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UNIVERSITY OF PUNE Department of Mathematics M.A./M.Sc./M.Tech. Entrance Examination-2012

Seat No.

Seat No.(In words) . . . . . . . . . . . . . . . . . . . . . . ......................................... Examination Center . . . . . . . . . . . . . . . . . . . . . ......................................... Signature of Invigillator . . . . . . . . . . . . . . . . .

[Total Questions: 25+10=35]

[Maximum Time:180 Minutes]

Instructions for Candidates: 1. Write your Seat Number in the space provided on the top of this page. 2. This booklet consists of twenty five objective type and ten subjective type questions on twenty pages. 3. Use answer sheet provided on Page No. 3 for objective type questions. 4. Objective type questions have four alternative response marked A, B, C and D. You have to darken one response. For example, if B is the correct answer then darken B as shown below A

B

C

D

5. Your responses to the items for this paper are to be indicated on the Answer Sheet only. Responses like (X),(×), (/) or light shaded responses will not be considered/evaluated. 6. Answer to the objective type questions, marked on the question paper will not be evaluated. 7. Answer(s) to subjective type questions are to be written on the space provided below each of the respective question(s) only. 8. There is no negative marking for wrong attempt/answer. 9. You have to return the booklet to the invigilator at the end of the examination compulsorily and should not carry it with you outside the examination hall.

.

[P.T.O.]

Entrance Examination 2012

Department of Mathematics This page is left intentionally blank.

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Department of Mathematics

University of Pune

Answer Sheet for Objective Type Questions

Q. 1

A

B

C

D

A Q. 14

B

C

D

Q. 2

A

B

C

D

A Q. 15

B

C

D

Q. 3

A

B

C

D

A Q. 16

B

C

D

Q. 4

A

B

C

D

A Q. 17

B

C

D

Q. 5

A

B

C

D

A Q. 18

B

C

D

Q. 6

A

B

C

D

A Q. 19

B

C

D

Q. 7

A

B

C

D

A Q. 20

B

C

D

Q. 8

A

B

C

D

A Q. 21

B

C

D

Q. 9

A

B

C

D

A Q. 22

B

C

D

A Q. 10

B

C

D

A Q. 23

B

C

D

A Q. 11

B

C

D

A Q. 24

B

C

D

A Q. 12

B

C

D

A Q. 25

B

C

D

A Q. 13

B

C

D

For office use only. Marks obtained for objective type questions: Marks obtained for subjective type questions: Total marks obtained:

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Objective type questions (Q. No. 1-25). Each question carries TWO marks. Q.1. The characteristic polynomial of a 3 × 3 matrix A is |λI − A| = λ3 + 3λ2 + 4λ + 3. Let x = trace(A) and y = det(A). Then A. C.

x y x y

= =

3 4 4 3

B. x = 3 and y = −3 D. x = y = −3.

Q.2. Let V be the span of n × n matrices over R and let W be the subspace of matrices with entries in each row adding up to zero.Then the dimension of W is A. n

B.

n(n−1) 2

C. n − 1

D. n(n − 1).

3 3 Q.3. For the standard basis {(1, 0, 0), (0, 1,0), (0, 0, 1)} of R  a linear transformation T from R 2 1 3 to R3 has the matrix representation  1 −1 1  The image under T of (2, 1, 2) is −1 1 −2

A. (11, 0, −1)

B. (11, 3, −5)

C. (7, 3, −1)

D. (7, 3, −5).

Q.4. Consider the system of equation M X = 0, where M is a n × n matrix. Then which of the following is incorrect statement? A. The system may not have a solution B. The system may have exactly one solution C. The system may have exactly two solutions D. The system mat have infinitely many solutions. 

 0 3 4 Q.5. Let M be a 3 × 3 matrix with determinant 10 and let N =  0 1 5  Then 0 0 2 A. det(M N ) = 0 B. det(M N ) = 10 C. det(M N ) = 13 D. det(M N ) = 20. Q.6. Solution of the differential equation xdy − ydx = 0 represents a A. straight line

B. hyperbola

C. parabola

D. circle

Q.7. Let y = x2 e3x + sinx be a solution of an initial value problem with constant coefficients. Then the least possible order of differential equation is A. 2

B. 3

C. 4

D. 5

dy + 2y = 0. Then dx A. every solution of this equation is identically zero.

Q.8. Consider the differential equation

B. all solutions of this equation are unbounded. C. every solution of this equation approaches zero as x → ∞. D. no solution of this equation approaches zero as x → ∞.

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Q.9. The partial differential equation corresponding to the integral z = eax f (x2 − y) is A. p + q = az

B. 2yp − q = az

C. p + 2xq = az

D. xp − yq = azy

Q.10. Complete solution of the partial differential equation p + q − pq = 0 is A. z = C. z =

ax a−1 ax a−1

ax a−1

+ ay + b

B. z =

+ ay

+ ay + b

D. (a − 1)z = ax + ay + b

Q.11. Consider the groups (Z, +), (Q, +)and (R, +). Which of the following statement is correct? A. All the three groups are naturally isomorphic. B. (Z, +) ∼ = (Q, +),but (Z, +) and (R, +) are not isomorphic. C. None of the groups is isomorphic to any other. D. (Z, +) ∼ = (R, +),but (Z, +) and (Q, +) are not isomorphic. cos θ − sin θ Q.12. Let T = : θ ∈ R , with matrix multiplication sin θ cos θ A. T is an abelian uncountable group B. T is an abelian countable group C. T is not closed under multiplication D. T with matrix addition is an abelian uncountable group Q.13. The set of permutations on {1, 2, 3, 4, 5} A. has 12 elements order 2.

