1. Which of the following sets is uncountable?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    We have, A set S is countable either, if it is finite or if is infinite.

    

    

    Logarithmic function is increasing function and so there is one to one corresponding between the given set and the set of rational number.

    Therefore, the given set is countable set.

    Option (B):

    For n = 2, (cos(x))² + (sin(x))² = 1

    It is true for all real number.

    Since the set of real number is uncountable set.

    

    

    Exponential function is increasing function and so there is one to one corresponding between the given set and set of rational number.

    Thus, the given set is countable set.

    

    Cosine function is periodic function and within in period there is a one to one corresponding between the given set and the set of rational number and it can be extended on whole real line.

    

    Since the set of rational number is countable. Thus, option (B) is correct.

2. Consider a sequence

    Let

    Then which of the following is true?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

    

    

    

    

    Therefore, option (a) and (d) are not correct.

    

    Hence, option (b) is correct.

3. Which of the following is true?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Test of convergence

    (i) We have test of convergence

    

    If finite and non-zero then both the converge or diverge simultaneously.

    (ii) p-test for positive term series: The series is convergent series, if p > 1 otherwise (p < 1) it is divergent series.

    (iii) Alternating series test/Leibnitz's test: The alternating is convergent if |a_n| is monotonically decreasing i.e.,

    Option (A): Alternating series test/Leibniz's test |a_n| is monotonically decreasing i.e., |a_{n + 1} | < |a_n|

    

    Therefore, the given series is converge.

    

    By the test, the given series is diverge.

    For option (C) and option (D):

    

    

    

    

    Hence, option (d) is correct.

4. For which of the following is true?

    (a) for all except possibly finitely many n.

    (b) for all except possibly finitely many n.

    (c) for all except possibly finitely many n.

    (d) for all except possibly finitely many n.

Ans.    (b)

Sol.    

    

    Thus, option (d) is not correct.

    

    Hence, option (b) is correct.

5. Let be a continuous and one-one function. Then which of the following is true?

    (a) f is onto

    (b) f is either strictly decreasing or strictly increasing

    (c)

    (d) f is unbounded

Ans.    (b)

Sol.    Given be a continuous and one-one function.

    

    

    f is bounded and f is either strictly decreasing or strictly increasing.

6.

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Given

    Let D be a subset of R and let {n} be a sequence of function defined on D. We say that converges pointwise on D.

    

    

    Thus, it is pointwise convergent.

    

    

    

    Hence, it is not uniform convergent.

    Thus, the given series is pointwise but not uniform.

7. Consider the vector space P_n of real polynomials in x of degree less than or equal to n. Define by (Tf)(x) = Then the matrix representation of T with respect to the bases {1, x, x²} and {1, x, x², x³} is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    Given, the vector space P_n of real polynomial in x of degree less than or equal to n and

    Then the matrix representation of T with respect of the base {1, x, x²} and {1, x, x², x³ + ....}

    

    

    

    So, option (b) is correct.

8. Let P_A(x) denote the characteristic polynomial of a matrix A. Then for which of the following matrices P_A(x) – P_A–1(x) is a constant?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    Given, P_A(x) denotes the characteristic polynomial of matrix A. Then for matrices P_A(x) – P_{A – 1}(x) is

    

    

    

    

    So, option (c) is correct.

9. Which of the following matrices is not diagonalizable over R?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Matrices is diagonalizable over R. Now algebraic multiplicity of an eigenvalue is a number of occurrence of as a roots of characteristic polynomial.

    Geometric multiplicity of an eigenvalue is the number of independent eigenvalue associated with on eigenvalue .

    (i)    Eigenvalues of upper/lower triangular matrix are diagonal values.

    (ii)    If the algebraic and geometric multiplicity of each eigenvalues are equal, then it is diagonalizable matrix.

    (iii)    Geometric multiplicity < Algebraic multiplicity.

    (iv) If a matrix has distinct eigenvalues, then it is diagonalizable matrix.

    

    Eigenvalues of upper triangular A are 2, 2, 3 Eigenvalues for repeated eigenvalue = 2.

    

    

    

    Eigenvalues of upper triangular A are 2, 3, 3 Eigenvector for repeated eigenvalue = 3

    

    

    

    

    

    Distinct eigenvectors DIAGONALIZABLE Option (A) is correct.

