PYQ 2013 - VedPrep

GATE MATHEMATICS 2013
Previous Year Question Paper with Solution.

1. The possible set of eigen values of a 4 × 4 skew-symmetric dorthogonal real matrix is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We know that the eigen values of skew symmetric matrix are either imaginary number of zero and that of orthogonal matrix are unit modulus real or imaginary number shown below.

So, the eigen values of 4 × 4 skew symmetric orthogonal matix are (± i).

2. The coefficient of (z – )² in the Taylor series expansion of

f(z) =

around is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Taylor's expression for f(z) around z = r is

3. Consider R² with the usual topology.

Which of the following statements are true for all A, B R²?

(a) P and R

(b) P and S

(d) Q and S

Ans. (b)

Sol.

Hence, Statement S is true.

Again, and implies that and and consequently.

Thus, is a closed set (being union of two closed sets) containing A B. Since, is the smallest closed set containing, then

From Eqs. (i) and (ii), we have

4. Let f : R R be a continuous function with f(1) = 5 and f(3) = 11. If g(x) = then is equal to

(a) 0

(b) 6

(d) 14

Ans. (b)

Sol. Consider the function f : R R is given by

f(x) = 3x + 2,

Then, f is continuous and

f(1) = 3 × 1 + 2 = 5

f(3) = 3 × 3 + 2 = 11

5. Let P be a 2 × 2 complex matrix such that tr(P) = 1 and det(P) = – 6. Then, tr(p⁴ – p³) is equal to

(a) 75

(b) 78

(d) 85

Ans. (b)

Sol. Consider the matrix,

Then, |P| = –6 and tr(P) = 1

tr(P⁴ – P³) = 36 + 42 = 78.

6. Suppose that R is a unique factorisation domain and that a, b R are distinct irreducible elements. Which of the following statements is true?

(a) The ideal (1 + a) is a prime ideal

(b) The ideal (a + b) is a prime ideal

(d) The ideal (a) is not necessarily a maximal ideal

Ans. (d)

Sol. Consider unique factorisation domain Z[x] and x Z[x]. The ideal < x > is a prime ideal in Z[x] but not a maximal ideal in Z[x]. To verify this, we begin with the observation that

<x> = {f(x) Z[x]/f(0) = 0}

Thus, if g(x) h(x) <x>, then g(0) h(0) = 0 and since g(0) and h(0) are integers, we have g(0) = 0 or h(0) = 0.

Hence, <x> is prime ideal in Z[x].

But <x> < x, 2 > Z[x].

Hence, <x> need not be maximal ideal in Z[x].

7. Let X be a compact Hausdorff topological space and Y be a topological space. Let f : X Y be a bijective continuous maping. Which of the following statements is true?

(a) f is closed map but not necessarily an open map

(b) f is an open map but not necessarily a closed map

(d) f need not be an open map or a closed map

Ans. (d)

Sol. Let X be a compact Hansderff topological space and left Y be a topological space. Left f : X Y be a bijective continuous mapping. Then, f need not be an open map or a closed map.

8. Consider the linear programming problem Maximise subject to constraints 2x + 3y 16,

x + 4y 18 and x 0, y 0

If S denotes the set of all solutions of the above problem, then

(a) S is empty

(b) S is a singleton

(d) S has positive area

Ans. (c)

Sol. Given linear programming problem maximise

z = x +

Subject to constraints

2x + 3y 16

x + 4y 18

x 0, y 0

whose graph is as follows

Hence, shaded region is feasible region.

The point of intersection of line 2x + 3y = 16 and x + 4y = 18 is B = (2, 4).

Therefore, the set S of all solutions is line BC, which is a line segment.

9. Which of the following groups has a proper subgroup that is not cyclic?

(a) Z₁₅ × Z₇₇

(b) S₃

(d) (Q, +)

Ans. (d)

Sol. (Q, +) has no proper cyclic subgroup.

10. The value of the integral is

(a) 3

(b) 2

(d) 4

Ans. (b)

Sol.

Changing the order of integration, we get

= (–2)(0 – 1) = 2.

