37.
a b c
(d) k
of unity. The p is
(a) p2
(b) p
(c) Identity matrix
(d) 0
Consider the system of equations P x = 0, where
If for a triangle ABC,
43.
sin3 A + sin3 B + sin3 C is equal to
(a) sin A + sin B + sin C
(b) 3 sin A sin B sin C
(c) sin 3A + sin 3B + sin 3C
(d) sin3 A sin3 B sin3 C
If ABC is not a right triangle, then the value of
c a b
(IIT - 2009)
2 c c
A c c c
8 7 c
38.
Let
3 0 0
A 0 1 1
0 1 1
(a)
(c)
39.
40.
1 0 0
0 2 0
0 0 3
0 0 1
0 1 0
0 0 2
44.
45.
If A = P-1 DP,
47.
IIT – 2008
(b)
(d)
tan B
1
1
1
tan C
1 0 0
0 3 0
0 0 2
0 0 0
0 2 0
0 0 3
The values of a and b for which the following system of linear
equation.
ax + y + 3z = a
2x + by – z = 3
5x + 7y + z = 7
has an infinite number of solutions, are
IIT – 2008
(a) a = 1, b = 1
(b) a = 1, b = 3
(c) a = 2, b = 3
(d) a = 2, b = 1
Let A and B be any arbitrary square matrices of order 3. then AB
and BA have
IIT – 2008
(a) the same eigen values and the same eigen vectors.
(b) the same eigen values but may have different eigen vectors.
(c) different eigen values but the same eigen vectors.
(d) different eigen values and different eigen vectors.
(b) 2
48.
JNU – 2007
is
(c) 3
(d) 0
q y rz
px
If
r z 0 , then the value of
q
JNU – 2007
1
px q y
(b) {1, 2, 8}
(d) None of the above
then the matrix D is equal to
1
p
46.
0 0 1
P 1 1 0 .
1 1 0
1
(a) – 1
(JN - 2009)
and
tan A
The value of k for which the
(IIT - 2009)
(a) 2 and – 2
(b) 2 and – 1
(c) –1 and – 2
(d) 1 and – 1
Let P and Q be two n n nonzero matrices such that P + Q = 0.
Which one of the following statements is NEVER true?
(IIT - 2009)
(a) P is nonsingular
(b) P = QT
(c) P = Q-1
(d) Rank (P)1 Rank (Q)
Fow which three values of c, the given matrix A is not invertible?
b c a 0 , then
42.
0
P
, where is a complex cube root
0
(a) {2, 8, 7}
(c) {0, 2, 7}
has
6 y z 9w 0
system will have a nontrivial solution are
36.
x y z w0
(a) no solution
(b) infinite number of solutions
(c) only one solution
(d) more than one but finite number of solutions.
(IIT – 2009)
1 k 4 4k 2
P 0 k 2 k 2 .
0 k 8 5k
3x z 6w 0
The system of linear equations
IIT – 2008
24
35.
41.
4
9 10 11 12
13 14 15 16
(c) 2k + 1
9x 3y z 0
r
p q r
x y z
is
JNU – 2005
(a) 0
(b) 1
(c) 2
(d) 4 pqr
The value of a for which the system of equations
a3x + (a + 1)3y + (a + 2)3z = 0
ax + (a – 1)y + (a + 2)z = 0
x+y+z=0
has a non-zero solution is
JNU – 2004
(a) 1
(b) 0
(c) – 1
(d) None of these
The equations 3x + y + 2z = 3, 2x – 3y – z = - 3, x + 2y + z = 4
have
JNU – 2002
(a) infinite number of solutions
(b) no solution
(c) a unique solution .
(d) None of these
The only integral root of the equation
2 y
2
3
2
5 y
6
3
4
10 y
0 , is
JNU – 2002
(a) y = 0
(b) y = 1
(c) y = 2
(d) y = 3
If A, B, C are angles of a triangle then the value of
sin 2 A cot A 1
sin 2 B cot B 1 is
sin 2 C cot C 1
(a) 0
a
49.
If A =
and
50.
17
(c)
(b) 1
2
JNU – 2002
b
2
(d) /2
c
2
a 1 b 1 c 1
2
2
2
a 1 b 1 c 1
2
2
a2
b2
c2
B a
b
1
1
c then
1
2
JNU – 2002
(a) A = 4B
(b) A = 2B
(c) A = B
(d) None of these
If A is a 3 3 matrix and let A = 2, then find the value of the
determinant det (adj (adj (adj (A-1))))
JNU – 1999
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
(a)
51.
1
512
(b)
1
1024
(c)
1
128
(d)
1
256
6.
The system of linear equations
x–y+z=0
x+y+z=2
x – 2y + 4z = 5
ex + fy + 2z = 2 is consistent if
53.
