CSIR Mathematical Sciences solved questions for Paper-1

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Let K be an extension of the field Q of rational numbers
1 If K is a finite extension then it is an algebraic extension
2 If K is an algebraic extension then it must be a finite extension
3 If K is an algebraic extension then it must be an infinite extension
4 If K is a finite extension then it need not be an algebraic extension

Let A, B and C be real n * n matrices such that AB +B2 = C. Suppose C is nonsingular. Which of the following is always true?
1. A is nonsingular
2. B is nonsingular
3. A and B are both nonsingular
4. A + B is nonsingular.

A simple symmetric random walk on the integer line is a Markov chain which is
1. recurrent
2. null recurrent
3. irreducible
4. positive recurrent.

21. The sequence an = 222111...(1)(2)nnn1 converges to 0 2 converges to 1/2 3 converges to 1/4 4 does not converge.

22. Let xn = n 1/n and yn = (n!)1/n , n1 be two sequences of real numbers. Then 1 (xn) converges, but (yn) does not converge 2 (yn) converges, but (xn) does not converge
3 both (xn) and (yn) converge 4 Neither (xn) nor (yn) converges

23. The set { x : x sin x 1 , x cos x 1 } is 1 a bounded closed set 2 a bounded open set 3 an unbounded closed set. 4 an unbounded open set.

24. Let f:[0,1] be continuous such that f(t) 0 for all t in [0, 1]. Define g(x) =0()xftdtthen 1 g is monotone and bounded 2 g is monotone, but not bounded 3 g is bounded, but not monotone 4 g is neither monotone nor bounded

25. Let f be a continuous function on [0, 1] with f(0) =1. Let G(a) = 01()afxdxa1 01lim()2aGa2 0lim()aGa1 3 0lim()aGa0 4 The limit 0lim()aGadose not exist

26. Let n = sin (21n) , n = 1,2, …. Then 1 1nnconverges 2 limsupliminfnnnn3 lim1nn4 1nndiverges

27. If, for x , φ(x) denotes the integer closest to x (if there are two such integers take the larger one), then 1210()xdxequals 1 22 2 11 3 20 4 12

28. Let P be a polynomial of degree k > 0 with a non-zero constant term. Let fn(x) = P(xn) x (0,∞) 1 lim()nnfxx (0, ∞) 2 x (0, ∞ ) such that lim()nnfx> P(0) 3 lim()nnfx=0 x (0, ∞) 4 lim() n
n
f x
= P(0) x (0, ∞)

29. Let C [0, 1] denote the space of all continuous functions with supremum norm. Then, 1 [0,1]:lim0nKffn®¥ìüæö=Î=íýç÷èøîþis a 1. vector space but not closed in C[0,1]. 2. closed but does not form a vector space. 3. a closed vector space but not an algebra. 4. a closed algebra.

30. Let u, v, w be three points in 3 not lying in any plane containing the origin. Then 1 1 u + 2 v + 3 w = 0 => 1 = 2 = 3 = 0 2 u, v, w are mutually orthogonal 3 one of u, v, w has to be zero 4 u, v , w cannot be pairwise orthogonal

31. Let x, y be linearly independent vectors in 2 suppose T: 2 2 is a linear transformation such that Ty = x and Tx =0 Then with respect to some basis in 2 , T is of the form 1 00aa, a > 0 2 00ab, a , b > 0; a b 3 01004 0000

32. Suppose A is an n x n real symmetric matrix with eigenvalues 12,,...,nthen 1 1det()niiA2 1det()niiA3 1det()niiA4 det()1ifAthen 1jfor j =1, … n.

33. Let f be analytic on D = { z : |z | < 1} and f(0) =0 . Define ();0()(0);0fzzgzzfzT hen 1 g is discontinuous at z = 0 for all f 2 g is continuous, but not analytic at z = 0 for all f 3 g is analytic at z = 0 for all f 4 g is analytic at z = 0 only if f ' (0) = 0

34. Let be a domain and let f(z) be an analytic function on such that |f(z)| = | sin z | for all z then 1 f(z) = sin z for all z 2 f(z) = sin z for all z . 3 there is a constant c with |c| = 1 such that f(z) = c sin z for all z
4 such a function f(z) does not exist

35. The radius of convergence of the power series 430(43)nnnzn is 1 0 2 1 3 5 4 ∞

36. Let be a finite field such that for every a the equation x2 =a has a solution in . Then 1 the characteristic of must be 2 2 must have a square number of elements 3 the order of is a power of 3 4 must be a field with prime number of elements 37. Let be a field with 512 elements. What is the total number of proper subfields of ? 1 3 2 6 3 8 4 5

38. Let K be an extension of the field Q of rational numbers 1 If K is a finite extension then it is an algebraic extension 2 If K is an algebraic extension then it must be a finite extension 3 If K is an algebraic extension then it must be an infinite extension 4 If K is a finite extension then it need not be an algebraic extension

39. Consider the group S9 of all the permutations on a set with 9 elements. What is the largest order of a permutation in S9 ? 1 21 2 20 3 30 4 14

40. Suppose V is a real vector space of dimension 3. Then the number of pairs of linearly independent vectors in V is 1 one 2 infinity 3 e3 4 3

41. Consider the differential equation 2,(,)dyyxydx. Then, 1. all solutions of the differential equation are defined on (–,). 2. no solution of the differential equation is defined on (–,). 3. the solution of the differential equation satisfying the initial condition y(x0) = y0, y0 > 0, is defined on 001,xy. 4. the solution of the differential equation satisfying the initial condition y(x0)=y0, y0>0, is defined on 001,xyæö-¥ç÷èø.

42. The second order partial differential equation 222221210uuuxyxyxxyy is 1. hyperbolic in the second and the fourth quadrants 2. elliptic in the first and the third quadrants
3. hyperbolic in the second and elliptic in the fourth quadrant 4. hyperbolic in the first and the third quadrants

43. A general solution of the equation (,)(,)xuxyuxyex-¶+=¶is 1. (,)f()xuxyey2. (,)f()xxuxyeyxe3. (,)f()xxuxyeyxe4. (,)f()xxuxyeyxe

44. Consider the application of Trapezoidal and Simpson‟s rules to the following integral 4320(2351)xxxdx1. Both Trapezoidal and Simpson‟s rules will give results with same accuracy. 2. The Simpson‟s rule will give more accuracy than the Trapezoidal rule but less accurate than the exact result. 3. The Simpson‟s rule will give the exact result. 4. Both Trapezoidal rule and Simpson‟s rule will give the exact results.

45. The integral equation βαg(x)y(x)=f(x)+λk(x,t)y(t)dtwith f(x), g(x) and k(x,t) as known functions, α and β as known constants, and λ as a known parameter, is a 1. linear integral equation of Volterra type 2. linear integral equation of Fredholm type 3. nonlinear integral equation of Volterra type 4. nonlinear integral equation of Fredholm type

46. Let bay(x) = f(x)+λk(x,t)y(t)dt, where f(x) and k(x,t) are known functions, a and b are known constants and λ is a known parameter. If λi be the eigenvalues of the corresponding homogeneous equation, then the above integral equation has in general, 1. many solutions for λ≠λi 2. no solution for λ≠λi 3. a unique solution for λ=λi 4. either many solutions or no solution at all for λ=λi, depending on the form of f(x)

47. The equation of motion of a particle in the x-z plane is given by ˆdvvkdt
with ˆvk, where = (t) and ˆkis the unit vector along the z-direction. If initially (i.e., t = 0) = 1, then the magnitude of velocity at t = 1 is 1. 2/e 2. (2+e)/3 3. (e–2)/e 4. 1

48. Consider the functional 22/20(,)2()()dudvFuvuxvxdxdxdx with (0)1,(0)1uvand 0,022uv . Then, the extremals satisfy 1. ()1,()1uv2. ()()0,()()2uvuv3. ()1,()1uvpp=-= 4. ()()2,()()0uvuv

49. The pairs of observations on two random variables X and Y are :257111319:01525455585XY Then the correlation coefficient between X and Y is 1 0 2 1/5 3 1/2 4 1

50. Let X1, X2, X3 be independent random variables with P(Xi = +1) = P(Xi = -1) = 1/2. Let Y1 = X2X3, Y2 = X1X3 and Y3 = X1X2. Then which of the following is NOT true? 1. Yi and Xi have same distribution for i = 1, 2, 3 2. (Y1, Y2, Y3) are mutually independent 3. X1 and (Y2, Y3) are independent 4. (X1, X2) and (Y1, Y2) have the same distribution

51. Let X be an exponential random variable with parameter. Let Y = [X] where [x] denotes the largest integer smaller than x. Then 1. Y has a Geometric distribution with parameter. 2. Y has a Geometric distribution with parameter 1.el-- 3. Y has a Poisson distribution with parameter 4. Y has mean [1/]

52. Consider a finite state space Markov chain with transition probability matrix P=((pij)). Suppose pii =0 for all states i. Then the Markov chain is 1. always irreducible with period 1. 2. may be reducible and may have period > 1. 3. may be reducible but period is always 1. 4. always irreducible but may have period > 1.

53. Let X1, X2, …. Xn be i.i.d. Normal random variables with mean 1 and variance 1. and let Zn = (X21+X2 +…. +Xn )/n Then 1. Zn converges in probability to 1 2. Zn converges in probability to 2 3. Zn converges in distribution to standard normal distribution 4. Zn converges in probability to Chi-square distribution.

54. Let X1, X2, …. Xn be a random sample of size n (4) from uniform (0,) distribution. Which of the following is NOT an ancillary statistic? 1. ()(1)nXX 2. 1nXX 3. 4132XXXX4. ()(1)nXX
55. Suppose X1, X2, … Xn are i.i.d, Uniform (0,),q {1,2....}. Then the MLE of is 1. X(n) 2. X 3. [X(n)] where [a] is the integer part of a. 4. [X(n)+1] where [a] is the integer part of a.

56. Let X1, X2, …., Xn be independent and identically distributed random variables with common continuous distribution function F(x). Let Ri = Rank(Xi), i= 1, 2, …, n. Then P ( | Rn – R1 | ³ n-1) is 1. 0 2. 1(1)nn3. 2(1)nn4. 1n

57. A simple random sample of size n is drawn without replacement from a population of size N (> n). If i(i=1,2,…N) and ijp (ij. i, j =1, 2, … N) denote respectively, the first and second order inclusion probabilities, then which of the following statements is NOT true? 1 1Niinp==å 2 (1)Nijijinpp¹=-å 3 ijijppp£for each pair ( i, j) 4 ijipp< for each pair ( i, j) .

Rest of the Questions are attached in the below file which is free of cost

Last edited by Aakashd; March 8th, 2020 at 05:59 PM.