11. Suppose the random variable U has uniform distribution on (0, 1) and X = –2 log U. The density of X is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

12. Let f be an entire function on C such that |f(z)| 100 kg |z| for each z with |z| 2. If f(i) = 2i, then f(1) must be

(a) 2

(b) 2i

(d) Cannot be determined

Ans. (b)

Sol.

13. The number of group homomorphisms from Z₃ to Z₉ is

(a) 4

(b) 5

(d) 6

Ans. (c)

Sol. The number of group homomorphisms from Z₃ to Z₉ is equal to GCD (3, 9) = 3.

14. Let u(x, t) be the solution to the wave equation (x, t), u(x, 0) = cos and (x, 0) = 0. Then, the value of u(1, 1) is

(a) 1

(b) 2

(d) 4

Ans. (a)

Sol. Given wave equation is

g(x) = 0, c = 1

15. Let f(x) = Then,

(a)

(b)

(c)

(d) does not exist

Ans. (a)

Sol.

16. Suppose X is a random variable with p(X = k) = (1 – p)^k p for k {0, 1, 2, ...} and some p (0, 1). For the hypothesis testing problem

H₀ : p =

Consider the test "Reject H₀, if X A or if X B", where A < B are given positive integers. The type I error of this test is

(a) 1 + 2^–B – 2^–A

(b) 1 – 2^–B + 2^–A

(d) 1 – 2^–B + 2^–A–1

Ans. (a)

Sol.

17. Let G be a group of order 231. The number of elements of order 11 in G is

(a) 11

(b) 12

(d) 13

Ans. (c)

Sol. We have, 0(G) = 231 = 3 × 7 × 11.

So, number of elements of order 11 in G = = 110.

18. Let f : R² R² be defined by f(x, y) = (e^{x + y}, e^{x – y}). The area of the image of the region {(x, y) R² : 0 < x, y < 1} under the mapping f is

(a) 1

(b) e – 1

(d)

Ans. (d)

Sol. Let f : R² R² be defined by

f(x, y) = (e^{x + y}, e^{x – y})

The region {(x, y) R² : 0 < x, y < 1} is as follows

Now, f(0, 0) = (e^{0 + 0}, e^{0 – 0}) = (1, 1)

f(1, 0) = (e^{1 + 0}, e^{1 – 0}) = (e, e)

f(1, 1) = (e^{1 + 1}, e^{1 – 1}) = (e², 1)

f(0, 1) = (e^{0 + 1}, e^{0 – 1}) =

19. Which of the following is a field?

(a) C(x)/(x² + 2)

(b) Z(x)/(x² + 2)

(d) R(x)/(x² – 2)

Ans. (c)

Sol. Since, ideal <x² – 2> is maximal ideal in Q[x].

Hence, if field.

20. Let x₀ = 0. Define x_{n + 1} = cos x_n for every n 0. Then,

(a) {x_n} is increasing and convergent

(b) {x_n} is decreasing and convergent

(d) {x_n} is not convergent

Ans. (c)

Sol.

21. Let C be the contour |z| = 2 oriented in the anti-clockwise direction. The value of the integral is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Let I =

The function ze^3/2 is analytic in the circle |z| = 1 except at z = 0.

By the Residue theorem,

where, R is the residue of the function at z = 0.

22. For each > 0, let be a random variable with exponential density . Then, Var

(a) is strictly increasing in

(b) is strictly decreasing in

(d) first increases and then decreases in

Ans. (c)

Sol.

23. Let {a_n} be the sequence of consecutive positive solutions of the equation tan x = x and {b_n} be the sequence of consecutive positive solutions of the equation tan Then,

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For the equation tan x = x, the first solution lies between , the second between (r/z',r/z + 2r) and so on.

Thus, we have a_n

24. Let f be an analyticfunction . Assume that |f(z)| 1 for each z . Then, which of the following is not a possible value of (0)?

(a) 2

(b) 6

(c)

(d)

Ans. (b)

Sol.

Let y = |f(0)|.

Then, |f(0)| 1 – y² by Schwarz-Pick theorem

= e^y(3 – 2y² + y⁴)

Now, e^y(3 – 2y² + y⁴) < 6 for y 6 for y [0, 1]

25. The number of non-isomorphic abelian groups of order 24 is

(a) 2

(b) 1

(d) 4

Ans. (c)

Sol. Factoring 24 into primes, we get 24 = 2³. 3¹

From the fundamental theorem of Abelian groups, there are three non-isomorphic Abelian groups of order 24.