JNU - 2007
(a) e – f = 2
(b) e – f = - 2
(c) e + f = 2
(d) e + f = - 2
Find all values of which satisfy the equation
x2
x
x tan 2log sec x / 2
2
2
x2 x2
x
tan 2log sec x
2
2
2
(c)
sin 2
sin
sin 3 0
(d) None of these
sin 2
sin 3
sin
The values are
JNU – 1999
(a)
(b)
(c)
(d) None of these
Find the general solution of the following second-order
F x
F ' x
y'
where F( ) and G( ) given.
(a)
(c)
7.
log x
x
(c)
JNU – 1998
2
JNU – 2008
(log x)
c
2
(log x) 2
(c)
c
4
1
x tan xdx is equal to
(b)
(c)
(d)
( x 1) tan x
xc
2
( x 2 1) tan 1 x x
c
2
( x 2 1) tan 1 x x
c
2
(c)
(c)
5.
(a) - x
(b) - + x
sin ln x
x
1
dx.
JNU-2010
The value of
JNU - 2008
(b)
sin(e x ) c
(d)
2
1
sin(e x ) c
2
is
(b)
(d)
is given by
(a) loglog x
(c) ex
x2
x
cot
2
2
x
x cot
2
1 3
If ( x) cot xdx cot x cot x
3
, then (x) is
2 2
4
2
(D) 2
3.
(a)
2
(b) x
(d) logx
/2
sin x
0
sin x cos x
(b)
3
4
(c)
4
dx
is
(d)
3
2
5.
The area bounded by the curves y = x and x2 = y is
6.
(a) 1/3 (b) 2/3(c) 4/3 (d) 5/3
The area bounded by y2 = 4 – x and y2 = x is
(IIT - 2009)
2
(IIT – 2009)
(JNU - 2009)
JNU - 2007
x2
x
cot
2
2
x
x cot
2
(C) 1
(a) 1 – sin 2
(b) 1 – cos (ln 2)
(c) 1 + cos (ln 2)
(d) 1 + ln 2
The integrating factor of the differential equation
2
2 sin(e x ) c
(B) −1
Calculate
4.
2
(a)
2
n
' ..... f ' equals
n
n
dy
x log x y 2 log x
dx
x
x
x e cos(e )dx is equal to
for all x R. Then
2.
2
2
1
cos(e x ) c
2
x sin x
dx
The value of
1 cos x
2
1 x2
1
' f
n
f x
JNU-2010
(d) None of the above
(a)
Let
(A) − 2
1
2
ln x
ln sin 1
c
2
IIT-2011
(log x)
c
3
(log x 2 ) 2
c
4
JNU – 2008
(b)
(d)
2
(a)
(a)
1 ln x
(b) ln tan
c
2
1 ln x
ln cot
c
2
1
lim f
n x n
dx is equal to
2
1 cos2 1 ln x
DEFINITE INTEGRAL
1.
2
dx
IIT– 2005
G ' x 0
(b)
(d) None of these
x
The value of
1 ln x
(a) ln tan
c
2
G x
INDEFINITE INTEGRAL
4.
(b)
sin 3
y " F " x G " x
3.
x x
x
tan 2log sec x
2 2
2
sin 2
differential equation
2.
(a)
sin
y
1.
x
2
JNU – 1999
52.
sec x 1 dx is
The value of
(a)
16
8
(b)
3 2
(c)
3 2
/2
7.
The value of the integral
(a)
and
8.
(a)
18
(b)
2
0
(d)
16 3
3
tan x
dx
tan x cot x
(c) 0
(d)
IIT – 2008
4
The area of the region bounded by the curves x2 = 2y and y2 = 2x
is
IIT – 2008
JNU - 2008
(c) /2 – x (d) None of these
6
16 2
3
1
3
(b)
2
3
(c)
4
3
(d) 4
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
9.
2
0 sin x cos 2 xdx is equal to
23.
b
JNU – 2008
10.
11.
e
x2
f ( x) g ( x)dx
b] then
(a) 1
(b) 2
(c) 4
(d) 0
The area included between the parabola y2 = 4ax and x
2
= 4ay is equal to
JNU – 2008
(a) 8a2/3
(b) 16a2/3 (c) 4a2/3
(d) None of these
For > 0, the value of the integral
If f(x) and g(x) are continuous in [a, b] and g(x) 0 for all x [a,
b
g ( ) f ( x)dx
(a)
a
b
f ( ) g ( x)dx
(b)
a
b
dx dx
f ( ) g ( x)dx
(c)
0
a
b
equals
IIT – 2007
1
(a)
(b)
2
12.
If
/2
0
2
2
2
(c)
(d)
2
g ( ) f ( x)dx
(d)
24.
13.
The value of integral
4
14.
(b)
2
(c)
The integral
dx is
(d) None of these
Area enclosed by the curves y2 = x and y2 = 2x – 1 lying in the
first quadrant is
IIT – 2005
(a) 1/6
(b) 1/4
(c) 1/2
(d) 1/3
min(sin x, cos x) dx equals
2 2
(a)
26.
27.