10. What is the rank of the following matrix?

    

    (a) 2

    (b) 3

    (c) 4

    (d) 5

Ans.    (d)

Sol.    Rank of matrix is the number of non-zero rows in row echelon from A.

    

    

    Row echelon from of A.

    Number of non-zeros is 5. Hence, rank is 5.

11. Let V denote the vector space of real valued continuous functions on the closed interval [0, 1]. Let W be the subspace of V spanned by {sin(x), cos(x), tan(x)}. Then the dimension of W over R is

    (a) 1

    (b) 2

    (c) 3

    (d) Infinite

Ans.    (c)

Sol.    Given V denote the vector space of real valued continuous functions on the closed internal [0, 1]. Given W be the substance of V spanned by f sin(x), cos(x), tan(x)}. Then dimension of W over R. The dimension of W is the number of independent vectors (functions) in the set [sin(x), cos(x), tan(x)}. We set {f₁(x), f₂(x), f₃(x)} of functions is independent set of function if the following. Wronskian determinant is non-zero for at least one x.

    

    = sin x[– sin(2 sec² tan x) + sec x + cos x (cos x(2 sec²x tan x) + sin x (sec²x) + tan x(–cos²x – sin²x)]

    = –2 tan²x tan x + tan x – 2 tan x – x – tan x – tan x

    = –2 tan 3x – 3 tan x

    = – tan x (2 tan² × + 3)

    

    Wronskian determinant is non-zero, so {sin(x) cos(x), tan(x) is linearly independent dim (W) = 3 rank is 3.

    Hence, the dimension of W over R is 3.

12. Let V be the vector space of polynomials in the variable t of degree at most 2 over R. An inner product on V is defined by

    for Let W = span{1 + t², 1 + t²} and be the orthogonal complement of W in V. Which of the following conditions is satisfied for all

    (a) h is an even function i.e. h(t) = h(–t)

    (b) h is an odd function i.e. h(t) = –h(–t)

    (c) h(t) = 0 has a real solution

    (d) h(0) = 0

Ans.    (c)

Sol.    Given V be the vector space of polynomials in the variable t of degree of most over R.

    For, Given W = span{1 – t², 1 + t²} and W be the orthogonal complement of W is V.

    The orthogonal vectors u and v are orthogonal if = 0.

    

    

    On solving, we get

    

    

    

    Thus, option (c) is correct.

13. Let C be the counter-clockwise oriented circle of radius 1/2 centered at Then the values of contour integral is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Let us consider C is simple closed curve and the function f(z) is analytic on a region containing C and its interior. We assume L is a oriented counter, clockwise

    

    Then, for any z₀ inside C

    

    

    Thus, correct option is (a).

14. Consider the function given by f(z) = e^z. Which of the following is false?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

               f(x) = e^x

    If | z | < 1, because | z | < 1 is not an open.

    Hence, option (A) is correct.

    Options (B), (C), (D) are not correct.

15. Given a real number a > 0 consider the triangle with vertices 0, a, a + ia. If is given the counter clockwise orientation then the contour integral (with Re(z) denoting the real part of z) is equal to

    (a) 0

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

    

    

    

    

    

    

    

16. Let be an entire function such that Then which of the following is true?

    (a) f is constant

    (b) f can have infinitely many zeros

    (c) f can have at most finitely many zeros

    (d) f is necessarily nowhere vanishing

Ans.    (c)

Sol.    

    

    

    

    Also, f is entire function and f has pole at infinity of order n, then f is a polynomial of degree n.    

               f(z) = a₀ + a₁z + a₂z² + ... a_nzⁿ

               f(z) = 0

    a₀ + a₁z + a₂z² + .... + a_nzⁿ

    f can have at most finitely many zeros.

17. For any integer n >1, let

    d(n) = number of positive divisors of n

    v(n) = number of distinct prime divisors of n

     = number of prime divisors of n counted with multiplicity

    [for example: If p is prime then d(p) = 2, v(p) = v(p²) = 1, = 2]    

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    For any integer N > 1.