(z_B) × z₃

(z₄ × z₂) × z₃

(z₂ × z₂ × z₂) × z₃

So, the answer is thus 3.

26. Let V be the real vector space of all polynomials in one variable with real coefficients and having degree atmost 20. Define the subspaces

Then, the dimension of W₁ W₂ is

(a) 16

(b) 17

(d) 19

Ans. (c)

Sol. Let V the real vector space of all polynomials in one variable with real coefficients and having degree atmost 20. Define the subspaces

p(3) = 0, p(4) = 0, p(5) = 0, p(7) = 0}

dim(W₁ W₂) = 20 – 6 + 1 = 15.

27. Let f, g : [0, 1] R be defined by

Then,

(a) Both f and g are Riemann integrable

(b) f is Riemann integrable and g is Lebesgue integrable

(d) Neither f nor g is Riemann integrable

Ans. (b)

Sol. Let f, g : [0, 1] R be defined by

Thus, the given function f is discontinuous at the points

Further, the above set of points of discontinuous of f has only one limit point viz. 0. Hence, f is Riemann integrable.

Let P = {a = x₀, x₁,...,x_{i – 1}, x_i,...,x_n = b} be any partition of an internal [a, b] (a b).

Then,

in any subinterval [n_{i – 1}, n_j], it is clear that

M_i = 1, m_i = 0. (where, i = 1, 2, ..., n)

So, f is not integrable on [a, b].

28. Consider the following linear programming problem

Maximize x + 3y + 6z – w

Subject to constriants 5x + y + 6z + 7w 20,

6x + 2y + 2z + 9w 40

and x 0, y 0, z 0, w 0.

Then, the optimal value is

(a) 90

(b) 60

(d) 80

Ans. (b)

Sol. Given linear programming problem is

Max z = x + 3y + 6z – w

Subject to constraints

5x + y + 6z + 7w 20

6x + 2y + 2z + 9w 40

x 0, y 0, 2 0, w 0

Standard form of LPP is

Max Z = x + 3y + 6z – w

Subject to constraints

5x + y + 6z + 7w + S₁ = 20

6x + 2y + 2z + 9w + S₂ = 40

x 0, y 0, z 0, w 0, S₁ 0, S₂ 0.

x = 0, y = 20, z = 0, w = 0

Max(x + 3y + 6z – w) = 3 × 20 = 60

29. Suppose X is a real valued random variable.

Which of the following values cannot be attained by E[X] and E[X²], respectively?

(a) 0 and 1

(b) 2 and 3

(c)

(d) 2 and 5

Ans. (b)

Sol.

30. Which of the following subsets of R³ is not compact?

(a) {(x, y) R²: –1 x 1, y = sin x}

(b) {(x, y) R²: – 1 y 1, y = x⁸ – x³ – 1}

(d)

Ans. (c)

Sol. Subset {(x, y) R² : y = 0, sin 1e^–x) = 0} of 1R² is not compact.

31. Let M be the real vector space of 2 × 3 matrices with real entries. Let T : M M be defined by

The determinant of T is

(a) –1

(b) –2

(d) –4

Ans. (a)

Sol. Let T : M M be linear transformation defined by

where, M be real vector space of 2 × 3 order.

Then,

= –(–1) [–1] [1[–1[–1]]]]

= –1.

32. Let H be a Hilbert space and {e_n : n 1} be an orthonormal basis of H. Suppose T : H H is a bounded linear operator. Which of the following cannot be true?

(a) T(e_n) e₁ for all n 1

(b) T(e_n) = e_{n + 1} for all n 1

(d) T(e_n) = e_{n – 1} for all n 2 and T(e₁) = 0

Ans. (A)

Sol. Let H be a Hilbert space and let {e_n : n ≥ 1} be an orthonormal basis of H. Suppose T : H H is a bounded linear operator. Then,

T(e_n) e₁ for all n 1.

33. The value of is

(a) 0

(b) some c (0, 1)

(d)

Ans. (a)

Sol.

34. Let f : C {3i} C be defined by f(z) = .

Which of the following statements about f is false?

