PYQ JUNE 2018 - VedPrep

CSIR NET MATHEMATICS (JUNE - 2018)
Previous Year Question Paper with Solution.

1. Given {a_n}, {b_n} two monotone sequences of real numbers and that is convergent, which of the following is true?

(a)

(b)

(d) At least one of {a_n}, {b_n} is bounded

Ans. (d)

Sol.

2. Let S = {(x, y) : x² + y² =

Here Q is the set of rational numbers and N is the set of positive integers. Which of the following is true?

(a) S is a finite non empty set

(b) S is countable

(d) S is empty

Ans. (b)

Sol.

3. Define the sequence {a_n} as follows:

Which of the following is true?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The given sequence is

Thus a₂, a₄, a₆, ...... are negative term and a₁ = 1, a₅ = 408.71 and a₅ < a₇ < a₉ < a₁₁ ...... and so on.

Hence, option (a) is correct.

4. If {x_n} is a convergent sequence in R and {y_n} is a bounded sequence in R and {y_n} is a bounded sequence in R then we can conclude that

(a) {x_n + y_n} is convergent

(b) {x_n + y_n} is bounded

(d) {x_n + y_n} has no bounded subsequence

Ans. (b)

Sol. Let {x_n} be a convergent sequence in R. Since we know that every convergent sequence is a bounded sequence.

Therefore {x_n} is bounded sequence.

Let {y_n} be a bounded sequence.

We know that sum of two bounded sequence is bounded sequence.

Option (B) is correct.

5. The difference log (2) = is

(a) a less than 0

(b) greater than 1

(d) greater than

Ans. (c)

Sol. We know that

Hence, option (c) is correct.

(a)

(b) 0

(d) cos(x + y)

Ans. (c)

Sol. Given

7. Let A be an (m × n) matrix and B be a (n × m) matrix over real numbers with m < n. Then,

(a) AB is always non-singular

(b) AB is always singular

(d) BA is always singular

Ans. (a)

Sol.

Again, suppose

8. If A is a (2 × 2) matrix over R with det(A + 1) = 1 + det(A) then we can conclude that:

(a) det(A) = 0

(b) A = 0

(d) A is non-singular

Ans. (c)

Sol.

9. The system of equations:

1·x + 2·x² + 3·xy + 0·y = 6

2·x + 1·x² + 3·xy + 1·y = 5

3·x – 1·x² + 0·xy + 1·y = 7

(a) has solution in rational numbers

(b) has solutions in real numbers

(d) has no solution

Ans. (d)

Sol. The given system of equation is

1·x + 2·x² + 3·xy + 0·y = 6

2·x + 1·x² + 3·xy + 1·y = 5

1·x – 1·x² + 0·xy + 1·y = 7

which is a non-homogeneous system by equations.

The augmented matrix is

10. The trace of the matrix is

(a) 7²⁰

(b) 2²⁰ + 3²⁰

(d) 2²⁰ + 3²⁰ + 1

Ans. (c)

Sol. Given matrix is

Alternative Method: Given matrix is

11. Given that there are real constants a, b, c, d such that the identity = (ax + by)² + (cx + dy)² holds for all This implies

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have

12. Let be equipped with standard inner product. Let {v₁, v₂, ..., v_n} be n column vectors forming an orthonormal basis of Rⁿ. Let A be the n × n matrix formed by the column vectors v₁, ... v_n. Then

(a) A = A^–1

(b) A = A^T

(d) Det(A) = 1

Ans. (c)

Sol.

13. Suppose the f is non-constant analytic function defined over C. Then which one of the following is false?

(a) f is unbounded.

(b) f sends open sets into open sets.

(d) The image of f is dense in C.

Ans. (c)

Sol.

14. The value of the integral is

(a) 0

(b)

(c)

(d) (e + e^–1)

Ans. (b)

Sol. We have

15. Let be a non-constant analytic function. Which of the following conditions can possible be satisfied by f?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

16. Consider the map

Which of the following is true?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

17. Let S₇ denote the group of permutations of the set {1, 2, 3, 4, 5, 6, 7}. Which of the following is true?

(a) There are no elements of order 6 in S₇

(b) There are no elements of order 7 in S₇

(d) There are no elements of order 10 in S₇

Ans. (c)

Sol.