(b)
(c) 2 2
(d) 2 2
The area bounded by the curve y = (x + 1)2, its tangent at (1, 4)
and the x-axis is
IIT – 2006
(a) 1/3
(b) 2/3
(c) 1
(d) 4/3
For the integral
e2
e
x
2 e dx
dx
and I 2
, then
1
log x
x
(a) I1 = I2
18.
JNU – 2005
(a) 2
(b) 1
(c) 0
(d) 3
x
The area of the figure bounded by the curves y = e , y = e-x and the
straight line x = 1
JNU – 2005
1
(a) e
e
1
(b) e
e
29.
x | x |dx is
The value of the integral
1
1
(c) e 2 (d) None of these
e
30.
1.5
0
[ x 2 ] dx
(b)
2 2
32.
(d) 3
20.
The value of
I log sin x dx is
(c) log 2
21.
If (n – m) is odd, then
22.
(a) 2n/ (n – m )
(c) 2m / (n2 – m2)
The value of the integral
2
1/ 2
1/ 2
0
(b) 0
(d) - log 2
JNU – 2002
cos mx sin nxdx is
(b) 2n / (m – n )
(d) 0
2
2
1 x
cos x.log e
dx
1 x
(a) 1/4
JNU – 2004
0
(a) – 2 log 2 (b) 2 log 2
2
The value of
sin 3
5 3cos d is equal to
e
1 1 ln x
e
dx is
x2
33.
Let
(b) 1/e
f t
t2
t
(c) e
(d) 0
sin tx
dx, t 0 . Then the value of f'(1) is
x
equal to
JNU – 2006
(a) sin (1) (b) 0
(c) – sin (1) ( d) 2 sin (1)
The area bounded by the curve y = f(x), x-axis and the coordinates
x = 1 and x = b is (b – 1) sin (3b + 4). The f(x) equals
JNU – 2004
(a) (x – 1) cos (3x – 4)
(b) sin (3x + 4) + 3(x – 1) cos (3x + 4)
(c) sin (3x + 4)
(d) (3x + 4) sin (x – 1) + cos (3x + 4)
Consider the following inequality
1
k
k
1
1
3
x 1
dx 4
x
2
The value of k for which are above inequality is satisfied, lie in
the interval
JNU – 2004
(a) (0, 4)
(b) (8, 12)
(c) (32, 48)
(d) (-, 0)
is
(c) 1
(d) 5
IIT – 2005
JNU – 2004
2 2
(c) 1 2
(c) 3
JNU– 2006
(a) infinity (b) 2/3
(c) 1/3
(d) None of these
The area bounded by the curve y = f(x), the x-axis and the
ordinates x = 1 and x = b is (b – 1) sin (3b + 4). Then f(x) is
JNU– 2006
(a) (x – 1) cos (3x + 4)
(b) sin (3x + 4)
(c) sin (3x + 4) + 3(x – 1) cos (3x + 4)
(d) None of these
(a) 1
31.
(a)
The value of
1
[.] denote the greatest integer function, then the value of
tan n x dx is equal to (-), the least positive
17.
19.
(b) 5/2
28.
JNU – 2005
(d) None of these
(b) 2I1 = I2 (c) I1 2I2
1
JNU– 2006
(a) 3/2
(b) 1/3 log 2 (c) 1/2 log 2 (d) 1/2
I1
If
2 2
value n is equal to
JNU – 2007
16.
IIT– 2007
0
3x
0 x 1 x 2 is given by
(a) log 2
(d) f(a)/2
0
1
15.
(c) f(a)
2
JNU– 2006
(a)
is equal to
1 e f ( x)
25.
cot x tan x
0
dx
JNU – 2008
(b) a/2
(d) 1/4
cot x
for all x (a, b)
a
JNU – 2007
/2
for some x (a, b)
Let f(x) be a continuous function such that f(a – x) + f(x) = 0 for
(a) a
(c) 1/3
for all x (a, b)
all x [0, a]. Then 0
given by
(b) 1/2
for exactly one x (a, b)
a
dx
1
tan 1 , then the value of is
5 3sin x
2
(a) 1
is (IIT - 2009)
a
JNU – 2001
(d) None of these
19
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
/4
34.
The value of
0
JNU – 2006
sin x cos x
dx is
25 144sin 2 x
JNU – 2002
(a) (1/78) loge (1/5)
(c) (1/78) loge5
JNU-2010
(a) infinite (b) two (c) 3001
i
13
The value of the sum
n
n 1
3.
4.
5.
Find z4, if
13.
(d) 4001
i n 1 , i 1
14.
is
JNU-2010
(b) i – 1
(a) i
x y
k a 2 b 2 , where k is equal to
a b
(b) (1/156) loge (5)
(d) None of these
COMPLEX NUMBERS
1.
The number of roots of x2.1 + x3.01 + x4.00 = 1 is
2.
12.
(c) 1 – i
(d) 0
z 1 3i, i 1.
4 cos
(b)
4
2
4 cos
i sin
3
3
(c)
4
4
16 cos
i sin
3
3
(d)
4
4
16 cos
i sin
3
3
15.
JNU-2010
4
4
i sin
3
3
(a)
16.
17.
18.
The region of the arg and plane defined by |z - i| + |z + i| 4 is
(JNU – 2009)
(a) interior of an ellipse
(b) exterior of a circle
(c) interior and boundary of an ellipse
(d) None of the above
Centre of the arc represented by arg
z 3i
4
z 2i 4
is
(c)
6.
1
5i 5
2
1
9i 5
2
(b)
(d)
1
5i 5
2
1
9i 5
2
If is a complex cube root of unity, then the value of the
expression
cos{(1 )(1 2 ) .... (10 )(10 2 )}
900
The value of real such that
20.
(a) n
(c) n /3
Solve z5 = 1, for z
7.
The value of
2k
2k
i cos
sin
11
11
k 1
(c) 1
3/2
(d)
10
8.
(b) 0
(c) – i
21.
22.
23.
The value of
If x r cos
sin r , then x1x2x3 …. to
r
2
2
1 2i
1 (1 i ) 2
24.
9.
(b) – 2
(c) – 1
The number of solutions to the equation
(d) 0
z 2 z 0 is
10.
(b) 2
(c) 3
25.
real number is
JNU – 2006
11.
is
are respectively.
JNU – 2000
z 5i
1
z 5i
lie on
If
1
x iy
, 2n , n I ,
1 cos 2i sin
then
maximum value of x is
1 i cos
26.
is a
1 2i cos
(a) 2n
(b) (2n + 1)
(c) 2n /2
(d) None of these
If | z2 – 1 | = | z |2 + 1, then z lies on a / an
4
JNU – 1998
(d) 4
The real value of for which the expression
JNU – 2004
the line -
JNU – 2006
(a) 1
1
(1 i) 4 1
i
The complex numbers z = x + iy satisfying
JNU – 2008
(a) – 3
(b) n /2
(d) n /6
JNU – 2002
(a) 12
(b) – 12
(c) 16
(d) – 16
If f(x) = (cos x + i sinx) (cos 3x + i sin 3x) …. Cos ((2n – 1)x + i
sin (2n – 1)x),
JNU – 2002
(a) n2 f(x)
(b) – n4 f(x)
(c) – n2 f(x)
(d) n4 f(x)
The modulus and the principal argument of the complex number
is
(d) i
is purely imaginary is
JNU – 2004
JNU – 2008
(a) – 1
3 2i sin
1 2i sin
(a) z = e2in, n = 0, 1, 2, ….
(b) z = e2in/5, n = 0, 1, 2,…
(c) z = ein/5, n = 0, 1, 2, ….
(d) z = e5in, n = 0, 1, 2, ….
(JNU – 2009)
(b) 0
Lie on a circle of radius 3
Form an equilateral triangle
None of these
19.
is
(a) –1
JNU – 2006
(a) 0
(b) 2
(c) 4
(d) None of these
If the area of a triangle on the complex plane formed by the point
z, z + iz and iz is 50, then | z | is
JNU – 2006
(a) 1
(b) 5
(c) 10
(d) 15
If z = ( + 3) + i(5 - 2)1/2, then the locus of z is a / an
JNU – 2006
(a) ellipse (b) circle
(c) plane
(d) None of these
If the complex numbers sin x + i cos 2x and cos x – i sin 2x are
conjugate to each other, then x is equal to
JNU - 2005
(a) n
(b) (n + 1/2)
(c) 0
(d) None of these
For any complex number z, the minimum value of |z| + |z – 1| is
JNU - 2005
(a) 1
(b) 0
(c) 1/2
(d) 3/2
If the complex numbers z1, z2, z3 are in AP, then they lie on a/an
JNU - 2005
(a) circle
(b) parabola
(c) line
(d) ellipse
the cube roots of unity
JNU – 2002
(a) are collinear
(b)
(c)
(d)
(JNU – 2009)
(a)
(a) straight line
(b) circle
(c) ellipse
(d) None of these
If z = z + iy, z1/3 = a – ib, a ba, b 0, then
27.
20
(JNU – 2009)
(a) 1
(b) 2
(c) 1/2
(d) 1/3
If z = x + iy, z1/3 = a – ib, a ab, b 0, then bx – ay = kab(a2 –
b2) where k is equal to
JNU – 2008
(a) 1
(b) 2
(c) 3
(d) 4
If 1, , 2, …, n-1 are the nth roots of unity, then (2 - ) (2 – 2)
… (2 – n-1) equals to
JNU – 2008
(a) 2n – 1
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
(b) C1 + C2 + … + Cn
(c) [2n+1C0 + 2n+1C1 + … + 2n+1Cn]1/2 - 1
(d) None of these
n
28.
n
n
4.
tan i (sin( / 2) cos( / 2))
If
is purely imaginary,
1 2i sin( / 2)
then is not given by
(a) n + /4
(b) n - /4
(c) 2n
(d) 2n + /4
Assume that either |z| = 1 or || = 1 and z 1 , where z, are
complex numbers and z is the conjugate of z. The value of
z
1 z
IIT-2010
(a) (7, 8, 5)
(c) (5, 8, 7)
5.
(a) 2
(b) 3
(c)
If eix = cos x + i sin x and
3/ 2
e i / 4
e i /3
1
e 2 i /3
e 2 i /3
e 2 i
5 a b
(d) None of these
VECTORS
7.
8.
9.
a b c . b c a c a b equals
IIT-2011
(b)
a . b c
(d)
6 a . b c
If
11.
12.
13.
2.5 (D) 5
2
is equal to
14.
1 a b
2
(c) 1 a b
2
2
(b)
a b 1
(d) a b
and
c 2 i j 2 k is.
(a)
8
3
(b)
17
3
(c)
IIT – 2008
11
3
(d) 1
The volume of the tetrahedron whose vertices are the points with
position vectors i – 6j + 10k, - i -3j + 7k, 5i – j + 1k and 7i – 4j +
7k is 11 cubic units if the value of is
JNU – 2008
(a) – 1 (b) 1
(c) – 7
(d) None of these
Let a, b and c be three non-zero vectors such that a + b + c = 0 and
|a| =3, |b| = 5 and |c| = 7. Then an angle between a and b is
JNU – 2008
(a) 15
(b) 30
(c) 45
(d) 60
The area of the parallelogram with sides
IIT – 2007
6
(a)
IIT-2010
(a)
is
3i j
IIT-2011
a and b , a b
| a b |
is
(IIT - 2009)
(a) –40
(b) –24
(c) 24 (d) 40
The vectors a = i + j + mk, b = i + j + (m + 1)k and c = i – j + mk
are coplanar if m is equal to
(JNU - 2009)
(a) 1
(b) 4
(c) 3
(d) None of these
Projection of a + b in the direction of c where
(d)
x i j k and y i j
The area of the parallelogram in R2 whose diagonals are
For any two unit vectors
a b
a b c 0, | a | 3, | b | 5 and | c | 8 then a c
i 3 j is
(b)
3.
are three vectors in R3, then
(B) 5 (C)
a i 2 j3k, b 2 i j3k
10.
(A) 2.5
(d) 12
a ( a b ) b is
JNU – 2000
and
be two non-collinear and non-orthogonal unit
| a b |
(c)
where n is a positive integer and
4 a . b c
(c) 13
a b
(a)
n
12 2
b
and
x 1, y 2
If the number (z – 1) / (z + 1) is purely imaginary, then
JNU – 2006
(a) |z| = 1
(b) |z| > 1
(c) |z| < 1
(d) |z| > 2
Which of the following is correct?
JNU - 2005
(a) 1 + i > 2 – i
(b) 2 + i > 1 + i
(c) 2 – i > 1 + i
(d) None of these
The imaginary part of tan-1 (5i/3) is
JNU - 2005
(a) 0
(b)
(c) log 2
(d) log 4
Find the modulus and argument of
and
a
Let
(2k + 1).
2.
a5 b
(IIT - 2009)
x 2, y 2
(c)
then the
vectors. Then
(c)
(a) 0
(b)
(b)
a, b, c
4
(IIT - 2009)
6.
x 1, y 2
If
is
13 2
(a)
(d) None of these
(a)
1 cos i sin
1 cos i sin
1.
| a | 1, | b | 1 and the angle between a, b is
If
JNU – 2007
34.
(b) (8, 7, 5)
(d) (8, 5, 7)
e i /3
33.
PQ 48 , then Q is
that
is
x iy e i / 4
32.
If Q is a point on L in the first octant such
area of the parallelogram with two adjacent ides as
1
32.
2
JNU – 2007
30.
i j k .
the vector
JNU – 2008
29.
Let P be the point (3, 4, 1). Let L be the line through P parallel to
Let
(b)
2 3
(c)
3 2
(d) 6
Then the volume of the parallelepiped with sides x, y and z is
IIT – 2007
(a) 1 + + 2
(b) 1 + – 2
(c) 1 – + 2
(d) 2 + – 1
If a = i + j + k, a.b = 1, a × b = j – k, then b is equal to
JNU – 2007
(a) 2i (b) i
(c) 2i – j
(d) 2i – k
15.
The value of p such that the unit vectors
2
b
a
2 i p j k
5 p2
&
i 2 j 3 k
(a) 2/5
21
x i j k , y i k and z i j .
JNU – 2007
are orthogonal is
14
(b) 5/2
(c) 3/7
(d) 2/7
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
16.
is equal to
U V W
(a) U .W V U .V W
(c) V .W U U .W V
IIT – 2006
(b)
U .W V V .W U
U .W V V .W U
(d)
The value of k for which the points A(1, 0, 3), B(-1, 3, 4), C(1, 2,
1) and D(k, 2, 5) are coplanar is
JNU -2006
(a) 1
(b) 2
(c) 0
(d) – 1
18.
The value of
[ a b , b c , c a ] is
(a) 0
(b) 1
19.
27.
28.
(c) 2
JNU -2006
(d) 3
A = (1, 1, 1) and C = (0, 1, -1) are given vectors, then a
If
vectors
20.
29.
4 m n, m, n being unit vectors, inclined at an angle 45 is
IIT – 2005
2
(b)
3
3 2
(a)
21.
a, b
Let
3
(c)
(d)
2
30.
b2
b3
c1
c2
c3
(b) 1
If a and b are two unit vectors, then the vector (a + b) (a b) is
parallel to the vector
JNU – 2008
(a) a – b
(b) a + b
(c) 2a – b
(d) 2a + b
Let u, v R3, v ≠ 0. Which of the following is FALSE?
IIT – 2007
u
(a)
b c
,
b
c]
v
is the length of the projection of u along v
v
q
and
[a
b
c]
(d)
31.
a b
. Then the value of the expression
[a
b
c]
a b . p b c . q c a . r
is equal to
32.
JNU – 2006
a 1,1, 0 and b 0,1,1 is
1
1
1
is equal to
1 a 1 b 1 c
(b) 2
(c) 1
33.
34.
(d) 0
a, b and a 2 b are collinear then the value of is
If
25.
(IIT - 2009)
(a) 0
(b) 1
(c) 2
(d) 3
Let u and v be two non zero parallel vectors with
w i j 2 k . Then (u v) w = 0 if
IIT – 2008
If
For
(b)
a b
(c)
JNU -2006
orthogonal to u
(b) u and v are orthogonal to each other but not orthogonal to w.
a i b j c k where a + b = 36.
2c
(d) v is orthogonal to u but not orthogonal to w.
Two sides of a triangle are formed by the vectors a = 3i + 6j – 2k
and b = 4i – j + 3k. One of the angle of the triangle is given by
JNU – 2008
22
2 a b (d) 2 a b
a , b and c to be unit vectors satisfying
b
a a c , the angles between a
2
u a i b j c k , a + b = 2c and v is any vector
is parallel to the vector
a b
35.
a and b are two unit vectors, then the vector a b
(a)
(c) u and v both are of the form
The word done by the force p 3i 2 j 4k acting on a
particle, if the particle is displaced from A(8, - 2, - 3) to B(-2, 0, 6)
along the line segment AB, is
IIT – 2006
(a) 0
(b) 2
(c) 3.5
(d) 4.2
a b
24.
If two forces at a given point, the resultant of these forces can
never have
JNU – 2007
(a) The magnitude of either of these forces
(b) The direction of either of these forces
(c) a magnitude that is less than that of either of these forces
(d) a magnitude that is greater than the algebraic sum of these
forces
Let a = i + 2j + k, b = i – j + k, c = i + j – k. A vector in the plane
JNU – 2006
(a) 3
JNU – 2006
(a) one
(b) two
(c) three
(d) None of these
If the vectors (a, 1, i), (1, b, 1) and (1, 1, c) (a b c 1) are
coplanar, then
of a and b whose projection on c is 1/ 3 , is
JNU – 2007
(a) 4i – j + 4k
(b) 2i + j – 2k
(c) 3i + j – 3k
(d) 4i + j – 4k
(a) 0
(b) 1
(c) 2
(d) 3
The number of vectors of unit length perpendicular to the vectors
1
|| u v || 2 || u v || 2
2
|| u v ||2 || u v ||2 2 u ||2 || v ||2
(c) u . v
c a
[a
r
JNU – 2008
is equal to
1 2
(a1 a22 a32 )(b12 b22 c32 )
4
3 2
(a1 a22 a32 )(b12 b22 c32 )(c12 c22 c32 )
4
c are three non-coplanar vectors, and let p, q
and
p
26.
b1
2
(b) If u.w = v.w for all w R3 then u = v
(a)
a3
r , be the vectors defined by the relations
and
23.
a2
2 3
22.
a1
(d)
2 m n and
Area of a parallelogram where diagonals are
If d = (a b) + (b c) + (c a) and [a b c] = 1/8, then + +
is equal to
JNU – 2008
(a) (a + b + c)
(b) (a b c)
(c) (a b c)
(d) None of the above
Let a = a1i + a2j + a3k, b = b1i + b2j + b3k and c = c1i + c2j + c3k be
three non-zero vectors such that c is a unit vector perpendicular to
both a and b, if the angle between a and b is p/6. then
(c)
JNU -2006
(b) (2/3, 5/3, 2/3)
(d) None of these
3
(d) None of the above
(a) 0
B satisfying A B C and A . B 3 is
(a) (5/3, 2/3, 2/3)
(c) (2/3, 2/3, 5/3)
cos 1
15
2
3
cos 1
(c)
(b)
75
17.
7
cos 1
(a)
and
b , a and c ,
respectively, are
IIT – 2005
(a) 90, 45
(b) 60, 90
(c) 90, 60
(d) 45, 90
The temperature T at a surface is given by T = x2 + y2 – z. In
which direction a mosquito at the point (4, 4, 2) on the surface
will fly so that it cools fastest?
IIT – 2005
(a) 8i + 3j – k
(b) -8i – 8j + k
(c) i – j + 2k
(d) i + j – k
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
x2 y2 z6
is
2
3
6
THREE DIMENSIONAL GEOMETRY
1.
The equation of the plane containing the line
x 1 y 3 z 2
and the point (0, 7, -7) is
3
2
1
2.
(JNU - 2009)
(a) x + y + z = 1
(b) x + y + z = 2
(c) x + y + z = 0
(d) None of the above
The point on the sphere x2 + y2 + z2 = 1 farthest from the point (1,
-2, 1) is
IIT – 2007
(a)
(c)
3.
4.
1 2 1
,
,
6 6 6
(b)
1 2 1
,
,
6 6 6
(d)
1
cos 1
5
(c)
1
cos 1
15
x2 + y2 + z2 = 1 and
5.
6.
cos 1
1
(d) cos
x2 y 3
8.
2
1
1
(b)
(i k )
2
(i k )
(d) None of the above
The maximum value of f(x, y, z) = xyz along all points lying on
the intersection of the planes x + y + z = 40 and z = x + y is
IIT – 2008
(a) 4000
(b) 3000
(c) 2000
(d) 1000
The volume of the tetrahdedron included between the plane 3x +
4y – 5z = 60 and the coordinate planes in cubic units is
JNU – 2007
(a) 60
(b) 600
(c) 720
(d) None of these
Which of the following is a unit normal to the surface z = xy at
P(2, - 1, -1) ?
IIT – 2006
9.
IIT – 2007
1
5
10.
3
15
z2 4
(i k )
2
(a)
i2jk
(b)
6
(c)
2
1
(a)
(c)
1 2 1
,
,
6 6 6
(b)
The spheres
7.
1 2 1
,
,
6 6 6
Let , 0 be the angle between the planes
x – y + z = 3 and 2x – z = 4.
The value of is
(a)
(JNU – 2009)
(a) 6 units (b) 5 units (c) 8 units (d) 7 units
A unit vector in XZ – plane making angles /4 and /3
respectively with u = 2i + 2j – k and v = j – k is
(JNU–2009)
intersect at an
11.
i 2 j k
(d)
i2j k
i 2 j k
6
If ax + hy + gz = 0, hx + by + fz = 0, gx + fy + cz = 0, then
JNU – 2007
angle
IIT – 2007
(a) 0
(b) /6
(c) /4(d) /3
If a line makes angles , , , with four diagonals of a cube, then
cos2 + cos2 + cos2 + cos2 is equal to
JNU – 2007
(a) 1/3 (b) 2/3(c) 4/3 (d) 8/3
The distance of the point (1, 0, - 3) from the plane x – y – z = 9
measured parallel to the line
(a)
x2
y2
z2
2
2
bc f
ca g
ab h 2
(b) (bc – f2) (ca – g2) (ab – h2) = (fg – ch) (gh – af) (hf – bg)
(c) (bc – f2) (ca – g2) (ab – h2) = (fg + ch) (gh + af) (hf + bg)
(d) (bc + f2) (ca + g2) (ab + h2) = (fg – ch) (gh – af) (hf – bg)
IIT-JNU QUESTIONS
SETS & RELATIONS
1
2
11
C
21
D
12
C
22
A
3
B
13
B
23
D
1
D
11
B
21
B
2
C
12
A
22
B
3
A
13
C
23
A
4
D
14
D
24
C
5
A
15
D
25
A
6
B
16
B
7
A
17
D
8
C
18
D
9
C
19
D
10
C
20
B
8
C
18
B
28
A
9
A
19
B
29
D
10
B
20
B
30
B
THEORY OF EQUATIONS
1
C
11
B
1
B
11
A
21
D
1
B
11
D
21
B
1
B
4
B
14
C
24
D
5
C
15
C
25
A
6
D
16
B
26
D
7
B
17
D
27
C
SEQUENCE & SERIES
4
5
6
7
8
9
10
C
D
C
B
A
C
C
14
15
B
D
BINOMIAL
2
3
4
5
6
7
8
9
10
D
D
B
B
D
B
A
B
A
12
13
14
15
16
17
18
19
20
C
A
C
A
C
C
B
B
B
22
23
D
D
EXPONENTIAL & LOGARITHMIC SERIES
1
2
3
4
C
B
B
B
PERMUTATION & COMBINATIONS
2
3
4
5
6
7
8
9
10
A
A
C
C
B
A
C
D
B
12
13
14
15
16
17
18
19
20
D
B
B
B
B
A
C
B
A
22
23
24
B
C
PROBABILITY
2
3
4
5
6
7
8
9
10
B
D
B
B
C
A
A
A
A
2
D
12
B
3
D
13
C
23
11
B
21
A
31
A
41
D
12
C
22
A
32
B
42
D
1
B
11
A
21
B
31
B
41
C
51
C
61
B
2
A
12
B
22
C
32
C
42
D
52
A
62
C
1
A
11
A
21
C
31
D
41
B
51
C
61
D
2
A
12
B
22
B
32
C
42
B
52
C
62
A
1
2
13
D
23
D
33
C
43
A
14
A
24
C
34
C
44
C
15
C
25
B
35
D
45
A
16
A
26
B
36
C
46
C
17
C
27
B
37
D
47
B
18
B
28
B
38
D
TRIGONOMETRY
3
4
5
6
7
8
B
D
C
B
B
C
13
14
15
16
17
18
A
A
D
B
N
B
23
24
25
26
27
28
A
B
C
A
D
C
33
34
35
36
37
38
C
D
B
A
D
A
43
44
45
46
47
48
C
D
A
C
B
A
53
54
55
56
57
58
D
A
D
C
B
B
63
64
D
A
TWO-DIMENSIONAL GEOMETRY
3
4
5
6
7
8
C
A
C
B
D
A
13
14
15
16
17
18
D
A
C
A
B
D
23
24
25
26
27
28
A
C
D
B
D
C
33
34
35
36
37
38
B
B
C
B
A
C
43
44
45
46
47
48
A
D
D
C
C
B
53
54
55
56
57
58
C
A
A
A
D
C
63
64
A
A
FUNCTIONS
3
4
5
6
7
8
19
A
29
A
39
D
20
B
30
C
40
C
9
A
19
A
29
D
39
N
49
C
59
A
10
C
20
C
30
D
40
A
50
B
60
D
9
B
19
B
29
D
39
C
49
C
59
D
10
D
20
A
30
A
40
D
50
C
60
D
9
10
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
INFOMATHS
D
11
A
B
12
A
A
13
D
1
D
11
D
21
B
2
D
12
C
22
D
3
C
13
A
23
A
1
C
11
C
21
C
2
C
12
C
22
A
3
D
13
C
1
C
2
D
1
D
11
C
21
-
2
C
12
A
22
A
1
D
11
C
21
B
31
C
41
C
51
4
D
14
A
DERIVATIVES
5
6
7
C
D
B
15
16
17
A
C
A
D
18
D
C
19
C
D
20
C
8
C
18
A
9
C
19
B
10
B
20
A
8
B
18
B
9
A
19
D
10
D
20
C
DIFFERENTIATION
4
5
6
7
8
B
A
B
D
C
APPLICATION OF DERIVATIVES
3
4
5
6
7
8
A
A
C
A
A
C
13
14
15
16
17
18
C
C
A
D
A
A
23
24
25
MATRICES
3
4
5
6
7
8
B
B
A
C
C
13
14
15
16
17
18
D
C
B
A
D
C
23
24
25
26
27
28
D
C
C
D
D
A
33
34
35
36
37
38
D
C
A
D
C
D
43
44
45
46
47
48
B
C
C
C
B
A
53
3
B
2
D
12
D
22
B
32
C
42
B
52
C
D
A
A
14
15
16
17
D
C
A
B
LIMITS & CONTINUITY
4
5
6
7
D
B
B
C
14
15
16
17
D
B
9
C
10
A
9
C
19
B
10
D
20
-
9
A
19
C
29
A
39
B
49
A
A
-
-
INDEFINITE INTEGRAL
4
5
6
7
C
D
B
A
DEFINITE INTEGRAL
1
2
3
4
5
6
7
8
9
10
D
B
D
C
A
C
D
C
D
B
11
12
13
14
15
16
17
18
19
20
A
B
A
D
D
A
C
C
A
D
21
22
23
24
25
26
27
28
29
30
A
B
C
B
B
B
A
D
C
C
31
32
33
34
B
A
B
COMPLEX NUMBERS (OLD QUESTIONS)
1
2
3
4
5
6
7
8
9
10
D
B
C
A
D
B
D
C
D
C
11
12
13
14
15
16
17
18
19
20
A
C
C
B
D
A
C
C
C
B
21
22
23
24
25
26
27
28
29
30
C
B
C
D
A
C
D
D
31
32
33
34
A
D
C
VECTORS (OLD QUESTIONS)
1
2
3
4
5
6
7
8
9
10
C
B
C
A
A
A
B
D
B
B
11
12
13
14
15
16
17
18
19
20
D
A
C
B
B
B
D
A
A
A
21
22
23
24
25
26
27
28
29
30
C
C
A
D
C
A
C
31
32
33
34
35
36
D
A
B
A
C
B
THREE DIMENSIONAL GEOMETRY (OLD QUESTIONS)
1
2
3
4
5
6
7
8
9
10
C
C
A
C
D
D
B
C
B
A
11
B
1
D
10
C
20
A
30
A
40
B
50
D
24
2
B
3
D
INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS
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