CSIR NET MATHEMATICS (DEC - 2018)
Previous Year Question Paper with Solution.
1.
(i) f(0) = 0
(ii) f(r) =
Then the map f is
(a) one-to-one and onto
(b) not one-to-one, but onto
(c) onto but not one-to-one
(d) neither one-to-one nor onto
Ans. (d)
Sol.
So, f is not onto.
Hence, f is neither one-one nor onto.
Option (d) is correct.
2. Let x be a real number such that |x| < 1. Which of the following is false?
(a)
(b)
(c)
(d)
Ans. (c)
Sol. Let x be a real number such that |x| < 1.
as difference of rationals is again rational and 1/rational is again rational.
3. Suppose the (xn) is a sequence of real numbers satisfying the following. For every ε > 0, there exists n0 such that The sequence {xn} is
(a) bounded but not necessarily Cauchy
(b) Cauchy but not necessarily bounded
(c) convergent
(d) not necessarily bounded
Ans. (d)
Sol.
4.
Then
(a) L = 0 if c > 3
(b) L = 1 if c = 3
(c) L = 2 if c = 3
(d)
Ans. (b)
Sol.
Option (b) is correct.
5. Consider the function tan x on the set
We say that it has a fixed point in S if such that tan x = x. Then
(a) there is a unique fixed point.
(b) there is no fixed point.
(c) there are infinitely many fixed points.
(d) there are more than one but finitely many fixed points.
Ans. (c)
Sol. Consider the function "tan x" on the set
as we draw y = tan x and y = x then all intersection points will be fixed points.
In the set S there are infinitely many fixed points.
Option (c) is correct.
6. Define f(x) = for x > 0. Then f is uniformly continuous
(a)
(b)
(c) on (0, r] for any r > 0
(d) only on intervals of the form [a, b] for
Ans. (b)
Sol.
if we select r > 0 fixed number then
7. Consider the subspaces W1 and W2 and R3 given by
(a) W = span {(0, 1, –1), (0, 1, 1)}
(b) W = span {(1, 0, –1), (0, 1, –1)}
(c) W = span {(1, 0, –1), (0, 1, 1)}
(d) W = span {(1, 0, –1), (1, 0, 1)}
Ans. (a)
Sol.
Hence, option (a) is correct.
8.
If T[C] represents the matrix of T with respect to the basis C then which among the following is true?
(a)
(b)
(c)
(d)
Ans. (c)
Sol.
Option (b) and (d) are incorrect.
Option (c) is correct.
9.
Then which among the following is true?
(a) dim(W1) = 1
(b) dim(W2) = 2
(c)
(d) dim(W1 + W2) = 3
Ans. (c)
Sol.
will contain common point and one free variable is common in both W1 and W i.e., W.
Option (c) is correct.
10. Let A be an n × n complex matrix. Assume that A is self-adjoint and let B denote the inverse of A + iIn. Then all eigen values of (A – iIn)B are
(a) purely imaginary
(b) of modulus one
(c) real
(d) of modulus less than one
Ans. (b)
Sol. Given A is n × n complex & self adjoint matrix.
i.e., 1 + is of modulus one.
Hence, option (b) is correct.
11. Let (u1, u2, ...., un) be an orthonormal basis of Cn as column vectors. Let M = (u1, ..., uk), N = (uk + 1, ..., un) and P be the diagonal k × k matrix with diagonal entries Then which of the following is true?
(a)
(b)
(c) Rank (M*N) = min (k, n – k)
(d) Rank (MM* + NN*) < n
Ans. (b)
Sol.
Option (c) is wrong.
12.
Which of the following is true?
(a) B is a linear transformation
(b) B is a positive definite bilinear form
(c) B is symmetric but not positive definite
(d) B is neither linear nor bilinear
Ans. (b)
Sol.
13. Let R denote the radius of convergence of the power series . Then
(a) R > 0 and the series is convergent on [–R, R].
(b) R > 0 and the series converges at x = –R.
(c) R > 0 and the series does not converge outside (–R, R).
(d) R = 0
Ans. (c)
Sol. Given series
Option (c) is correct.