(a) f is conformal on C{3i}

(b) f maps circles in C{3i} onto circles in C

(d) There is no straight line in C{3i} which is mapped onto a straight line in C by f

Ans. (c)

Sol. Let f : C|{3i} C be fined by

f(z) =

Let z be the fixed point.

Then, f(z) = z

35. The matrix A = can be decomposed uniquely into the product A = LU, where

The solution of the system LX = [1 2 2]^t is

(a) [111]^t

(b) [110]^t

(d) [101]^t

Ans. (a)

Sol. We have,

Now, A = LU implies that

implies u₁₁ = 1, u₁₂ = 2, u₁₃ = 0

l₂₁u₁₁ = 1

l₂₁u₁₂ + u₂₂ = 3

u₂₂ = 3 – l₂₁u₁₂ = 3 – 1 × 2 = 1

l₂₁u₁₃ + u₂₃ = 1

u₂₃ = 1 – l₂₁u₁₃ = 1 – 1 × 0 = 1

l₃₁u₁₁ = 0

l₃₁ = 0

l₃₁u₁₂ + l₃₂u₂₂ = 1

l₃₂ =

l₃₁u₁₃ + l₃₂u₂₃ + u₃₃ = 3

u₃₃ = 3 – l₃₁u₁₃ – l₃₂u₂₃

= 3 – 0 × 0 – 1 × 1 = 2

Therefore, the solution is [1 1 1]^t.

36. Let S = Then, the supremum of S is

(a) 1

(b)

(d)

Ans. (a)

Sol. The supremum of S =

37. The image of the region

{z C : Re(z) > Im(z) > 0}

under the mapping is

(a) {w C : Re(w) > 0, Im (w) > 0}

(b) {w C : Re(w) > 0, Im(w) > 0, |w| > 1}

(d) {w C : Im(w) > 0, |w| > 1}

Ans. (c)

Sol. The given region is D = {Z C : Re(z) > Im(z) > 0}

and the mapping is

f(z) = e^z2

Let z = 9 + 10i, then z D

Now, w = f(z) = f(9 + 10i)

= e^{(09 + 10i)2}

= e^{–19 + 180i}

= e^–19.e¹⁸⁰ⁱ

= e^–19 (cos 180 + sin 180i)

= –e^–19

Im W = 0

38. Which of the following groups contains a unique normal subgroup of order four?

(a) Z₂ Z₄

(b) The dihedral group, D₄ of order eight

(d) Z₂ Z₂ Z₂

Ans. (c)

Sol. (a, b, c) { 1, i}, { 1, j}, { 1, k} are normal subgroup of Q₈ of order 4.

39. Let B be a real symmetric positive definite n × n matrix. Consider the inner product on Rⁿ defined by (X, Y) = Y^t BX. Let A be an n × n real matrix and let T : Rⁿ Rⁿ be the linear operator defined by T(X) = AX for all X Rⁿ. If S is the adjoint of T, then S(X) = CX for all X Rⁿ, where C is the matrix

(a) B^–1 A^t B

(b) B A^t B^–1

(d) A^t

Ans. (a)

Sol. Let B be a real symmetric positive definite n × n matrix. Consider the inner product on IRⁿ defined by <X, Y> = y^t BX. Let A be an n × n real matrix and T : IRⁿ IRⁿ be the linear operator defined by T(X) = AX for all X IRⁿ. If S is the adjacent of T. Then, S(X) = CX for all X 1Rⁿ, where C = B^–1A^tB.

40. Let X be an arbitrary random variable that takes values in {0, 1, ..., 10}. The minimum and maximum possible values of the variance of X are

(a) 0 and 30

(b) 1 and 20

(d) 1 and 25

Ans. (b)

Sol.