B. has 120 elements and is non abelian

C. has no normal subgroup

D. has 60 elements and a normal subgroups.

Q.14. The order of the permutation (1, 2, 3)(4, 5, 6) on S = {1, 2, 3, 4, 5, 6} is A. 9 B. 2 C. 6 D. 3 Q.15. Z/3Z × Z/5Z with component wise addition A. has a subgroup of order 10 and is abelian. B. is a cyclic group with 15 elements. C. has a subgroup isomorphic to Z/3Z and is nonabelian. D. has a subgroup of order 5 and is nonabelian. Q.16. Let (X, d) be a metric space and {xn }, {yn } be two sequences in X such that {yn } is a Cauchy sequence in X and d(xn , yn ) → 0 as n → ∞. Consider the following statements: i. {xn } is cauchy sequence in X. ii. xn → x if and only if yn → y. A. Both i and ii are true

B. only i is true

C. only ii is true

D. Both are false. Page 6

Entrance Examination 2012 Q.17. The sum of the series

∞ X n=1

A. 0

B. 1

C. 2


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2n + 1 is equal to n2 (n + 1)2 D. 3

Q.18. Let f (x) = 2x + 1. Then |f (x) − 1| < 0.01 if |x − 0| is less than A. 0.01

B. 0.02

C. 0.005

D. 0.006.

, 1 )}. Then S is Q.19. Let f : R → R be f (x) = sin x and S = f −1 {( −1 2 2 A. a connected set

B. a finite union of disjoint open interval

C. a closed set

D. an infinite union of disjoint open interval.

Q.20. The Cantor set C ⊆ R A. is not compact B. is not contained in an interval C. does not contain a non trivial interval. D. does not have uncountably many elements. Q.21. The real and imaginary part of the complex number i25 are A. 1 and 0

B. -1 and 1 C. -1 and 0 D. 0 and 1

Q.22. If z is a complex number, the real part of z is given by z−z z−z z+z z+z A. B. C. D. 2 2i 2 2i R 1 Q.23. The value of the integral γ z−a dz where γ(t) = a + reit , 0 ≤ t ≤ 2π is A. i

B. 2π

C. πi D. 2πi

Q.24. The function f (x + iy) = x3 + ax2 y + bxy 2 + cy 3 is analytic in the complex plane C only if A. a = 3i, b = −3 and c = −i

B. a = 3, b = −3i and c = 1

C. a = −3, b = 3i and c = −1

D. a = 3, b = i and c = −i

Q.25. If f : C → C is a boumded and entire function then A. f need not be analytic at some point. B. f is constant. C. f takes only real values.

D. f need not be constant.

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Subjective type questions (Q. No. 26-35). Each question carries FIVE marks. Q.26. Let M be an n × n matrix. Show that the determinant of Adjoint of M is n or 1 or 0.

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Q.27. Let T : R2 → R2 be a linear map such that T (1, 1) = (2, 3), T (1, 2) = (3, 4). Determine T (4, 5).

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Q.28. Consider the nonlinear differential equation y 0 + α(x)y = β(x)y k , where k is a constant. Then i. transform this equation into the linear equation. ii. find all solutions of y 0 − 2xy = xy 2 .

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Q.29. Find a function φ which has a continuous derivative on 0 ≤ x ≤ 2 which satisfies φ(0) = 0, φ0 (0) = 1 and y 00 − y = 0, for 0 ≤ x ≤ 1, y 00 − 9y = 0, for 1 ≤ x ≤ 2.

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Entrance Examination 2012 Q.30.


i. Find all homomorphism from (R, +) to (Z, +). ii. Find all homomorphism from (Z, +) to (R, +).

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Q.31. Find all subgroups of Z/5Z × Z/5Z.

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Q.32. Prove that the only continuous functions f : R → R satisfying the equation f (x + y) = f (x) + f (y) are the linear functions of the form f (x) = ax, where a ∈ R.

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Q.33. Suppose that (X, d) is a metric space. For any two non-empty disjoint subsets A and B of X, we define d(A, B) = inf {d(a, b) : a ∈ A, b ∈ B}. i. Prove that if A and B are compact, then d(A, B) = d(a, b) for some a ∈ A and b ∈ B. ii. Give an example of a metric space (X, d) with disjoint closed subsets A and B of X such that d(A, B) = 0.

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Q.34. Prove that if p(z) is a non constant polynomial then there is a complex number a with p(a) = 0.

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Q.35. Use Cauchy’s theorem or Cauchy’s integral formula to evaluate the following integrals. Z z4 i. dz 3 |z|=4 (z − i) Z z3 dz ii. |z|=2 (z − 3)

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Department of Mathematics Space for rough work

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