    Given, d(n) is number of positive divisors of n.

    u(n) is number of distinct prime divisors of n = number of prime divisors of n counted with multiplicity.

    

    

    

    

    So, option (c) is true.

18. Consider the set of matrices

    G = .

    Then which of the following is true?

    (a) G forms a group under addition.

    (b) G forms an abelian group under multiplication.

    (c) Every element in G is diagonalisable over C.

    (d) G is a finitely generated group under multiplication.

Ans.    (d)

Sol.    Given, the set of matrices

    

    Hence, option (a) is wrong.

    

    Hence, every element in G is diagonalisable over C

    

    G is finitely generated group under multiplication.

19. Let R be a commutative ring with unity. Which of the following is true?

    (a) If R has finitely many prime ideals then R is a field.

    (b) If R has finitely many ideals then R is finite.

    (c) If R is a P.I.D. then every subring of R with unity is as P.I.D.

    (d) If R is an integral domain which has finitely many ideals then R is a field.

Ans.    (d)

Sol.    Given be commutative ring with unity then R has finitely many prime ideals then R is field and R has finitely many ideals then R is finite which are not true. If R is P.I.D. then every subring of R with unity is P.I.D, which is wrong statement.

    

20. Let A be a non-empty subset of a topological space X. Which of the following statements is true?

    (a) If A is connected then its closure is not necessarily connected

    (b) If A is path connected then its closure is path connected

    (c) If A is connected then its interior is not necessarily connected.

    (d) If A is path connected then its interior is connected.

Ans.    (c)

Sol.    Given, A be a non-empty subset of topological space X.

    

    A is connected then A also connected

    A is connected, then its closure A is not necessarily connected which is not true so option (A) is not connected.

    

    

    So, A is connected then its interior is not necessarily connected.

    Hence, option (C) is connected.

21. Let y(x) be the solution of

    Then the values of y(3) is

    (a)

    (b) 1

    (c)

    (d)

Ans.    (d)

Sol.

22. The positive values of for which the equation has non-trivial satisfying y(0) = are

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

    

    

    

    

23. Consider the PDE

    

    where P and Q are polynomials in two variables with real coefficients. Then which of the following is true for all choices of P and Q?

    (a) There exists R > 0 such that the PDE is elliptic in

    (b) There exists R > 0 such that the PDE is hyperbolic in

    (c) There exists R > 0 such that the PDE is parabolic in

    (d) There exists R > 0 such that the PDE is hyperbolic in

Ans.    (b)

Sol.    The positive value of for which the equation has non-trivial solution

    

    

    

    Hence, option (b) is correct.

24. Let u be the unique of solution of

    

    Then which of the following is true?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    Given, the PDE

    

    Here, P and Q are polynomials in two variables with real coefficients.

Then there exists R > 0 such that the PDE is hyperbolic in

25. Consider solving the following system by Jacobi iteration scheme

        x + 2my – 2mz = 1

           nx + y + nz = 2

        2mx + 2my + z = 1

    where With any initial vector, the scheme converges provided m, n satisfy    

    (a) m + n = 3

    (b) m > n

    (c) m < n

    (d) m = n

Ans.    (d)

Sol.    Given, u be the unique solution of

    

    

    

    

    

    Thus, options (a), (b) and (d) are correct.

    

    Hence, option (c) is correct.

26. Let x^*(t) be the curve which minimizes the functional

 

    satisfying x(0) = 0, x(1) = 1. Then the value of x^*(1/2) is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Given be the curve which minimizes the functional

    

    

    

    

    

    

    Hence, option (a) is correct.

27. If y is a solution of then which of the following is true?

    (a) y is bounded but not periodic in R

    (b) y is periodic in R

    (c)

    (d)

Ans.    (d)

Sol.    Given y is a solution of

    

    Hence, this having value of R. Option (b) is wrong.

    

    

    So, options (a), (b), (c) are not correct.

    

    Hence, option (d) is correct.

28. Suppose a point mass m is attached to one end of a spring of spring constant k. The other end of the spring is fixed on a mass less cart that is being moved uniformly on a horizontal plane by an external device with speed v₀. If the position q of the mass in the stationary system is taken as the generalized coordinate then the Lagrangian of the system is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.

29. There are 30 questions in a certain multiple choice examination paper. Each question has 4 options and exactly one is to be marked by the candidate. Three candidates A, B, C mark each of the 30 questions at random independently. The probability that all the 30 answers of the three students match each other perfectly is

    (a) 60^–4

    (b) 30^–4

    (c) 4^–30

    (d) 4^–60

Ans.    (d)

Sol.

30. Let X₁, X₂, X₃, X₄, X₅ be i.i.d. random variables having a continuous distribution function. Then P(X₁ > X₂ > X₃ > X₄ > X₅|X₁ = max(X₁, X₂, X₃, X₄, X₅)) equals

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.

31. Consider a Markov Chain with state space {0, 1, 2, 3, 4} and transition matrix

    

    Then equals

    (a)

    (b)

    (c) 0

    (d) 1

Ans.    (c)

Sol.

32. Consider the function f(x) defined as

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.

33. Suppose (X, Y) follows bivariate normal distribution with means standard deviations and correlation coefficient where all the parameters are unknown. Then testing is equivalent to testing the independence of

    (a) X and Y

    (b) X and X – Y

    (c) X + Y and Y

    (d) X + Y and X – Y

Ans.    (d)

Sol.

34. A random sample of size 7 is drawn from a distribution with p.d.f.

    

    and the observations are 12, –54, 26, –2, 23, 17, –39.

    What is the maximum likelihood estimate of

    (a) 12

    (b) 24

    (c) 26

    (d) 27

Ans.    (d)

Sol.

35. Let X₁, X₂ be a sequence of independent normally distributed random variables with mean 1 and variance. Let N be a Poisson random variable with mean 2, independent of X₁, X₂. Then, the variance of X₁ + X₂ + ... + X_{N + 1} is

    (a) 3

    (b) 4

    (c) 5

    (d) 9

Ans.    (c)

Sol.    Given, X₁, X₂ ..... be a sequence of independent normally distributed random variable with mean 1 and variance 1. Let N be a Poisson random variable with mean 2, independent of X₁, X₂ ..... then the variance of X₁ + X₂ ..... X_{N + 1} is 5.

    Thus, correct option is (c).

36. There are two sets of observations on a random vector (X, Y). Consider a simple linear regression model with an intercept for regressing Y on X. Let be the least squares estimate of the regression coefficient obtained from the i^th (i = 1, 2) set consisting of n_i observations (n₁, n₂ > 2). Let be the least squares estimate obtained from the pooled sample of size n₁ + n₂. If it is known that which of the following statements is true?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.

37. Suppose r_1.23 and r_1.234 are sample multiple correlation coefficients of X₁ on X₂, X₃ and X₁ on X₂, X₃, X₄ respectively. Which of the following is possible?

    (a) r_1.23 = –0.3, r_1.234 = 0.7

    (b) r_1.23 = 0.7, r_1.234 = 0.3

    (c) r_1.23 = 0.3, r_1.234 = 0.7

    (d) r_1.23 = 0.7, r_1.234 = –0.3

Ans.    (c)

Sol.

38. A sample of size n = 2 is drawn from a population of size N = 4 using probability proportional to size without replacement sampling scheme, where the probabilities proportional to size are

    i:  1    2    3    4

    p_i: 0.4  0.2  0.2  0.2

    The probability of inclusion of unit 1 in the sample is    

    (a) 0.4

    (b) 0.6

    (c) 0.7

    (d) 0.75

Ans.    (c)

Sol.

39. Suppose that has a bivariate density f = where f₁ and f₂ are respectively, the densities of bivariate normal distributions with

    

    identity matrix. Then which of the following is correct?    

    (a) X and Y are positively correlated.

    (b) X and Y are negatively correlated.

    (c) X and Y are uncorrelated but they are not independent.

    (d) X and Y are independent.

Ans.    (a)

Sol.

40. The maximum value of the objective function z = 5x + 2y under the linear constraints x > 0, y > 0, x > y and 2 < x + y < 4 is

    (a) 14

    (b) 20

    (c) 25

    (d) 27

Ans.    (b)

Sol.    Given the maximum value of the objective z = 5x + 2y, x > 0, y > 0, x > y, x + y < 4

    

    

    Hence, option (b) is correct.

41. Let {a_n}_{n >0} be sequence of positive real numbers. Then, for which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (c)

Sol.

42. For denote the greatest integer smaller than or equal to by Then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (d)

Sol.

43. Consider a function Then which of the following are true?

    (a) f is not one-one if the graph of f intersects some line parallel to X-axis in at least two points.

    (b) f is one-one if the graph of f intersects any line parallel to the X-axis in at most one point.

    (c) f is surjective if the graph of f intersects every line parallel to X-axis.

    (d) f is to surjective if the graph of f does not intersect at least one line parallel to X-axis.

Ans.    (a), (b), (c), (d)

Sol.

44. Let f(x) = Then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (b)

Sol.

45. Suppose that {x_n} is a sequence of positive reals. Let y_n = . Then which of the following are true?

    (a) {x_n} is convergent if {y_n} is convergent

    (b) {y_n} is convergent if {x_n} is convergent

    (c) {y_n} is bounded if {x_n} is bounded

    (d) {x_n} is bounded if {y_n} is bounded

Ans.    (b), (c)

Sol.    Given that {x_n} is sequence of positive reals.

    Given

    If a_n is converse than

    

    Hence, option (a) is not correct.

    

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

46. Let f(x) = and g(x) = xf(x) for

    Then which of the following are true?

    (a) f is of bounded variation

    (b) f is not of bounded variation

    (c) g is of bounded variation

    (d) g is not of bounded variation

Ans.    (b), (c)

Sol.    

    

    

    Hence, option (a) is wrong.

    

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

47. Let a < c < b, be continuous. Assume that f is differentiable at every point of (a, b)\{c} and has a limit at c. Then which of the following are true?

    (a) f is differentiable at c.

    (b) f need not be differentiable at c.

    (c) f is differentiable at c and

    (d) f is differentiable at c but

Ans.    (a), (c)

Sol.

48. Let be a non-decreasing function. Which of the following can be the set of discontinuities of F?

    (a) Z

    (b) N

    (c) Q

    (d) R/Q

Ans.    (a), (b), (c)

Sol.    

    

    In the function, non-decreasing function.

    So, Z, N, Qare correct.

    Hence, options (a), (b) and (c) are correct.

49. Let

    Consider there exists an open subset U around (x₁, x₂, x₃) such that f|_U is an open map}. Then which of the following are true?

    (a)

    (b) E is countable

    (c)

    (d)

Ans.    (c), (d)

Sol.

50. Let X be a countable set. Then which of the following are true?

    (a) There exists a metric d on X such that (X, d) is complete.

    (b) There exists a metric d on X such that (X, d) is not complete.

    (c) There exists a metric d on X such that (X, d) is compact.

    (d) There exists a metric d on X such that (X, d) is not compact.

Ans.    (a), (b), (c), (d)

Sol.

51. Let L(Rⁿ) be the space of ¡-linear maps from Rⁿ to Rⁿ. If Ker(T) denotes the kernel (null space) of T then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (c)

Sol.    

    

    

    

    

    

    

    

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

52. Let V be a finite dimensional vector space over R and be a linear map. Can you always write T = T₂ º T₁ for some linear maps where W is some finite dimensional vector space and such that

    (a) both T₁ and T₂ are onto

    (b) both T₁ and T₂ are one to one

    (c) T₁ is onto, T₂ is one to one

    (d) T₁ is one to one, T₂ is onto

Ans.    (c), (d)

Sol.    Given V be a finite dimension vector space over R and be a linear map. Now, T = T₂ º T₁ for some linear maps T₁ = where W is some finite dimensional vector.

    

    

    Hence, options (a) and (b) are incorrect.

    

    

    

    

    

    T is one-to-one, T₂ is onto.

    Hence, options (c) and (d) are correct.

53. Let A = ((a_ij)) be a 3 × 3 complex matrix. Identify the correct statements:

    (a) det(((–1)^{i + j}a_ij)) = det A

    (b) det(((–1)^{i + j}a_ij)) = –det A

    (c)

    (d)

Ans.    (a), (c)

Sol.    Given A = [(a_ij) be 3 × 3 complex matrix)

    

    Hence, option (a) is correct.

    Now, check option (b) is not correct.

    

    Now, (c) is correct.

    Option (d) is not correct.

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

54. Let p(x) = a₀ + a₁x + ... + a_nxⁿ be a non-constant polynomial of degree n > 1.

Consider the polynomial:

    

    Let V denote the real vector space of all polynomials in x. Then which of the following are true?

    (a) q and r are linearly independent in V

    (b) q and r are linearly dependent in V

    (c) xⁿ belongs to the linear span of q and r

    (d) x^{n + 1} belongs to the linear span of q and r

Ans.    (a)

Sol.

55. Let M_n(R) be the ring of n × n matrices over R. Which of the following are true for every n > 2?

    (a) there exists matrices such that AB – BA = I_n where I_n denotes the identity n × n matrix.

    (b) if and AB = BA then A is diagonalisable over R if and only if B is diagonalizable over R.

    (c) if then AB and BA have same minimal polynomial

    (d) if then AB and BA have the same eigenvalues in R.

Ans.    (d)

Sol.    

             AB – BA =    I_n

          Tr(AB –BA) = Tr(I_n)

        Tr(AB) – Tr (BA) = n

                 n = 0

    Hence, option (A) is not correct.

    

    Option (b) is not correct.

    

56. Consider a matrix A = (a_ij)_{5 × 5}, 1 < i, j < 5 such that a_ij = where Then in which of the following cases A is a positive definite matrix?

    (a) n_i = i for all i = 1, 2, 3, 4, 5

    (b) n₁ < n₂ < ... < n₅

    (c) n₁ = n₂ = ... = n₅

    (d) n₁ > n₂ > ... > n₅

Ans.    (a), (b), (d)

Sol.

57. Let denote the standard inner product on Rⁿ. For a non-zero Which of the following are true?

    (a) det (T_w) = 1

    (b)

    (c) T_w = T_w^–1

    (d) T_2w= 2T_w

Ans.    (b), (c)

Sol.    Given denote the standard inner product on by

    

    u₁(1, 0) and v₁

    

    

    

    

    

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

58. Consider the matrix A = over the field Q of rationals. Which of the following matrices are of the form P^t AP for a suitable 2 × 2 invertible matrix P over Q? Here P^t denotes the transpose of P.

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (c), (d)

Sol.    

    

    

    

    

    

    

    Hence, option (d) is correct.

59. Let f(z) = (z³ + 1) sin z² for Let f(z) = u(x, y) + iv(x, y) where z = x + iy and u, v are real valued functions. Then which of the following are true?

    (a)

    (b) u is continuous but need not be differentiable

    (c) u is bounded

    (d) f can be represented by an absolutely convergent power series

Ans.    (a), (d)

Sol.               f(z) = (z³ + 1) sin z²

               f(z) = u(x, y) + iv (x, y)

               f(z) = (x³ –3xy² + 1) + i(3x²y – y³)

    sin(x² – y² cosh 2xy + i sin h 2xy cos(x² – y²).

    Hence, options (A) is correct and (B) is false

            u(x, y) < m

              g(2j) = ef(z)

             |g(z)| = eⁿ

    Hence, options (c) is false and (d) is correct.

60. Let Re(z), Im(z) denote the real and imaginary parts of respectively. Consider the domain = and let f_n(z) = log zⁿ where and where defines the principal branch of logarithm. Then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (b), (c), (d)

Sol.    

    

    

    

    Then, correct options are

    

61.

    

    Then which of the following are true?

    (a) F is a finite set

    (b) F is an infinite set

    (c)

    (d)

Ans.    (b), (d)

Sol.

62. Let

Define

Then which one of the following are true?

    (a) F is one-to-one

    (b) F is not one-to-one

    (c) F is onto

    (d) F is not onto

Ans.    (a), (c)

Sol.

63. Let be such that a = b² + c² where Then a cannot be written as

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (c), (d)

Sol.    

64. For any prime p, consider the group .

    Then which of the following are true?

    (a) G has an element of order p.

    (b) G has exactly one element of order p.

    (c) G has no p-Sylow subgroups.

    (d) Every element of order p is conjugate to a matrix

Ans.    (a), (d)

Sol.

65. