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INFOMATHSWORK SHEET(IIT-JNU QUESTIONS) 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 IIT-2011(A) (Q\ N) × (Q\ N) (B) [(Q\ N)×Q] U [Q×(Q\ N)]

(C) [(N×Q) \ (Q×N)] U[(Q×N) \ (N×Q)] (D) (Q×N) \ (N×Q)

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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)

11. 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

12. 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

13. 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

14. 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

JNU – 2003(a) 1400 (b) 6000 (c) 3300 (d) 4000

15. The expressions A (B C) is the same as JNU – 2001

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

16. 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 is at most (IIT – 2009)

(a) (b) n – 1 (c) 2n (d) n2

17. 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

18. 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)

(a) (b) (c) (d)

19. 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. 20. 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

21. 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

22. 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

23. 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

24. 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

25. 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?

1 INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS

INFOMATHSIIT – 2008

(a) P(A B) ≠ P(A) P(B) (b) P(A B) = P(A) P(B) (c) {} P(A)(d) P(A)

THEORY OF EQUATIONS 1. The only value of x satisfying the equation

, where x R is

(MCA : JNU - 2009)(a) 4/35 (b) –4/35 (c) 16/3 (d) –16/3

2. The value of a for which the quadratic equation 3x2 + 2(1 + a2)x + (a2 – 3a + 2) = 0 possesses roots of opposite sign lies in

(MCA : JNU – 2008)(a) (-, 1) (b) (-, 0) (c) (1, 2) (d) (1.5, 2)

3. If x is real and k = (x2 – x + 1) / (x2 + x + 1), then (MCA : JNU – 2008)

(a) (1/3) k 3 (b) k 5 (c) k 0 (d) None of the above

4. The quadratic equation whose roots are reciprocal of the roots of the equation x3 – 3x + 2 = 0 is

(JNU : MCA – 2007)(a) 3x2 – 2x + 1 = 0 (b) 2x2 – x – 1 = 0 (c) x2 – 3x + 2 = 0 (d) None of these

5. The roots of the equation 6x3/4 = 7x1/4 – 2x-1/4 are (JNU : MCA – 2007)

(a) 4/9 and 1/9 (b) 9/4 and 1/4 (c) 4/9 and 1/4 (d) None of these

6. If sin and cos are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation

(JNU : MCA – 2006)(a) a2 + b2 + 2ac = 0 (b) a2 – b2 + 2ac = 0 (c) a2 + c2 + 2ab = 0 (d) a2 – b2 – 2ac = 0

7. If then x = (JNU : MCA – 2006)

(a) 4 (b) 2 (c) 3.14 (d) None of these8. The number of real solutions of sin (ex) = 5x + 5-x is

(JNU : MCA solved – 2006)(a) infinite (b) 5 (c) 0 (d) None of above

9. Solution of the equation esinx – e-sin x = 4 is JNU – 2004

(a) (b)

(c) (d)

10. If and are the root of 4x2 + 3x + 7 = 0, then the value of (1/) + (1/) is

JNU – 2002 (a) -3/4 (b) -3/7 (c) 3/7 (d) 4/7

11. If the roots of x2 – bx + c = 0 are two consecutive integers, then b2 – 4c is

JNU – 2002 (a) 0 (b) 1 (c) 2 (d) None of these

12. If 1, a1, a2, …., an-1 are the roots of unity, then (1 – a1) (1 – a2) ...... (1 – an-1) equals ……

JNU – 2001 (a) n (b) n2

(c) n – 1 (d) None of these13. If a2 + b2 + c2 = 1, then ab + bc + ca lies in the

interval JNU – 1999

(a) (b)

(c) (d)

14. Let f(x) = 2x3 – x2 + 2x – 5. Consider the following statements about the roots of

f(x) = 0

P. At least one root is positive Q. At least one root is negative R. There is a root between x = 1 and x = 2 Which one of the following is TRUE?

(IIT : MCA – 2007)(a) P, Q and R are valid statements (b) P and Q are valid statements but R is NOT a

valid statement(c) P and R are valid statements but Q is NOT a

valid statement (d) P is a valid statement but Q and R NOT valid

statements15. The number of solutions of

is

(JNU : MCA – 2007)(a) zero (b) four (c) two (d) infinite

16. The number of irrational solutions of the equation is

(JNU : MCA – 2007)(a) 0 (b) 2 (c) 4 (d) indefinite

17. The set of real x such that is

(JNU : MCA – 2006)(a) (- , - 1) (b) (-, 0) (c) (-, ) (d) None of above

18. The equation 3x – 1 + 5x – 1 = 34 has (JNU : MCA – 2006)

(a) no solution (b) one solution (c) two solutions (d) more than two solutions

19. S is defined as S =

Find the

value of x for which S is minimum JNU – 2006

(a) 1/2 (b) 1/3 (c) 2/3 (d) 78/80 20. The number of solutions of the equation 5x – 5-x =

log10 25, (x R) is JNU – 2005

(a) 0 (b) 1 (c) 2 (d) infinitely many 21. If every pair from among the equations x2 + px + qr

= 0, x2 + qx + rp = 0 has a common root, then the sum of the three common root is

JNU – 2005 (a) 2(p + q + r) (b) p + q + r (c) –(p + q + r) (d) pqr

22. If a and b ( 0) are the roots of the quadratic equation x2 + ax + b = 0, then the least value of x2 + ax + b (x R) is

JNU – 2005 (a) 9/4 (b) -9/4 (c) -1/4 (d) 1/4


INFOMATHS23. The number of real roots of the equation ex-1 + x – 2

= 0 is JNU – 2005

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

24. If , then

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 - n1

n-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 these30. 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)

SEQUENCE & SERIES 1. 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

2. The greatest value of the positive integer n so that the sum to n

terms of the series is less than

is JNU-2010(a) 5 (b) 7 (c) 8 (d) 10

3. For – 1 x 1, the infinite power series

converges to JNU-2010

(a) ex (b) sin x (c) ln x (d) ln (1 + x)4. 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

5. The harmonic mean of the roots of the equation is

(JNU–2009)(a) 2 (b) 6 (c) 8 (d) 4

6. For 0 < a < /2, if

then

(JNU–2008)(a) xyz = xz + y (b) x + y + z + xyz = 0 (c) xyz = xy + z (d) xy2 + x2y = z

7. Let f(1) = 1 and f(n) = Then is equal

to (JNU–2008)(a) 3m – 1 – 1 (b) 3m – 1 (c) 3m – 1 (d) None of the above

8. 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

9. If the sum of m terms to the sum of n terms in an AP is m 2 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

10. If A1, A2 be two AMs and G1, G2 be two GMs between a and B

then is equal to JNU – 2005

(a) (b) (c) (d)

11. If pth, qth and rth terms of a GP are x, y, z respectively, then xq-r

yr-p zp-q is equal to JNU– 2005(a) 0 (b) 1 (c) – 1 (d) None of these

12. 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

13. 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

14. 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

15. If , the value of n is

JNU– 2004(a) 49 (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

(a) (b) (c) (d)

7. The term independent of x in the expansion of (2x2 – 1/x)12 is JNU– 2007

(a) 7910 (b) 7920 (c) 7930 (d) 79008. The remainder obtained on dividing 21680 by 1763 is

IIT– 2006(a) 1 (b) 3 (c) 13 (d) 31

9. The inequality n! > 2n-1 is true for JNU– 2006

(a) all n N (b) n > 2 (c) n > 1 (d) n N 10. 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) -500011. If n is even and nC0 < nC1 < nC2 < … < nCr > nCr+1 > nCr+2 > … >

nCn, then r is equal to JNU– 2005


INFOMATHS(a) (b) (c) (d)

12. The coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 + ……. + (1 + x)30

JNU– 2005(a) 51C5 (b) 9C5 (c) 31C6 – 21C6 (d) 30C5 + 20C6

13. For n N, 32n + 2 – 8n – 9 is divisible by JNU– 2004

(a) 81 (b) 72 (c) 64 (d) 4914. The coefficient of x2 in the trinomial expansion of (1 + x + x2)10

JNU– 2003

(a) (b)

(c) (d)

15. 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

16. n-1C3 + n-1C4 > nC3 if n > ……. JNU– 2002

(a) 5 (b) 6 (c) 7 (d) 817. The value of

is

(JNU - 2009)(a) 0 (b) 1 (c) (-1)n (d) 2n

18. We define , when k is non-

negative and when k is negative. Thus

equals to JNU – 2008

(a) 0 (b) 29.52 (c) 1.52 (d) 19. For n 5, the expression

1 + 2x + 3x2 + 4x3 + … + nxn – 1, x 1, is equal to IIT– 2007

(a) (b)

(c) (d)

20. The sum of the series is equal

to JNU– 2007

(a) (b) (c) (d) None of these

21. The coefficient of in the expansion of (X1 + X2 + X3 – X4 + X5)15 is JNU– 2004(a) 1!2!3!4!5 (b) 15.14.13.12.11

(c) (d)

22. , where A is JNU– 2004

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

23. The value of 4{nC1 + 4. nC2 + 42. nC3 + …. + 4n-1} is JNU– 2002

(a) 0 (b) 5n + 1 (c) 5n (d) 5n - 1EXPONENTIAL & LOGARITHMIC SERIES 1. The sum of the series

is

JNU – 2005

(a) 3e (b) (c) (d)

2. is equal to JNU – 2005

(a) logex (b) x (c) logxe (d) None of these3. The coefficient of xn in the expansion of loga(1 + x) is

JNU – 2005

(a) (b)

(c) (d)

4. The coefficient of x2 in the expansion of e3x + 4 is JNU – 2002(a) 9e2/2 (b) 9e4/2 (c) 3e4/2 (d) 3e2/2

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) 6256. 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

(a) (b) n3 (c) 3n (d)

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. A polygon has 90 diagonals then it has JNU– 1999(a) 10 sides (b) 15 sides (c) 20 sides (d) 25 sides


INFOMATHS16. The number of zeroes that 1000! ends up with, when expanded out

is JNU– 1999(a) 237 (b) 249 (c) 261 (d) 280

17. 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

18. A polygon has 44 diagonals then the number of its sides are JNU– 1998

(a) 7 (b) 9 (c) 11 (d) None of these19. 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

20. 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

21. 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

22. 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

(a) and (b) and

(c) and (d) and

23. 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

24. 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

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

(A) (B) (C) (D)

2. An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is IIT-2011

(A) (B) (C) (D)

3. 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) (b) (c) (d) 1

4. 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

5. If A and B are two independent events and

, then P(A B) is IIT-2010

(a) (b) (c) (d)

6. Let and .Consider

the following lists:

List I List II

1. P.

2. Q.

3. R.

4. S.

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)

7. 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

(a) (b) (c) (d)

8. Let . An element of S is

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

9. 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)

(a) (b) (c) (d)

10. 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) (b) (c) (d)

11. 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) (b) (c) (d)

12. 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

13. 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 14. 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) (b) (c) (d)


INFOMATHS15. Let A and B be events with and

. Match Lists I and II and select the correct

answer: IIT– 2006

List I List II

1. P.

2. Q.

3. R.

4. S.

(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)

16. 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

17. 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

18. Let A, B and C be independent and mutually exclusive events

with probability of occurrences ,

respectively, then p lies in IIT – 2005

(a) [-1/4, 5/6] (b) [-1/4, 1/3] (c) [-1/4, 1/2] (d) [-1/2, 1/3]

19. 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

20. 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

21. 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)

22. A random variate has the following distribution: x : 0 1 2 3 4 5 6 7

p(x) : 0 k 2k 2k 3k K2 2k2 7k2 +k

The value of k is JNU – 2004

(a) 0.1(b) -0.1 (c) – 1 (d) 123. 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

24. 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

25. 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

26. The probability that three men hit a target are respectively

and . If only one hits the target, what is the probability that

it was the first man? JNU – 1999

(a) 3/31 (b) 6/31 (c) 7/30 (d) None of these27. 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


28. 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

29. Let where a, b, c are chosen randomly from the

set {1, 2, 3, 4, 5}. The probability that P is singular is (IIT - 2009)

(a) (b) (c) (d)

30. 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

31. 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

32. 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) (b) (c) (d)

33. 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

34. 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) (b) (c) (d)

35. 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


36. 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

37. 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

38. 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


INFOMATHSfunctions correctly is higher than the probability that a three-components system functions correctly?

IIT – 2005(a) [0.4, 0.6] (b) [0, 0.5] (c) [0, 1] (d) [0.5, 1]

39. 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

40. 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 p2 the probability that their sum be 8, then (p1 + p2) is

JNU – 2002(a) 7/729 (b) 137/729 (c) 16/81 (d) 137/81

41. 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

42. 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

43. 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 in the cafeteria is 40% and , 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

44. 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) (b) (c) (d)

45. 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.


46. 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

47. 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

TRIGONOMETRY 1. sinh ix equals JNU-2010

(a) coshx (b) i sin x (c) cos x (d) 12. If sin-1 x + sin-1 y + sin-1z = 3/2, then the value of

is

(JNU : - 2009)(a) 0 (b) 1 (c) 2 (d) 3

3. The maximum value of =

for (JNU : - 2009)

(a) 3 (b) 4 (c) 5 (d) None of the above 4. The value of tan 100 + tan 125 + tan 100 tan 125 is

(JNU-2009)(a) 0 (b) 1/2 (c) –1 (d) 1

5. The most general values for which are (JNU-2009)

(a) n + 7/4 (b) n + (-1)n 7/4 (c) 2n + 7/4 (d) 2n + (-1)n 7/4

6. 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 these7. 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

8. 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

9. 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

10. The value of sin is JNU– 2007

(a) 12/13 (b) 13/14 (c) 14/15 (d) None of these11. The angle of the elevation of the sum when the length of the

shadow of the pole is times the height of the pole is JNU – 2007

(a) 30(b) 45(c) 60(d) 13512. 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

13. 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) (b)

(c) (d) None of these

14. 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

15. 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

16. If , then one of the values of y is

JNU :– 2006(a) tan A (b) cot A (c) – tan (2A) (d) – cot A

17. The expression lies in the interval

JNU :– 2006(a) (-4, 4)

(b)

(c)

(d)

18. 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

19. If tan( cos ) = cot ( sin ), then cos( - /4) is equal to JNU – 2006

(a) (b) (c) (d)

20. The value of tan 1 tan 2 … tan 89 is JNU– 2006(a) – 1 (b) 0 (c) 1 (d) N.O.T.

21. If cosA + cosB = m and sinA + sinB = n, where, m, n 0, then sin (A + B) is equal to

JNU– 2005

(a) (b)

(c) (d)

22. 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) km (b) km


INFOMATHS(c) (d) 6 km

23. The principal value of sin-1 [sin(2/3)] is JNU - 2004(a) /3 (b) -2/3 (c) 2/3 (d) 5/3

24. The value of is JNU - 2004

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

25. If , then B equals JNU - 2004

(a) (A - n)/2 (b) n - 4 (c) 2(A - n) (d) A/2 - n

26. 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(a) (b) 5 (c) (d)

27. If A = 45 and B = 75, what is the value of side b? JNU - 2004

(a) (b)

(c) (d)

28. The value of is

JNU - 2004(a) 1/8 (b) 3/16 (c) 1/16 (d) 3

29. The greatest value of sin cos is JNU– 2002(a) – 1 (b) 1 (c) – 1/2 (d) 1/2

30. Let tan = m/(m + 1) and tan = 1/ (2m + 1), then the value of ( + ) is JNU– 2002(a) /3 (b) /6 (c) /2 (d)

31. The smallest positive root of the equation tan x – x = 0 lies in JNU– 2000

(a) (b)


32. The general solution of the trigonometric equation sin x + cos x = 1 is given by JNU– 1999(a)

(b)

(c)

(d) None of these33. Compute : (sin 15 + cos 15)6 JNU– 1998


34. The function 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, …

35. 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

36. 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

37. 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

38. 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)

30. 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

40. The sum of the series + ….. will be

equal to JNU– 2007

(a) (b)


41. The number of real solutions of the equation

JNU :– 2007(a) zero (b) one (c) two (d) infinite

42. Let the two statements (I) sin 100 sin 500 sin 700 = 1/8

(II) If , then

Of the following, identify the correct statement 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

43. 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

44. 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)

45. If then

roots of f(x) = 0 are (JNU–2007) (a) 1/2, - 1 (b) 1/2, - 1, 0 (c) – 1/2, 1, 0 (d) – 1/2, - 1, 0

46. If sin(x + 3) = 3 sin( – x), then (JNU– 2007)

(a) tan x = tan (b) tan x = tan2 (c) tan x = tan3 (d) tan x = 3 tan

47. tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A is equal to JNU – 2007

(a) tan 2A (b) cot A (c) sin 3A (d) None of these48. 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) (b) (c) (d)

49. If sin ( + ) = 1 and sin ( – ) = 1/2 where , [0, /2], then

is equal to JNU–2006

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

50. If , then is equal to

JNU – 2006(a) 1/y (b) y (c) 1 – y (d) 1 + y


INFOMATHS51. 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

52. If , then which of the following are true? (I)

(II)

(III) JNU – 2006(a) (I) and (II) only (b) (II) and (III) only (c) (III) and (I) only (d) (I), (II) and (III)

53. 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)

54. If x cos + y sin = x cos + y sin = 2 (0 < < / 2), then it is also true that

JNU :– 2006

(a)

(b)

(c)

(d)

55. If , then the value of satisfying 0 < <

is JNU– 2005

(a) 3/2 (b) /6 (c) 5/6 (d) /2 56. The general solution of sin x – 3 sin 2x + sin 3x = cos x – 3 cos 2x

+ cos 3x is JNU– 2005

(a) (b)

(c) (d)

57. 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

58. 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

59. 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

60. 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

61. 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

62. , then

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

63. 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) (b)

(c) (d) 64. The number of solutions of the equation

cos4x + sin4x = snx cosx (0 x 2)is ……………….. JNU– 2001

(a) (b) (c) (d)

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

4. The equation

will represented a hyperbola for (JNU-2009)(a) K (0, 2) (b) K (-2, 1) (c) K (1, ) (d) K (0, )

5. 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

6. The equation of the circle through (1, 1) and the points of intersection of x2 + y2 + 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

7. The line y = x + 5 does not touch JNU – 2008

(a) the parabola y2 = 20x(b) the ellipse 9x2 + 16y2 = 144 (c) the hyperbola 4x2 – 29y2 = 116 (d) the circle x2 + y2 = 25

8. 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

9. If the foci of the ellipse and the hyperbola

coincide, then the value of b2 is

JNU - 2007(a) 3 (b) 16 (c) 9 (d) 12

10. Equation of a common tangent to the curve y2 = 8x and xy = - 1 is JNU– 2007

(a) 3y = 9x + 2 (b) y = 2x + 1 (c) 2y = x + 8 (d) y = x + 2

11. 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

12. 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


INFOMATHS(a) x2 + y2 – 6x + 4 = 0 (b) x2 + y2 – 3x + 1 = 0 (c) x2 + y2 – 4y + 2 = 0 (d) None of these

13. The straight line y = 4x + c is tangent to the ellipse

. Then C is equal to JNU: - 2006

(a) (b) (c) (d) 14. 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 15. 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

16. 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

17. If the foci of the ellipse and the hyperbola

coincide, then the value of b2 is

JNU– 2007

(a) 3 (b) 16 (c) 9 (d) 12

18. The straight line y = 4x + c is tangent to the ellipse

. Then C is equal to JNU – 2006(a) (b) (c) (d)

19. 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

20. 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

21. 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 these22. Area of the quadrilateral formed by the lines |x| + |y| = 1 is


23. The parametric coordinates of any point on the parabola y2 = 4ax can be JNU – 2005(a) (-at2, -2at) (b) (-at2, 2at)(c) (- a sin2t, - 2a sin t) (d) (- a sin t, 2a cos t)

24. 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

25. The straight lines7x – 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

26. 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

27. 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

28. 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

29. 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)

30. 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

31. 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

32. 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

33. The eccentricity of a rectangular hyperbola is always JNU – 2002 (a) 1 (b) (c) (d) 2

34. 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

35. 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

36. 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 (c) a2 = 3h2 (d) 3a2 = h2

37. 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 these38. 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

39. 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

40. The length of the line joining two points on the parabola y2 = x which is bisected at (1, 2) is

JNU– 2007(a) (b) (c) (d)

41. The shortest distance of (0, 0) from the curve

JNU– 2007(a) 1/2 (b) 1 (c) 2 (d) None of these

42. If a, b, c are the sides of a triangle, then the value of the

expression is equal to

JNU– 2007(a) 1 (b) 3/2 (c) 2 (d) 5/2

43. 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) (b) (c) (d)

44. 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

45. The parametric equations

where is a parameter, represents JNU– 2007

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

46. The centre of a circle passing through the point (0, 1) and touching the curve y = x2 at (2, 4) is

JNU - 2006

(a) (b)


INFOMATHS(c) (d) None of these

47. 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

48. 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


49. 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

50. 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

51. 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

52. 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

53. The center of a circle passing through the point (0, 1) and touching the curve y = x2 at (2, 4) is

JNU – 2006

(a) (b)


54. 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

55. 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)

56. 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

57. 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

58. 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

59. 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

60. 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)

61. 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)

62. 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

63. In a rectangular hyperbola, the asymptotes are JNU – 2001

(a) at right angles (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

64. x – 2y + 4 = 0 is a common tangent to y2 = 4x and

. Then the value of b and the other common

tangent are given by JNU – 2001 (a) b = ; x + 2y + 4 = 0 (b) b = 3; x + 2y + 4 = 0 (c) b = ; x + 2y – 4 = 0 (d) None of these

FUNCTIONS1. 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]

2. Suppose that

Computer fgh (x) at x = 2. JNU-2010(a) 113(b) 11-3/2 (c) 113/2 (d) None of these

3. What is the range of the function f that maps R toR2 by means of

the formula ? 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

4. 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)

5. If f(x) = cos (ln x) then f(x) f(y) has

the value JNU – 2000

(a) – 1 (b) 1/2(c) – 2 (d) None of these6. 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)

(c) f(x) = x cos x (d) None of these

7. Suppose

The value of T(125) is (IIT - 2009)


INFOMATHS(a) 500 (b) 400 (c) 375 (d) 380

8. Consider the function

. For

positive integers m and n, f(m, n) is (IIT - 2009)

(a) m + n (b)

(c) mn (d) mn 9. 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

10. 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

11. 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

12. Consider the following functions of a complex variable

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

13. 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

14. 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

15. Let f(x) = x3, x [a, b] and the value of the determinant

is equal to (-16) Then b – a is equal to (JNU – 2006)

(a) 0 (b) 1 (c) 2 (d) 416. If an odd function increase for x > 0, then for x < 0 it

IIT - 2005(a) increases (b) decreases (c) remains constant (d) oscillates

17. The domain of the real valued function f(x, y) defined by

is

IIT - 2005(a) Points inside y ≤ |x| (b) Points inside |y| ≤ |x|(c) Points inside y ≤ x (d) Points inside |y| ≤ x

18. If the function f : R A given by is a

surjection, then A is equal to JNU – 2005

(a) R (b) [0, 1] (c) (0, 1] (d) [0, 1) 19. 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

20. 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 (c) f(c) = z, f-1 (x) = b (d) f(c) = x, f-1 (x) = b

LIMITS & CONTINUITY1. Given f(x) is differentiable and f ' (4) = 5, find

JNU-2010(a) (b) 0 (c) 5 (d) – 20

2. is

JNU-2010

(a) 0 (b) 1 (c) (d) Does not exist

3. If a is a repeated root of px2 + px + r = 0, then

is

JNU-2010

(a) 0 (b) r (c) p (d)

4. The value of f(0,) for which is

continuous, is JNU-2010

(a) 51 (b) 59 (c) 61 (d) None of these

5. is equal to

JNU – 2008 (a) n/2 (b) n(n + 1)/2 (c) 1 (d) None of these

6. is equal to

JNU – 2008 (a) 0 (b) – 1/2 (c) 1/2 (d) None of these

7. equals

JNU – 2005(a) e (b) (c) e2 (d) 1/e

8. The value of a so that the function

Be continuous at x = 0 is JNU – 2005

(a) 1 (b) – 1 (c) 1 (d) 0

9. The value of is JNU – 2004

(a) 0 (b) 1 (c) 5 (d)

10. The value of is

JNU – 2004(a) (b) 1/120 (c) 1/20 (d) 1/2

11. If f(9) = 9, f'(9) = 4, then equals


INFOMATHSJNU – 2004

(a) 0.50 (b) 1 (c) 2 (d) 4

12. is equal to

JNU – 2004(a) ½ log 5 (b) 1/5 log 2 (c) 2 log 5 (d) 5 log 2

13. If m and n are positive numbers, then the limit is

equal to JNU – 2003

(a) (b) (c) (d) Does not exist

14.

JNU – 2003(a) 1 (b) (c) (d) 0

15.

JNU : – 2001

16. If then JNU– 2001

17. The function is

(JNU : - 2009)(a) continuous (b) discontinuous (c) differentiable such that f'(0) = 1(d) differentiable such that f'(0) = - 1

18. Let Then

is IIT – 2008

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

19. 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. The value of is equal to

JNU – 2008 (a) -/2 (b) /2 (c) 0 (d) None of these

21. If

Then the value of is IIT - 2007

(a) – 1 (b) 0 (c) 1 (d) 2

22. The limits where x and 0 < A < 1 is

JNU - 2007(a) B (b) 1 (c) A (d) 0

23. is equal to

JNU– 2006(a) k2 (b) 2k (c) 2 In (k) (d) None of these

DERIVATIVES

1. Let for all x R. Then

IIT-2011 (A) f is continuous but not differentiable on R(B) f ' is bounded on R(C) f ' has exactly three zeroes(D) f is continuous and bounded on R

2. Let

Which one of the following is TRUE?IIT-2011

(A) f '(0) = 1 and f ''(0) = 2(B) f '(0) = 1 but f ''(0) is not defined(C) f '(0) does not exist

(D) f is not continuous at x = 0

3. It is given that and

. If then is

equal to JNU-2010

(a) 2 (b) 2 (c) 4 (d) 84. 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


5. Derivative of w.r.t. is

JNU – 2006(a) – 2 (b) – 1 (c) 1 (d) 2

6. If xy = ex-y, then dy/dx is equal to JNU – 2005

(a) (1 + log x)-1 (b) (1 + log x)-2 (c) log x. (1 – log x)-1 (d) None of these

7. If y2 = ax2 + bx + c, where a, b, c are constants, then 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

8. 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

9. 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

10. If , x = t cos t, y = t sin t then at is

(IIT - 2009)

(a) (b) (c) (d)

11. 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.

12. 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


INFOMATHS(a) continuous and differentiable at origin (b) not continuous but differentiable (c) continuous but not differentiable (d) not continuous and not differentiable

13. The function

is differentiable at x = 0 for all a in the interval IIT – 2006

(a) (-, 1] (b) (-1, )(c) (1, ) (d) (1, )

14. The function is

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)

15. 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

16. For a real number y, let [y] denote the greatest less than or equal

to y. Function f(x) is given by JNU – 2006

(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

17. 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)

18. 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

19. Let f(x) is a function differentiable at x = c, then

equals JNU – 2005 (a) f'(c) (b) f"(c) (c) 1/f(c) (d) None of these

20. If , then equals

JNU – 2005 (a) 0 (b) y (c) k f(x) (d) k2 f(x)

21. If , then equals

JNU – 2005 (a) 2/c (b) -2/c2 (c) 2/c2 (d) None of these

22. If , then dy / dx is

JNU – 2004

(a) (b)

(c) (d)

APPLICATION OF DERIVATIVES

1. Let

Which one of the following is TRUE?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(D) f is not differentiable at x = 0 but has a local maximum at x = 0

2. 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 (b)

(c) (d)

3. if

IIT-2010 (a) if has a local maximum at x = 0 and a local minimum at x = 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

(d) f has local maxima at x = 1 and x = 0 4. If f(x) = ax3 +bx2 + x +1 has a local maximum value 3 at x = -2 ,

then IIT-2010

(a) (b)

(c) (d)

5. 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

6. 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

7. If , then the

function a0xn + a1xn-1 + a2xn-2 + … + an has in (0, 1) JNU – 2008

(a) at least one zero (b) at most one zero (c) only 3 zeros (d) only 2 zeros

8. The slope of the tangent line to the curve x = a (t – sin t), y = a(1 – cos t), t R

at is IIT - 2007

(a) – 1 (b) 0 (c) 1 (d) 9. 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) 110. If 8x – y = 15 is a tangent at (2, 1) to the curve y = x 3 + ax2 + b,

then (a, b) is IIT– 2006

(a) (1, 3) (b) (-1, 3) (c) (1, - 3) (d) (-1, - 3) 11. 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

12. If f(x) = kx – sin x is monotonically increasing, then JNU – 2005

(a) k > 1 (b) k > - 1 (c) k < 1 (d) k < - 1 13. 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) 1/214. The normal to a given curve is parallel to x-axis if

JNU – 200514 INFOMATHS/MCA/MATHS/IIT-JNU QUESTIONS

INFOMATHS(a) (b) (c) (d)

15. 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

16. 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

17. 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

18. The length of the subnormal to the parabola y2 = 4ax at any point is equal to JNU – 2004 (a) 2a (b) (c) (d)

19. 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

20. The critical points of the function f(x) = (x – 2)2/3, (2x + 1) are ……………. JNU – 2001(a) (b) (c) (d) None of these

21. If 2x + 5y = 3 Find maximum value of x3 y4. JNU – 2000(a) (b) (c) (d) None of these

22. The set of all points where is differentiable, is

JNU – 1999(a) (-,) (b) (0, ) (c) (-, 0) (0, ) (d) None of these

23. 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

24. 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

25. The maximum value of is equal to - JNU – 1998

MATRICES

1. If , then P50 equals

IIT-2011

(a) (b)

(c) (d)

2. Consider the following system of equations2x + 3y + 4z = 13 5x + 7y + 7z = 27 9x + 13y + 15z = 13The value of for which the system has infinitely many solutions is

IIT-2011 (A) 1 (B) 2 (C) 3 (D) 4

3. 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

4. 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

5. 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

6. For what value of k, will the equations 2x + 3y = 5

and 6x + ky = 15 have an infinite number of solutions?

(JNU : MCA - 2009)(a) 7 (b) 8 (c) 9 (d) 10

7. If a + b + c 0 and then the

total number of different values of x is equal to (JNU - 2009)

(a) 1 (b) 2 (c) 3 (d) None of these

8. The multiplicative inverse of the matrix is

given by IIT – 2008(a) A + 4I (b) A – 4I

(c) (d)

9. The value of the determinant

is zero if.

JNU – 2008(a) sin x = 0 (b) cos x = 0 (c) a = 0 (d) N.O.T

10. 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

11. If and I be 3 3 unit matrix, then rank of

I – A is JNU– 2007

(a) 0 (b) 1 (c) 2 (d) 312. Which of the following is false?

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 equals

JNU– 2007(a) 7 + 4i (b) 2 – 2i (c) – 7 – 4i (d) – 2 + 2i

14. The inverse of the matrix


INFOMATHS

is

IIT– 2006

(a)

(b)

(c)

(d)

15. 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

16. The determinant

is independent of JNU– 2006

(a) n (b) a (c) x (d) None of these17. 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

18. For the matrix is given by

IIT– 2005(a) A2 – 2A (b) A2 + 2A + 3I (c) A2 – 2A – I (d) A – 3I

19. The system of equations Ax = b, where A

and has

IIT– 2005(a) no solution (b) unique solution (c) infinitely many solutions

(d) more than one but finitely many solutions 20. Let P be a 2 2 matrix such that P102 = 0. Then

IIT– 2006 (a) P2 = 0 (b) (I – P)2 = 0 (c) (I + P)2 = 0 (d) P = 0

21. Let

then 8p-1 is equal to IIT– 2006

(a) (b)

(c) (d)

22. The determinant is equal to zero,

if JNU– 2005

(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

23. The inverse of a diagonal matrix is JNU– 2005

(a) a symmetric matrix (b) a skew-symmetric matrix (c) a diagonal matrix (d) None of these

24. The rank of a null matrix JNU– 2005

(a) is 0 (b) is 1 (c) does not exist (d) None of these

25. 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

26. 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 (c) 3A2 – 2A – 5 (d) None of these

27. If a, b, c are all different from zero, and

is equal to zero, then the value of a-1 + b-1 + c-1 is JNU– 2004

(a) abc (b) a-1 b-1 c-1

(c) – a – b – c (d) None of these28. If a, b, c are distinct and

then x is equal to JNU– 2004

(a) 0 (b) a (c) b (d) abc29. The value of the determinant

is independent of JNU– 2004(a) n (b) a (c) x (d) None of these


INFOMATHS

30. Let then the value of

is JNU– 2001(a) 0 (b) a + b + c (c) ab + bc + ca (d) None of these

31. The value of is JNU– 1998

(a) (a + b + c)2 (a2 + b2 + 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)