14.
Then
(a) Then interior of image (f) is empty.
(b) Image (f) intersects very line passing through the origin.
(c) There exists a disc in the complex plane, which is disjoint from Image (f)
(d) Image (f) contains all its Image (f)
Ans. (b)
Sol.
15. Consider the polynomials p(z), q(z) in the complex variable z and let
(a)
(b)
(c) Ip, 1 = 0 for all polynomials p
(d)
Ans. (c)
Sol.
16. Let be the positively oriented of radius 3 centered at the origin. The value of
for which
is
(a)
(b)
(c)
(d)
Ans. (a)
Sol.
17. The number of group homomorphisms from the alternating group A5 to the symmetric group S4 is
(a) 1
(b) 12
(c) 20
(d) 6
Ans. (a)
Sol.
18. Let p > 23 be a prime number such that the decimal expansion (base 10) of 1/p is periodic with period p – 1 that is with
{for all t and for any m, 1 < m < p – 1,
.
Let (Z/pZ)* denote the multiplicative group of integers modulo p. Then which of the following is correct?
(a)
(b)
(c)
(d)
Ans. (c)
Sol.
19. Given integers a and b, let Na, b denote the number of positive inters k < 100 such that k = a (mod 9) and k = b (mod 11). Then which of the following statements is correct?
(a) Na, b = 1 for all integers a and b.
(b) There exist integers a and b satisfying Na, b > 1.
(c) There exist integers a and b satisfying Na, b = 0.
(d) There exist integers a and b satisfying Na, b = 0 and there exist integers c and d satisfying Nc, d > 1.
Ans. (a)
Sol.
20. Let X be a topological space and U be a proper dense open subset of X. Pick the correct statement from the following:
(a) If X is connected then U is connected.
(b) If X is compact then U is compact.
(c) If X\U is compact then X is compact.
(d) If X is compact then X\U is compact.
Ans. (d)
Sol.
21. If y1(x) and y2(x) are two solutions of the differential equation ,
Then the Wronskian of y1(x) and y2(x) at x = is
(a)
(b) 6
(c) 3
(d)
Ans. (c)
Sol.
22. The critical point (0, 0) for the system
is
(a) stable spiral point
(b) unstable spiral point
(c) saddle point
(d) stable node
Ans. (c)
Sol.
23. Let u(x, t) be a function that satisfies the PDE uxx – utt = ex + 6t, t > 0 and the initial conditions u(x, 0) = sin(x), ut(x, 0) = 0 for very
.
Here subscripts denote partial derivatives corresponding to the variables indicated.
Then the value of is
(a)
(b)
(c)
(d)
Ans. (c)
Sol.
24. Let u(x, t) satisfy the IVP
Then the value of equals
(a) e
(b) p
(c) 1/2
(d) 1
Ans. (c)
Sol.
25. Let f(x) be a polynomial of unknown degree taking the values
All the fourth divided differences are –1/6.
Then the coefficient of x3 is
(a)
(b)
(c) 16
(d) –1
Ans. (a)
Sol.
26. Consider the function defined on y is piecewise C1 and y(0) = y(2) = 0. Let ye be a maximizer of the above functional. Then ye has
(a) a unique corner point
(b) two corner points
(c) more than two corner points
(d) no corner points
Ans. (a)
Sol.
27.
(a)
(b)
(c)
(d) 2e4
Ans. (b)
Sol.
28. Consider the two dimensional motion of a mass m attached to one end of a spring whose other end is fixed. Let k be the spring constant. The kinetic energy T and the potential energy V of the system are given by
Then which of the following statements is correct?
(a) r is an ignorable coordinate.
(b) q is not an ignorable coordinate
(c) remains constant throughout the motion.
(d) remains constant throughout the motion.
Ans. (c)
Sol.
29. Let (un)n > 1 be a sequence of real numbers satisfying the following conditions:
Which of the following statements are necessarily true?
(a)
(b)
(c)
(d)
Ans. (c), (d)
Sol.
For example,
30. Let S be an infinite set. Which of the following statements are true?
(a) If there is an injection from S to N then S is countable
(b) If there is a surjection from S to N then S is countable.
(c) If there is an injection from N to S then S is countable.
(d) If there is a surjection from N to S then S is countable.
Ans. (a), (d)
Sol. Let S be the infinite set.
Therefore, this statement does not imply firmly that S is countable, if may or may not be.
This is also imply, S may be countable or uncountable. So, this is also not a correct option.
31. Let pn denote the nth prime number, when we enumerate the prime numbers in the increasing order. For example, p1 = 2, p2 = 3, p3 = 5 and so on.
Then which of the following are correct?
(a)
(b)
(c)
(d)
Ans. (a), (b), (c), (d)
Sol.
32. For n > 1, consider the sequence of functions fn(x) = on the open interval (0, 1). Consider the statements:
(I) The sequence [fn] converges uniformly on (0, 1).
(II) The sequence [gn] converges uniformly on (0, 1).
Then,
(a) (I) is true
(b) (I) is false
(c) (I) is false and (II) is true
(d) Both (I) and (II) are true
Ans. (b), (c)
Sol.
superimum of will exist when 2nx + 1 will be minimum and
minimum (2nx
i.e., sup Mn will exist when 2n + 1/x will be minimum or 1/x will be minimum on (0, 1) and him
33. Suppose that {fn} is a sequence of continuous real valued functions on [0, 1] satisfying the following:
Then
(a) {gn} is Cauchy with respect to the sup norm.
(b) {gn} is uniformly convergent.
(c) {gn} need not converge pointwise.
(d)
Ans. (a), (b), (d)
Sol.
34. Given f : a strictly increasing function, we put g(x) = f(x) + f(1/x),
Consider a partition P of [1, 2] and let U(P, g) and L(P, g) denote the upper Riemann sum and lower Riemann sum of g. Then
(a) for a suitable f we can have U(P, g) = L(P, g)
(b)
(c)
(d) U(P, g) < L(P, g) for all choices of f
Ans. (a), (b), (c)
Sol.
35. Let f be a real valued continuously differentiable function of (0, 1).
Set is the derivative of f. Let
be two consecutive zeros of f. Which of the following statements are necessarily true?
(a) If g(a) > 0, then g crosses the real line from upper half plane to lower half plane at a
(b) If g(a) > 0, then g crosses the real line from lower half plane to upper half plane at a
(c)
(d)
Ans. (b), (d)
Sol.
36. Let A be an invertible real n × n matrix. Define a function by F(x, y) = (Ax, y) where (x, y) denotes the inner product of x and y. Let DF(x, y) denotes the derivative of F at (x, y) which is a linear transformation from
Then
(a)
(b)
(c)
(d) If x = 0 or y = 0, then DF(x, y) = 0
Ans. (a), (b), (c)
Sol. Given A is n × n invertible matrix
So, when we will differentiable F(x, y) then to find the derivative w.r.t. to x, we will use first result and for y we will use second result.
37. Let be a function given by f(x, y) = (x3 + 3xy2 – 15x – 12y, x + y).
Let : f is locally invertible at (x, y)}. Then
(a)
(b)
(c)
(d)
Ans. (b), (c)
Sol. For locally invertible, we will find point the Jacobian J where J it becomes zero.
That means S will not point (x, y) such that (x – y) = ±1 i.e. two lines y = x ± 1.
Means if (x, y) then we can draw a disc of e-epsilon radius then it will consist infinite point of S.
38. Let X = N, the set of positive integers. Consider the matrices d1, d2 on X given by d1(m, n) = |m – n|, m, n ∊ X
d2(m, n) = , m, n ∊ X
Let X1, X2 denote the metric spaces (X, d1), (X, d2) respectively. Then
(a) X1 is complete
(b) X2 is complete
(c) X1 is totally bounded
(d) X2 is totally bounded
Ans. (a), (d)
Sol.
39. Let be a linear map that satisfies T2 = T – In. Then which of the following are true?
(a) T is invertible
(b) T – In is not invertible
(c) T has a real eigenvalue
(d) T3 = –In
Ans. (a), (d)
Sol.
40. Let M =
Then which of the following are true?
(a) both systems MX = b1 and MX = b2 are inconsistent.
(b) both systems MX = b1 and MX = b2 are consistent.
(c) the system MX = b1 – b2 is consistent.
(d) the systems MX = b1 – b2 is inconsistent.
Ans. (a), (c)
Sol.
In the question; he is talking about the solution of MX = b1, MX = b2 & MX = b1 – b2
For consistent system, we have to check the rank of M with rank of
[M : b1], [M : b2] & [M : b1 – b2]
So, we will try to solve the augmented matrix [M : b1 b2 b1 – b2]
41. Let M = Given that 1 is an eigenvalue of M then which among the following are correct?
(a) The minimal polynomial of M is (X – 1)(X + 4).
(b) The minimal polynomial of M is (X – 1)2(X + 4).
(c) M is not diagonalizable.
(d)
Ans. (b), (c)
Sol.
Suppose other two eigenvalues are 11 and I2 then
Hence, eigenvalue of M are 1, 1 and –4
In Jordan canonical form, we have only one block of order 2 for eigenvalue 1.
So, in minimal polynomial one factor will be (X – 1)2
Complete minimal polynomial will be
m(X) = (X – 1)2(–X + 4)
Option (A) is false and (B) is true.
and every matrix satisfies its ch. polynomial.
42. Let A be a real matrix with characteristic polynomial (X – 1)3. Pick the correct statements from below:
(a) A is necessarily diagonalizable.
(b) If the minimal polynomial of A is (X – 1)3 then A is diagonalizable.
(c) Characteristic polynomial of A2is (X – 1)3.
(d) If A has exactly two Jordan blocks then (A – 1)2is diagonalizable.
Ans. (c), (d)
Sol. Given ChA(X) = (X – 1)3
43. Let P3 be the vector space of polynomials with real coefficients and of degree at most 3. Consider the linear map defined by T(p(x)) = p(x + 1) + p(x – 1)). Which of the following properties does the matrix of T (with respect of the standard basis B = {1, x, x2, x3} of P3} satisfy
(a) det T = 0
(b)
(c)
(d) 2 is an eigenvalue with multiplicity 4
Ans. (d)
Sol.
Form matrix representation, we will find the image of basis elements and try to write them again in their linear combination of basis element of P3.
So, T(1) = 1 + 1 = 2
= 2.1 + 0.x + 0.x2 + 0.x3
T(x) = (x + 1)(x – 1) = 2x
= 0.1 + 2.x + 0.x2 + 0.x3
T(x2) = (x + 1)2 + (x – 1)2
= x2 + 1 + 2x + x2 + 1 – 2x
= 2x2 + 2
= 2.1 + 0.x + 2x2 + 0.x3
T(x3) = (x + 1)3 + (x – 1)3
= x3 + 1 + 3x2 + 3x + x3 – 1 – 3x2 + 3x
= 2x3 + 6x
= 0.1 + 6.x + 0.x2 + 2.x3
44. Let M be an n × n Hermitian matrix of rank k, is an eigenvalue of M with corresponding unit column vector u with
then which of the following are true?
(a)
(b)
(c)
(d)
Ans. (a), (d)
Sol.
45. Define a real valued function B on as follows. If v = (x1, x2), w = (y1, y2) belong to R2 define B(u, w) = x1y1 – x1y2 – x2y1 + 4x2y2. Let v0 = (1, 0) and let
Then W:
(a) is not a subspace of R2.
(b) equals {(0, 0)}.
(c) is the y-axis.
(d) is the line passing through (0, 0) and (1, 1) .
Ans. (d)
Sol.
46. Consider the Quadratic forms
Q1(x, y) = xy
Q2(x, y) = x2 + 2xy + y2
Q3(x, y) = x2 + 3xy + 2y2
on R2. Choose the correct statements from below:
(a) Q1 and Q2 are equivalent
(b) Q1 and Q3 are equivalent
(c) Q2 and Q3 are equivalent
(d) all are equivalent
Ans. (b)
Sol. Given Q1 (x, y) = xy, Q2 (x, y) = x2 + 2xy + y2
Q3(x, y) = x2 + 3xy + 2y2
If we form the matrix for each quadratic form, then it will be
47. Let H denote the upper has plane that is H = {z = x + iy; y > 0}.
For z which of the following are true?
(a)
(b)
(c)
(d)
Ans. (d)
Sol.
48. Let be an analytic function. Then which of the following statements are true?
(a)
(b) If f is onto then the function f(cos z) is onto.
(c) If f is onto then the function f(ez) is onto
(d) If f is one-one then the function f(z4 + z + 2) is one-one.
Ans. (a), (b)
Sol.
49. Consider the entire functions f(z) = 1 + z + z20 and g(z) = ez, Which of the following statements are true?
(a)
(b)
(c) is bounded for every R > 0
(d) is bounded for every R > 0
Ans. (a), (c)
Sol.
50. Which of the following statements are true?
(a) tan z is an entire function.
(b) tan z is a meromorphic function on C.
(c) tan z has an isolated singularity at
(d) tan z has a non-isolated singularity at
Ans. (b), (d)
Sol.
51. Let a1 < a2 < ... < a51 be given distinct natural numbers such that 1 ≤ ai ≤ 100 for all i = 1, 2, ...., 51. Then which of the following are correct?
(a)
(b)
(c)
(d) There exist i < j such that |ai – aj| > 51.
Ans. (a), (b), (c)
Sol.
52. For any group G, let Aut(G) denote the group of automorphisms of G. Which of the following are true?
(a) If G is finite then Aut(G) is finite.
(b) If G is cyclic then Aut(G) is cyclic.
(c) If G is infinite then Aut(G) is infinite.
(d) If Aut(G) is isomorphic to Aut(H) where G and H are two groups, then G is isomorphic ot H.
Ans. (a)
Sol.
53. Let G be a group with the following property:
Given any positive integers m, n and r there exist elements g and k in G such that order (g) = m, order (h) = n and order (gh) = r. Then which of the following are necessarily true?
(a) G has to be an infinite group.
(b) G cannot be a cyclic group.
(c) G has infinitely many cyclic subgroups.
(d) G has to be a non-abelian group.
Ans. (a), (b), (c), (d)
Sol.
54. Let R be the ring C[x]/(x2 + 1). Pick the correct statements from below:
(a) dimCR = 3
(b) R has exactly two prime ideals
(c) R is a UFD
(d) (x) is a maximal ideal of R
Ans. (b)
Sol.
55. Let f(x) = x7 – 105x + 12. Then which of the following are correct?
(a) f(x is reducible over Q.
(b) There exists an integer m such that f(m) = 105.
(c) There exists an integer m such that f(m) = 2.
(d) f(m) is not a prime number for any integer m.
Ans. (d)
Sol.
56.
Pick the correct statements from below:
(a)
(b)
(c)
(d)
Ans. (a), (b), (c), (d)
Sol.
57. Let with the metric
Let f : x
be the function defined by
Choose the correct statements from below:
(a) f is continuous
(b) f is onto
(c) f is one-to-one
(d) f is open
Ans. (a), (b)
Sol.
58. Let A be a subset of R satisfying where for each n > 1, Vn is an open dense subset of R. Which of the following are correct?
(a) A is non-empty set
(b) A is uncountable
(c) A is uncountable
(d) A is dense in R
Ans. (a), (b)
Sol.
59. Three solutions of a certain second order non-homogeneous linear differentiable equation are
Which of the following is/are general solution(s) of the differential equation?
(a) (C1 + 1)y1 + (C2 – C1)y2 – C2y3 where C1 and C2 are arbitrary constants.
(b) C1(y1 – y2) + C2(y2 – y3) where C1 and C2 are arbitrary constants.
(c) C1(y1 – y2) + C2(y2 – y3) + C3(y3 – y1) where C1, C2 and C3 are arbitrary constants.
(d) C1(y1 – y3) + C2(y3 – y2) + y1 where C1 and C2 are arbitrary constants.
Ans. (a), (d)
Sol.
60. The method of variation of parameters to solve the differential equation where
and p(x), q(x), r(x) are non-zero continuous functions on an interval I, seeks a particular solution of the form yp(x) = v1(x)y1(x) + v2(x)y2(x) where y1 and y2 are linearly independent solutions of
and v1(x) and v2(x) are functions to be determined. Which of the following statements are necessarily true?
(a) The Wronskian of y1 and y2 is never zero in I.
(b) v1, v2 and v1y1 + v2y2 are twice differentiable.
(c) v1 and v2 may not be twice differentiable, but v1y1 + v2y2 is twice differentiable.
(d) The solution set of consists of functions of the form ay1 + by2 + yp where
are arbitrary constants.
Ans. (a), (c), (d)
Sol.
61. Consider the eigenvalue problem
Which of the following statements are true?
(a) All eigenvalues are strictly positive.
(b) All eigenvalues are non-negative.
(c) Distinct eigenfunctions are orthogonal in L2[–1, 1].
(d) The sequence of eigenvalues is bounded above.
Ans. (b), (c)
Sol.
62. Consider the IVP:
Then
(a) the solution is singular at (0, 0).
(b) the given space curve is not a characteristic curve at (0, 0).
(c) there is no base-characteristic curve in the (x, t) plane passing through (0, 0).
(d) a necessary condition for the IVP to have a unique C1 solution at (0, 0) does not hold.
Ans. (a), (b), (c), (d)
Sol.
63. Let u(x, t) be a function that satisfies the PDE : ut + uux = 1, t > 0, and the initial condition
Then the IVP has
(a) only one solution.
(b) two solutions.
(c) an infinite number of solutions.
(d) solutions none of which is differentiable on the characteristic base curve.
Ans. (b), (d)
Sol.
64. Let be twice continuously differentiable function with a unique fixed point (fx*) = x*. For given
consider the iteration xn + 1 = f(xn) for n ≥ 0.
(a) If L < 1, then xn converges to x*
(b)
(c) The error en = xn – x* satisfies |en + 1| < L|en|
(d) If ft(x*) = 0, then |en + 1| < C|en|2 for some C > 0
Ans. (a), (c), (d)
Sol.
65. Let u(x) satisfy the boundary value problem
Consider the finite difference approximation to (BVP)
Here, Uj is an approximation to u(xj) where xj = jh, j = 0, ...., N is a partition of [0, 1] with h = 1/N for some positive integer N. Then which of the following are true?
(a) There exists a solution to (BVP)h of the form Uj = arj + b for some and r satisfying (2 + h)r2 – 4r + (2 – h) = 0.
(b) Uj = (rj – 1)/(rN – 1) where r satisfies (2 + h)r2 – 4r + (2 – h) = 0 and
(c) u is monotonic in x.
(d) Uj is monotonic is j.
Ans. (a), (b), (c), (d)
Sol.
66. Consider the functional subject to y(0) = 0, y(1) = 0. A broken extremal is a continuous extremal whose derivative has jump discontinuities at a finite number of points. Then which of the following statements are true?
(a) There are no broken extremals and y = 0 is an extremal.
(b) There is a unique broken extremal.
(c) There exist more than one and finitely many broken extremals.
(d) There exist infinitely many broken extremals.
Ans. (c)
Sol.
67. The extremals of the functional
(a) x6 + 2x3 – 3x2
(b) x5 + 4x4 – 5x3
(c) x5 + x4 – 2x3
(d) x6 + 4x3 – 6x2
Ans. (a)
Sol.
68. If g is the solution of
(a) 2
(b) 4
(c) 6
(d) 8
Ans. (a)
Sol.
69. A characteristic number and the corresponding eigenfunction of the homogeneous Fredholm integral equation with kernel
are
(a)
(b)
(c)
(d)
Ans. (a), (d)
Sol.
70. Consider a point mass of mass m which is attached to a mass-less rigid rod of length a. The other end of the rod is made to move vertically such that its downward displacement from the origin at time t is given by The mass is moving in a fixed plane and its position vector at time t is given by
Then the equation of motion of the point mass is
(a)
(b)
(c)
(d)
Ans. (a)
Sol.