41. Let M be the space of all 4 × 3 matrices with entries in the finite field of three elements. Then, the number of matrices of rank three in M is

(a) (3⁴ – 3)(3⁴ – 3²)(3⁴ – 3³)

(b) (3⁴ – 1)(3⁴ – 2)(3⁴ – 3)

(d) 3⁴(3⁴ – 1)(3⁴ – 2)

Ans. (a)

Sol. Consider the linear transformation

T : I R³ I R³ defined by

T(x, y, z) = (y, z, 0)

Then, T(y, z, 0) = (z, 0, 0)

and T(z, 0, 0) = (0, 0, 0)

Hence, T³(x, y, z) = 0

but T²(x, y, z) (0, 0, 0)

Now, T(1, 0, 0) = (0, 0, 0)

T(0, 1, 0) = (1, 0, 0)

T(0, 0, 1) = (0, 1, 0)

So, the matrix associated to T is

[T] =

tr(T) = 0, also n m and T is not diagonalisable.

42. Let V be a vectror space of dimension m 2. Let T : V V be a linear transformation such that T^{n + 1} = 0 and Tⁿ 0 for some n 1. Then, which of the following is necessarily true?

(a) Rank (Tⁿ) Nullity (Tⁿ)

(b) tr (T) 0

(d) n = m

Ans.

Sol.

If S has n elements, then n can be 4.

43. Let X be a convex region in the plane bounded by straight lines. Let X have 7 vertices. Suppose f(x, y) = ax + by + c has maximum value M and minimum value N on X and N < M. Let S = {P : P is a vertex of X and N < f(P) < M)}. If S has n elements, then which of the following statements is true?

(a) n cannot be 5

(b) n can be 2

(d) n can be 4

Ans. (d)

Sol. Let X be a convex region in the plane bounded by straight lines. Let X have 7 vertices. Suppose f(x, y) = ax + by + c has maximum value M and minimum value N on X and N < M. Let S = {P : P is a vertex of X and M < f(P) < M}

44. Which of the following statements are true?

P. If f L¹ (R), then f is continuous.

Q. If f L¹ (R) and f(x) exists, then the limit is zero.

R. If f L¹( R), then f is bounded.

S. If f L¹ (R) is uniformly continuous, then f(x) exists and equals to zero.

(a) Q and S

(b) P and R

(d) R and S

Ans. (a)

Sol.

45. Let u be a real valued harmonic function on C.

Let g : R² R be defined by

g(x, y) =

Which of the following statements is true?

(a) g is a harmonic polynomial

(b) g is a polynomial but not harmonic

(d) g is neither harmonic nor a polynomial

Ans. (a)

Sol.

46. Let S = {z C : |z| = 1} with the induced topology from C and let f : [0, 2] S be defined asf(t) = Then, which of the following is true?

(a) K is closed in [0, 2] f(K) is closed in S

(b) U is open in [0, 2] f(U) is open in S

(d) f(Y) is open in S Y is open in [0, 2]

Ans. (a)

Sol. Let S = {Z C : |Z| = 1}

With the induced topology from and let f:[0, 2] S be defined by

f(t) =

Then, k is closed in [0, 2].

Hence, f(k) is closed in S.

47. Assume that all the zeroes of the polynomial a_nxⁿ + a_{n – 1}x^{n – 1} + ... + a₁x + a₀ have negative real parts. If u(t) is any solution to the ordinary differential equation

then is equal to

(a) 0

(b) 1

(d)

Ans. (a)

Sol. Consider the polynomial x² + 2x + 5 = 0

having zeroes x = –1 + 2i whose real part is negative.

Now, corresponding differential equation is

= 0

The auxiliary equation is

48. Which of the following is false?

(a) f is a continuous linear functional

(b)

(d) N¹ N{0}

Ans. (d)

Sol.

49. Which of the following is false?

(a) c₀₀ N

(b) N is closed

(d) c₀₀ = N N¹

Ans. (d)

Sol.

50. Which of the following is not an unbiased estimate of µ?

(a) X₁

(b)

(d)

Ans. (d)

Sol.

51. Consider the problem of estimating µ. the MSE (Mean Square Error) of the estimate

(a) µ²

(b)

(c)

(d)

Ans. (a)

Sol.

52. The value of n_o is

(a) 4

(b) 6

(d) 8

Ans. (a)

Sol.

53. Let 0_q1, ..., q_n0 + 1 be n₀ + 1 distinct points and Y = X\{q₁, ..., q_{n0 + 1}}. Let m be the number of connected components of Y. The maximum possible value of m is

(a) 8

(b) 7

(d) 6

Ans. (a)

Sol.

54. The product W(y₁, y₂) P(x) equals to

(a)