18. The number of group homomorphisms from Z₁₀ to Z₂₀ is

(a) 0

(b) 1

(d) 10

Ans. (d)

Sol. The number of group homomorphisms from Z₁₀ to Z₂₀

= G.C.D. (10, 20)

= 10

Option (D) is correct.

19. Let f(x) = x⁵ – 5x + 2. Then

(a) f has no real root

(b) f has exactly one real root

(d) all roots of f are real

Ans. (c)

Sol. We have f(x) = x⁵ – 5x + 2 which is odd degree polynomial with rational coefficients. Hence, f has real roots.

Option (A) is incorrect.

Now, f(x) = x⁵ – 5x + 2 has two sign change.

Hence it has two positive real roots and f(–x) = –x⁵ + 5x + 2 has one sign change. Hence it has one negative roots are complex roots. Hence, f has exactly 3 real roots.

Hence, option (C) is correct.

20. Consider the space where Q is the set of rational numbers. Then

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

21. Consider the ordinary differential equation

Which of the following statements is true?

(a) If y(0) = 0.5 then y is decreasing

(b) If y(0) = 1.2 then y is increasing

(d) If y(0) < 0 then y is bounded below

Ans. (c)

Sol. The given ordinary differential equation

On integrating, we get

which shows that y is unbounded.

Option (c) is correct.

22. Consider the ordinary differential equation where P and Q are smooth functions. Let y₁ and y₂ be any two solutions of the ODE. Let W(x) be the corresponding Wronskian. Then which of the following is always true?

(a) If y₁ and y₂ are linearly dependent then such that W(x₁) = 0 and

(b) If y₁ and y₂ are linearly independent then

(d) If y₁ and y₂ are linearly independent then

Ans. (d)

Sol. The given differential equation is P(x) and Q(x) are smooth function.

Let y₁ and y₂ be any two solutions of the ODE. Let W(x) be the corresponding Wronskian. Then

Where c is the constants depends on y₁ and y₂ but not on x.

Further W(y₁, y₂)(x) is either zero for all x (if c = 0) or else never zero

If y₁ and y₂ are linearly dependent then W(y₁, y₂)(x) = 0 but converse need not be true.

If then y₁ and y₂ are linearly independent.

Hence, option (D) is correct.

23. The Cauchy Problem has

(a) exactly one solution

(b) exactly two solution

(d) no solution

Ans. (d)

Sol. The Cauchy problem is 2u_x + 3u_y = 5 where u = 1 on the line 3x – 2y = 0.

24. Let u be the unique solution of

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Let u be the unique solution of

Option (c) is correct.

25. The value of a, b, c such that

is exact for polynomial f of degree as high as possible are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We have

which is exact for polynomials f of degree as high as possible.

On solving equations (1), (2) and (3), we get a = 0,

Option (a) is correct.

26. Consider subject to y(0) = 0, y(1) = 1. Then inf J[y] is

(a)

(b)

(c)

(d) does not exist

Ans. (a)

Sol.

27. The resolvent kernel for the integral equation

(a) e^{t – x}

(b) 1

(d) x² + e^{x – t}

Ans. (b)

Sol. The given Voltera integral equation is which is homogeneous.

Here

In similar manner, we get

Hence, option (b) is correct.

28. Given that the Lagrangian for the motion of a simple pendulum is where m is the mass of the pendulum bob suspended by a string of length l, g the acceleration due to gravity and is the amplitude of the pendulum from the mean position, then Hamiltonian corresponding to L is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The Lagrangian for the motion of a simple pendulum is

where m is the mass of the pendulum bob suspended by a string of length l, g is the acceleration due to gravity and  is the amplitude of the pendulum from the mean position.

Option (b) is correct.

29. Which of the following sets are uncountable?

(a) The set of all functions from R to {0, 1}.

(b) The set of all functions from N to {0, 1}.

(d) The set of all subsets of N.

Ans. (a), (b), (d)

Sol.

30. Let A = Which of the following statements are true?

(a)

(b)

(d) inf(A) = –1

Ans. (a), (b)

Sol.

31. Let | f is continuous and there exists a compact set K such that f(x) = 0 for all Which of the following statements are true?

(a) There exists a sequence {f_n} in

(b) There exists a sequence {f_n} in

(d) There does not exist any sequence in converging pointwise to g

Ans. (a), (b)

Sol.

32. Given that

Which of the following statements are true?

(a) a(n) > c(n) for all sufficiently large n

(b) b(n) > c(n) for all sufficiently large n

(d) a(n) > b(n) for all sufficiently large n

Ans. (a), (d)

Sol.

33.

Which of the following are true?

(a) f is uniformly continuous on compact intervals of R for all values of a and b.

(b) f is uniformly continuous on R and is bounded for all values of a and b.

(d) f is uniformly continuous on R and unbounded if

Ans. (a), (b)

Sol.

34.

Which of the following are true?

(a)

(b) a is a rational number

(d) sin(a) = 1

Ans. (d)

Sol. We have

35. Which of the following functions are of bounded variation?

(a)

(b)

(c)

(d)

Ans. (a), (c), (d)

Sol. Option (a): The given function is

Here f(x) = x² + x + 1 is continuously differentiable function with a bounded derivative on (–1, 1). Hence f(x) = x² + x + 1 is a function of bounded variation in (–1, 1).

Option (A) is correct.

Option (C):The given function is

Here f(x) = sin is continuously differentiable function with a bounded derivative on

Hence f(x) = sin is a function of bounded variation in

Option (c) is correct.

Option (d): The given function is

The graph of f(x) is as follows

from graph it is clear that the function f(x) = is monotonic function in (–1, 1).

Hence f(x) = is a function of bounded variation in (–1, 1).

Hence, option (d) is correct.

36. Let M_n(R) denote the space of all n × n real matrices identified with the Euclidean space Fix a column vector Define by f(A) = (A²x, x). Then

(a) f is linear

(b) f is differentiable

(d) f is unbounded

Ans. (b), (d)

Sol.

37. For any let [y] denote the greatest integer less than or equal to y.

Define by f(x, y) = x^[y] then

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

38. Let V denote the vector space of all sequences a = (a₁, a₂, ...) of real numbers such that S2ⁿ|a_n| converges. Define

Which of the following are true?

(a) V contains only the sequence (0, 0, ...)

(b) V is finite dimensional

(d) V is a complete normed space

Ans. (d)

Sol.

39. Let V be a vector space over C with dimension n. Let be a linear transformation with only I as eigenvalue. Then, which of the following must be true?

(a) (T – I) = 0

(b) (T – I)^{n – 1} = 0

(d) (T – I)²ⁿ = 0

Ans. (c), (d)

Sol.

40. If A is a (5 × 5) matrix and the dimension of the solution space of Ax = 0 is at least two then

(a)

(b)

(d) Det(A²) = 0

Ans. (a), (d)

Sol. Here A is a 5 × 5 matrix and the dimension of the solution space of Ax = 0 is at least two.

41.

(a) minimal polynomial of A can only be of degree 2

(b) minimal polynomial of A can only be of degree 3

(d) there are uncountably many A satisfying the above

Ans. (d)

Sol.

Then A is involuntary matrix.

Then A⁸ = (A²)⁴ = I⁴ = I.

But involuntary matrix are uncountably many in M₃(R).

Hence, option (d) is correct.

42. Let A be an n × n matrix (with n > 1) satisfying A² – 7A + 12 I_{n × n} = O_{n × n} where I_{n × n} and O_{n × n} denote the identify matrix and zero matrix or order n respectively. Then which of the following statements are true?

(a) A is invertible

(b) t² – 7t + 12n = 0 where t = tr(A)

(d)

Ans. (a), (d)

Sol. Let A be a 6 × 6 matrix over R with characteristic polynomial (x – 3)²(x – 2)⁴ and minimal polynomial (x – 3)(x – 2)².

Here 3, 3, 2, 2, 2, 2 are roots of characteristic polynomial.

2 is algebraic multiplicity of 3, and 4 is algebraic multiplicity of 2.

Here 3, 2, 2 are roots of minimal polynomial.

1 is geometric multiplicity of 3 and 2 is geometric multiplicity of 2.

Jordan canonical form are

Hence, option (b) and (c) are correct.

43. Let A be a (6 × 6) matrix over R with characteristic polynomial = (x – 3)²(x – 2)⁴ and minimal polynomial = (x – 3)(x – 2)². Then Jordan canonical form of A can be

(a)

(b)

(c)

(d)

Ans. (b), (c)

Sol.

44. Let V be an integer product space and S be a subset of V. Let denote the closure of S in V with respect to the topology induced by the metric given by the inner product. Which of the following statements are true?

(a)

(b)

(c)

(d)

Ans. (c), (d)

Sol.

45.

Which of the following statements are true?

(a) The matrix of second order partial derivatives of the quadratic form Q is 2A.

(b) The rank of the quadratic form Q is 2.

(d) The quadratic form Q takes the value 0 for some non-zero vector (x, y, z).

Ans. (d)

Sol.

46.

(a) The Lebesgue measure of E is infinite.

(b) E contains a non-empty open set.

(d) Every open set containing E^c has infinite Lebesgue measure.

Ans. (a)

Sol.

Then the Lebesgue measure of is infinite.

Hence, option (a) is correct.

47. Which of the following statements are true?

(a) If {a_k} is bounded then defines an analytic function on the open unit disk.

(b) If defines an analytic function on the open unit disk then {a_k} must converge to zero.

(c) If f(z) = are two power series functions whose radii of convergence are 1 then the product f.g has a power series representation of the form on the open unit disk.

(d) If f(z) = has a radius of convergence 1 then f is continuous on .

Ans. (a), (c)

Sol.

48. Let D be the open unit disk centered at 0 in C and be an analytic function. Let f = u + iv where u, v are the real and imaginary parts of f. If f(z) = is the power series of f then f is constant if

(a)

(b)

(c)

(d) For any closed curve 1 in D,

Ans. (a), (b), (d)

Sol. Let D be the open unit disk centered at O in C and be an analytic function. Let f = u + i where u, v are the real and imaginary parts of f. If f(z) = is the power series of f. Then domain D is symmetric.

Option (a) is correct.

Since f is analytic function. Therefore u and v are harmonic function. Then by maximum modulus theorem.

Hence, option (b) is correct.

Also cos x is not a constant function.

Option (c) is incorrect.

49. Suppose that is an analytic function. Then f is a polynomial if

(a) for any point is a power series expansion at a then a_n = 0 for at least on n.

(b)

(c)

(d)

Ans. (a), (b), (c), (d)

Sol.

is a power series expansion at a then there exists atleast one such that a_n = 0

Therefore f is polynomial.

Hence, option (a) is correct.

must have pole at z = 0

50. Let be an open connected subset of C. Let E = is a function such that is analytic. Then f is analytic on if

(a)

(b)

(d) for every j, if is Laurent series expansion of f at z_j then a_–1 = 0

Ans. (a), (c)

Sol.

51. Let S be the set of polynomial f(x) with integer coefficients satisfying

Which of the following statements are true?

(a) S is empty

(b) S is a singleton

(d) S is countably infinite

Ans. (a)

Sol.

52. Let G = S₃ be the permutation group of 3 symbols. Then

(a) G is isomorphic to a subgroup of a cyclic group

(b) there exists a cyclic group H such that G maps homomorphically onto H

(d) homomorphism from G to the additive group (Q, +) of rational numbers

Ans. (b)

Sol. Here, G = S₃ = Permutation group of 3 symbols.

Which is non-abelian group. We know that every subgroup of cyclic group is cyclic.

Hence, S₃ is non isomorphic to a subgroup of a cyclic group.

Option (A) is incorrect.

Hence, there exists a cyclic group H such that G maps homomorphically onto H.

Hence option (B) is correct.

Which is possible but it is trivial.

Hence there does not exists any non-trivial group homomorphism from G to the additive group (Q, +) of rational numbers.

Hence, the (D) is incorrect.

53. Let G be a group with |G| = 96. Suppose H and K are subgroups of G with |H| = 12 and |K| = 16. Then

(a)

(b)

(c)

(d)

Ans. (b), (c)

Sol. Let G be a group with |G| = 96. Suppose H and K are subgroups of G with |H| = 12 and |K| = 16.

Let G be an abelian group.

Then H and K are normal subgroup of G.

Then HK < G.

54. Which of the following statements are true?

(a) A subring of an integral domain is an integral domain

(b) A subring of a unique factorization domain (UFD) is a UFD

(d) A subring of an Euclidean domain is an Euclidean domain

Ans. (a)

Sol. We know that a subring may not have unit of either unique factorization domain (UFD), principal ideal domain (PID) or euclidean domain (ED).

A subring of an integral domain is an integral domain.

Option (A) is correct.

55.

(a)

(b)

(c)

(d)

Ans. (a), (c)

Sol.

Hence option (a) is correct but option (d) is incorrect.

Now take monic polynomial

Hence, option (b) is incorrect.

Option (c) is correct.