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 (c) NM-1x (d) N-1Mx

33. If then

(IIT – 2009)(a) 4k + 3 (b) 4k – 3 (c) 2k + 1 (d) k

34. Matrices Let where is a complex cube root

of unity. The p24 is (IIT - 2009)(a) p2 (b) p(c) Identity matrix (d) 0

35. Consider the system of equations P x = 0, where

. The value of k for which the

system will have a nontrivial solution are (IIT - 2009)

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

36. 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)

37. Fow which three values of c, the given matrix A is not invertible?

(JN -

2009)(a) {2, 8, 7} (b) {1, 2, 8}(c) {0, 2, 7} (d) None of the above

38. Let and . If A = P-

1 DP, then the matrix D is equal to IIT – 2008

(a) (b)

(c) (d)

39. 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

40. 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.

41. The system of linear equations has

IIT – 2008(a) no solution (b) infinite number of solutions (c) only one solution (d) more than one but finite number of solutions.

42. If for a triangle ABC, , then

sin3 A + sin3 B + sin3 C is equal to JNU – 2007(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

43. If ABC is not a right triangle, then the value of

is JNU – 2007

(a) – 1 (b) 2 (c) 3 (d) 0

44. If , then the value of

is JNU – 2005(a) 0 (b) 1 (c) 2 (d) 4 pqr

45. 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

46. 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

47. The only integral root of the equation

, is JNU – 2002

(a) y = 0 (b) y = 1 (c) y = 2 (d) y = 3 48. If A, B, C are angles of a triangle then the value of

is JNU – 2002

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


INFOMATHS

49. If A =

and then JNU – 2002

(a) A = 4B (b) A = 2B (c) A = B (d) None of these

50. 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

(a) (b) (c) (d)

51. The system of linear equations x – y + z = 0 x + y + z = 2 x – 2y + 4z = 5 ex + fy + 2z = 2 is consistent if

JNU – 1999(a) e – f = 2 (b) e – f = - 2 (c) e + f = 2 (d) e + f = - 2

52. Find all values of which satisfy the equation

The values are JNU – 1999(a) (b) (c) (d) None of these

53. Find the general solution of the following second-order

differential equation

where F( ) and G( ) given. JNU – 1998(a) (b) (c) (d) None of these

INDEFINITE INTEGRAL

1. dx is equal to

JNU – 2008

(a) (b)

(c) (d)

2. is equal to

JNU – 2008

(a)

(b)

(c)

(d) None of the above

3. is equal to JNU

- 2008

(a) (b)

(c) (d)

4. The value of is

JNU - 2007

(a) (b)

(c) (d)

5. If and

, then (x) is JNU - 2008

(a) - x (b) - + x (c) /2 – x (d) None of these

6. The value of dx is

JNU - 2007

(a)

(b)

(c)

(d) None of these

7. The value of

IIT– 2005

(a) (b)

(c) (d)

DEFINITE INTEGRAL

1. Let for all x R. Then

equals

IIT-2011 (A) − 2 (B) −1 (C) 1 (D) 2

2. Calculate

JNU-2010(a) 1 – sin 2 (b) 1 – cos (ln 2) (c) 1 + cos (ln 2) (d) 1 + ln 2

3. The integrating factor of the differential equation

is given by

JNU-2010(a) loglog x (b) x (c) ex (d) logx

4. The value of is

(IIT - 2009)

(a) (b) (c) (d)

5. The area bounded by the curves y2 = x and x2 = y is


INFOMATHS(IIT – 2009)

(a) 1/3(b) 2/3(c) 4/3(d) 5/36. The area bounded by y2 = 4 – x and y2 = x is

(JNU - 2009)

(a) (b) (c) (d)

7. The value of the integral IIT – 2008

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

8. The area of the region bounded by the curves x2 = 2y and y2 = 2x is IIT – 2008

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

9. is equal to

JNU – 2008(a) 1 (b) 2 (c) 4 (d) 0

10. 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

11. For > 0, the value of the integral

dx

equals IIT – 2007

(a) (b) (c) (d)

12. If , then the value of is

given by JNU – 2007

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

13. The value of integral is

JNU– 2006


14. 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

15. is given by

JNU – 2007 (a) log 2 (b) 1/3 log 2 (c) 1/2 log 2 (d) 1/2

16. If and , then

JNU – 2005(a) I1 = I2 (b) 2I1 = I2 (c) I1 2I2 (d) None of these

17. The value of the integral is

JNU – 2005(a) 2 (b) 1 (c) 0 (d) 3

18. The area of the figure bounded by the curves y = ex, y = e-x and the straight line x = 1

JNU – 2005


19. [.] denote the greatest integer function, then the value of

JNU – 2004

(a) (b)

(c) (d) 3

20. The value of is JNU – 2004

(a) – 2 log 2 (b) 2 log 2 (c) log 2 (d) - log 2

21. If (n – m) is odd, then is JNU – 2002

(a) 2n/ (n2 – m2) (b) 2n / (m2 – n2) (c) 2m / (n2 – m2) (d) 0

22. The value of the integral

is

JNU – 2001(a) 1/4 (b) 0 (c) 1 (d) None of these

23. If f(x) and g(x) are continuous in [a, b] and g(x) 0 for all x [a,

b] then is (IIT - 2009)

(a) for exactly one x (a, b)

(b) for all x (a, b)

(c) for some x (a, b)

(d) for all x (a, b)

24. Let f(x) be a continuous function such that f(a – x) + f(x) = 0 for

all x [0, a]. Then is equal to

JNU – 2008(a) a (b) a/2 (c) f(a) (d) f(a)/2

25. The integral equals

IIT– 2007(a) (b)

(c) (d) 26. 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

27. For the integral is equal to (-), the least positive

value n is equal to JNU– 2006

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

28. The value of is equal to

JNU– 2006 (a) infinity (b) 2/3 (c) 1/3 (d) None of these

29. 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

30. The value of is

IIT – 2005(a) 1 (b) 1/e (c) e (d) 0


INFOMATHS

31. Let . Then the value of f'(1) is equal to JNU – 2006(a) sin (1) (b) 0 (c) – sin (1) ( d) 2 sin (1)

32. 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)

33. Consider the following inequality

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)

34. The value of is

JNU – 2002(a) (1/78) loge (1/5) (b) (1/156) loge (5) (c) (1/78) loge5 (d) None of these

COMPLEX NUMBERS 1. The number of roots of x2.1 + x3.01 + x4.00 = 1 is

JNU-2010(a) infinite (b) two (c) 3001 (d) 4001

2. The value of the sum is

JNU-2010(a) i (b) i – 1 (c) 1 – i (d) 0

3. Find z4, if

JNU-2010

(a)

(b)

(c)

(d)

4. 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

5. Centre of the arc represented by arg is

(JNU – 2009)

(a) (b)

(c) (d)

6. If is a complex cube root of unity, then the value of the expression

is (JNU – 2009)

(a) –1 (b) 0 (c) 1 (d)

7. The value of is

JNU – 2008(a) – 1 (b) 0 (c) – i (d) i

8. If then x1x2x3 …. to

JNU – 2008(a) – 3 (b) – 2 (c) – 1 (d) 0

9. The number of solutions to the equation is JNU – 2006

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

10. The real value of for which the expression is a

real number is JNU – 2006

(a) 2n (b) (2n + 1)(c) 2n /2 (d) None of these

11. If | z2 – 1 | = | z |2 + 1, then z lies on a / an JNU – 2006

(a) straight line (b) circle (c) ellipse (d) None of these

12. If z = z + iy, z1/3 = a – ib, a ba, b 0, then

, where k is equal to


13. 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

14. 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 these15. 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

16. For any complex number z, the minimum value of |z| + |z – 1| is JNU - 2005

(a) 1 (b) 0 (c) 1/2 (d) 3/217. 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

18. the cube roots of unity JNU – 2002

(a) are collinear (b) Lie on a circle of radius (c) Form an equilateral triangle(d) None of these

19. The value of real such that is purely imaginary

is JNU – 2004

(a) n (b) n /2(c) n /3 (d) n /6

20. Solve z5 = 1, for z JNU – 2004

(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, ….

21. The value of is

JNU – 2002(a) 12 (b) – 12 (c) 16 (d) – 16

22. 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)


INFOMATHS23. The modulus and the principal argument of the complex number

are respectively. JNU

– 2000

24. The complex numbers z = x + iy satisfying lie on

the line - JNU – 1998

25. If

then maximum value of x is (JNU – 2009)

(a) 1 (b) 2 (c) 1/2 (d) 1/3 26. 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) 427. 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 (b) nC1 + nC2 + … + nCn

(c) [2n+1C0 + 2n+1C1 + … + 2n+1Cn]1/2 - 1(d) None of these

28. If is purely

imaginary, then is not given by JNU – 2008

(a) n + /4 (b) n - /4 (c) 2n (d) 2n + /4

29. Assume that either |z| = 1 or || = 1 and , where z, are

complex numbers and is the conjugate of z. The value of

is

JNU – 2007(a) (b) (c) (d) None of these

30. If eix = cos x + i sin x and

JNU – 2007

(a) (b)


32. 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 32. 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

33. The imaginary part of tan-1 (5i/3) is JNU - 2005

(a) 0 (b) (c) log 2 (d) log 4 34. Find the modulus and argument of

where n is a positive integer and

(2k + 1).JNU – 2000

VECTORS

1. If are three vectors in R3, then

equals

IIT-2011

(a) 0 (b)

(c) (d)

2. The area of the parallelogram in R2 whose diagonals are

and is IIT-2011

(A) 2.5 (B) 5 (C) (D) 5

3. For any two unit vectors and , is equal to

IIT-2010

(a) (b)

(c) (d)

4. Let P be the point (3, 4, 1). Let L be the line through P parallel to

the vector If Q is a point on L in the first octant such

that , then Q is

IIT-2010(a) (7, 8, 5) (b) (8, 7, 5) (c) (5, 8, 7) (d) (8, 5, 7)

5. If and the angle between is then

the area of the parallelogram with two adjacent ides as

and is (IIT -

2009)(a) (b) (c) 13 (d) 12

6. Let and be two non-collinear and non-orthogonal unit

vectors. Then is

(IIT - 2009)

(a) (b)

(c) (d)

7. If and then

is (IIT -

2009)(a) –40 (b) –24 (c) 24 (d) 40

8. 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

9. Projection of a + b in the direction of c where

and is. IIT – 2008

(a) (b) (c) 311

(d) 110. 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

11. 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

12. The area of the parallelogram with sides

and is IIT – 2007


INFOMATHS(a) (b) (c) (d) 6

13. Let , and .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

14. 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 &

are orthogonal is JNU – 2007

(a) 2/5 (b) 5/2 (c) 3/7 (d) 2/7

16. is equal to IIT – 2006

(a) (b)

(c) (d) 17. 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 is JNU -2006(a) 0 (b) 1 (c) 2 (d) 3

19. If = (1, 1, 1) and = (0, 1, -1) are given vectors, then a

vectors satisfying and is

JNU -2006(a) (5/3, 2/3, 2/3) (b) (2/3, 5/3, 2/3) (c) (2/3, 2/3, 5/3) (d) None of these

20. Area of a parallelogram where diagonals are and

being unit vectors, inclined at an angle 45 is

IIT – 2005

(a) (b) (c) (d)

21. Let and are three non-coplanar vectors, and let

and , be the vectors defined by the relations

, and

. Then the value of the expression

is equal to

JNU – 2006 (a) 0 (b) 1 (c) 2 (d) 3

22. The number of vectors of unit length perpendicular to the vectors

and is

JNU – 2006 (a) one (b) two (c) three (d) None of these

23. If the vectors (a, 1, i), (1, b, 1) and (1, 1, c) (a b c 1) are

coplanar, then is equal to

JNU – 2006 (a) 3 (b) 2 (c) 1 (d) 0

24. If and are collinear then the value of is (IIT - 2009)

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

25. Let u and v be two non zero parallel vectors with

Then (u v) w = 0 if IIT –

2008

(a) a + b = 2c and v is any vector orthogonal to u (b) u and v are orthogonal to each other but not orthogonal to w.

(c) u and v both are of the form where a + b = 2c (d) v is orthogonal to u but not orthogonal to w.

26. 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

(a) (b)

(c) (d) None of the above 27. 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

28. 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

is equal to JNU – 2008

(a) 0 (b) 1

(c)

(d) 29. 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

30. Let u, v R3, v ≠ 0. Which of the following is FALSE? IIT – 2007

(a) is the length of the projection of u along v (b) If u.w = v.w for all w R3 then u = v

(c) u . v

(d) 31. 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

32. Let a = i + 2j + k, b = i – j + k, c = i + j – k. A vector in the plane of a and b whose projection on c is , is JNU – 2007(a) 4i – j + 4k (b) 2i + j – 2k (c) 3i + j – 3k (d) 4i + j – 4k

33. The word done by the force 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

34. If and are two unit vectors, then the vector

is parallel to the vector JNU -2006

(a) (b) (c) (d)


INFOMATHS

35. For , and to be unit vectors satisfying

, the angles between and , and , respectively,

are IIT – 2005(a) 90, 45 (b) 60, 90(c) 90, 60 (d) 45, 90

36. 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

THREE DIMENSIONAL GEOMETRY 1. The equation of the plane containing the line

and the point (0, 7, -7) is

(JNU - 2009)(a) x + y + z = 1 (b) x + y + z = 2 (c) x + y + z = 0 (d) None of the above

2. The point on the sphere x2 + y2 + z2 = 1 farthest from the point (1, -2, 1) is IIT – 2007

(a) (b)

(c) (d)

3. Let , 0 be the angle between the planes x – y + z = 3 and 2x – z = 4. The value of is IIT – 2007

(a) (b)

(c) (d)

4. The spheres

x2 + y2 + z2 = 1 and intersect at an

angle IIT – 2007

(a) 0 (b) /6 (c) /4(d) /35. 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) 8/36. The distance of the point (1, 0, - 3) from the plane x – y – z = 9

measured parallel to the line

is

(JNU – 2009)(a) 6 units (b) 5 units (c) 8 units (d) 7 units

7. A unit vector in XZ – plane making angles /4 and /3 respectively with u = 2i + 2j – k and v = j – k is (JNU–2009)

(a) (b)

(c) (d) None of the above

8. 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

9. The volume of the tetrahdedron included between the plane 3x + 4y – 5z = 60 and the coordinate planes in cubic units is


10. Which of the following is a unit normal to the surface z = xy at P(2, - 1, -1) ? IIT – 2006

(a) (b)

(c) (d)

11. If ax + hy + gz = 0, hx + by + fz = 0, gx + fy + cz = 0, then JNU – 2007

(a)

(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 QUESTIONSSETS & RELATIONS

1 2 3 4 5 6 7 8 9 10B D A B A C C C

11 12 13 14 15 16 17 18 19 20C C B D D B D D D B21 22 23 24 25D A D C A

THEORY OF EQUATIONS1 2 3 4 5 6 7 8 9 10D C A B C D B C A B11 12 13 14 15 16 17 18 19 20B A C C C B D B B B21 22 23 24 25 26 27 28 29 30B B A D A D C A D B

SEQUENCE & SERIES1 2 3 4 5 6 7 8 9 10C D D C D C B A C C11 12 13 14 15B B C B D

BINOMIAL1 2 3 4 5 6 7 8 9 10B D D B B D B A B A11 12 13 14 15 16 17 18 19 20A C A C A C C B B B21 22 23D D D

EXPONENTIAL & LOGARITHMIC SERIES1 2 3 4C B B B

PERMUTATION & COMBINATIONS

1 2 3 4 5 6 7 8 9 10B A A C C B A C D B11 12 13 14 15 16 17 18 19 20D D B B B B A C B A21 22 23 24B B C

PROBABILITY1 2 3 4 5 6 7 8 9 10B B D B B C A A A A11 12 13 14 15 16 17 18 19 20B C D A C A C B A B21 22 23 24 25 26 27 28 29 30A A D C B B B B A C31 32 33 34 35 36 37 38 39 40A B C C D C D D D C41 42 43 44 45 46 47D D A C A C B

TRIGONOMETRY1 2 3 4 5 6 7 8 9 10B A B D C B B C A C11 12 13 14 15 16 17 18 19 20A B A A D B N B A C21 22 23 24 25 26 27 28 29 30B C A B C A D C D D31 32 33 34 35 36 37 38 39 40B C C D B A D A N A41 42 43 44 45 46 47 48 49 50C D C D A C B A C B51 52 53 54 55 56 57 58 59 60C A D A D C B B A D61 62 63 64


INFOMATHSB C D A

TWO-DIMENSIONAL GEOMETRY1 2 3 4 5 6 7 8 9 10A A C A C B D A B D11 12 13 14 15 16 17 18 19 20A B D A C A B D B A21 22 23 24 25 26 27 28 29 30C B A C D B D C D A31 32 33 34 35 36 37 38 39 40D C B B C B A C C D41 42 43 44 45 46 47 48 49 50B B A D D C C B C C51 52 53 54 55 56 57 58 59 60C C C A A A D C D D61 62 63 64D A A A

FUNCTIONS1 2 3 4 5 6 7 8 9 10D B A C D A A D C D11 12 13 14 15 16 17 18 19 20A A D D C A B D C C

LIMITS & CONTINUITY1 2 3 4 5 6 7 8 9 10D D C D B B C C C B11 12 13 14 15 16 17 18 19 20D C A D - - B A B A21 22 23B D A

DERIVATIVES1 2 3 4 5 6 7 8 9 10C C D D C D B B A D11 12 13 14 15 16 17 18 19 20C C C A A C A B D C21 22C A

DIFFERENTIATION1 2 3 4 5 6 7 8 9 10C D B B A B D C C A

APPLICATION OF DERIVATIVES1 2 3 4 5 6 7 8 9 10D C A A C A A C C D11 12 13 14 15 16 17 18 19 20C A C C A D A A B -21 22 23 24 25- A - - -

MATRICES1 2 3 4 5 6 7 8 9 10

D D B B A C C - A C11 12 13 14 15 16 17 18 19 20C D D C B A D C C A21 22 23 24 25 26 27 28 29 30B B D C C D D A A A31 32 33 34 35 36 37 38 39 40C C D C A D C D B B41 42 43 44 45 46 47 48 49 50C B B C C C B A A D51 52 53A - -

INDEFINITE INTEGRAL1 2 3 4 5 6 7D B D C D B A

DEFINITE INTEGRAL1 2 3 4 5 6 7 8 9 10D B D C A C D C D B11 12 13 14 15 16 17 18 19 20A B A D D A C C A D21 22 23 24 25 26 27 28 29 30A B C B B B A D C C31 32 33 34- B A B

COMPLEX NUMBERS (OLD QUESTIONS)1 2 3 4 5 6 7 8 9 10D B C A D B D C D C11 12 13 14 15 16 17 18 19 20A C C B D A C C C B21 22 23 24 25 26 27 28 29 30C B C D A C D D31 32 33 34A D C -

VECTORS (OLD QUESTIONS)1 2 3 4 5 6 7 8 9 10C B C A A A B D B B11 12 13 14 15 16 17 18 19 20D A C B B B D A A A21 22 23 24 25 26 27 28 29 30- - - C C A D C A C

31 32 33 34 35 36D A B A C B

THREE DIMENSIONAL GEOMETRY (OLD QUESTIONS)

1 2 3 4 5 6 7 8 9 10C A C D C D B C B A11B


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