56. Which of the following statements are true?

(a) The multiplicative group of a finite field is always cyclic

(b) The additive group of a finite field is always cyclic

(d) There exists at most one finite field (upto isomorphism) of any given order

Ans. (a), (d)

Sol. We know that the multiplicative group of a finite field always cyclic.

Hence, option (A) is correct.

But additive group of a finite field is not cyclic.

For example, |F| = pⁿ

Now we cannot find any field or order 10.

Hence option (C) is incorrect.

In option (D), there exist 'at most' one finite field (upto isomorphism) of any given order.

For any given order either there does not exist field or exists uniquely (upto isomorphism).

Hence, option (D) is correct.

57. Let X be a topological space and A be a non-empty subset of X. Then one can conclude that

(a) A is dense in X, if (X\A) is nowhere dense in X

(b) (X\A) is nowhere dense in X, if A is dense in X

(d) the interior of (X\A) is empty, if A is dense in X

Ans. (a), (c), (d)

Sol. Let X be topological space and A be non-empty subset of X.

Then,

(i) A is dense in X, if (X\A) is nowhere dense in X.

(ii) A is dense in X, if the interior of (X\A) is empty.

(iii) The interior of (X\A) is empty, if A is dense in X.

Hence, options (A), (C) and (D) are correct.

58. Which of the following statements are true?

(a) Every compact metric space is separable

(b) If a metric space (X, d) is separable, then the metric d is not the discrete metric

(d) Every first countable topological space is separable

Ans. (a), (c)

Sol.

59. Consider the Sturm-Lioville problem

Which of the following statements are true?

(a) There exist only countably many characteristic values

(b) There exist uncountably many characteristic values

(d) Each characteristic function corresponding to the characteristic value has exactly zeros in

Ans. (a), (c)

Sol. The given Sturm-Liouville problem is

We know that the characteristic value of Sturm-Liouville problem is always real.

where A and B are arbitrary constants.

which is again a trivial solution.

which is a characteristic function corresponding to characteristic value

But Z is countable.

Hence, there exist only countably many characteristic values.

Option (A) is correct.

But option (B) is incorrect.

Now we have y_n(x) = B_n sin nx.

For n = 1, y₁(x) = sin x, B₁ = 1

which has zero solution is

i.e.,

Similarly, we can verify for other values of n.

Each characteristic function corresponding to the characteristic value l has exactly zero in

Hence, option (C) is correct but option (D) is incorrect.

60. Consider the system of differential equations

Then the critical point (0, 0) of the system is an:

(a) asymptotically stable node

(b) unstable node

(d) unstable spiral

Ans. (a)

Sol. The given system of equation is

We know that the critical points of system of equations

are given by P(x₀, y₀) = 0 = Q(x₀, y₀).

Now, Coefficient matrix is

Now since

Therefore it is asymptotically stable nodes.

Hence, option (a) is correct but option (b) is incorrect.

61. Assume that is continuous function. Consider the ordinary differential equation

Which of the following statements are true?

(a)

(b)

(c)

(d)

Ans. (a), (b), (c)

Sol.

62. If u(x, t) is the solution of

then

(a)

(b)

(c)

(d)

Ans. (a), (b), (c), (d)

Sol.

where u(x, t) is the solution of the above equation.

63. Let a be a fixed real constant. Consider the first order partial differential equation R, t > 0 with the initial data u(x, 0) = u₀(x), where u₀ is a continuously differentiable function. Consider the following two statements:

S₁: There exists a bounded function u₀ for which the solution u is unbounded.

S₂: If u₀ vanishes outside a compact set then for each fixed T > 0 there exists a compact set K_T R such that u(x, T) vanishes for

(a) S₁ is true and S₂ is false

(b) S₁ is true and S₂ is also true

(d) S₁ is true and S₂ is also false

Ans. (c)

Sol.

64. Assume that a non-singular matrix A = L + D + U where L and U are lower and upper triangular matrices respectively with all diagonal entries are zero and D is a diagonal matrix. Let x* be the solution of Ax = b. Then the Gauss-Seidel iteration method x^{(k + 1)} = Hx^(k) + c, k = 0, 1, 2, ... with ||H|| < 1 converges to x* provided H is equal to

(a) –D^–1(L + U)

(b) –(D + L)^–1U

(d) –(L – D)^–1U

Ans. (b)

Sol.

65. The forward difference operator is defined as DU_n = U_{n + 1} – U_n. Then which of the following difference equations has an unbounded general solution?

(a)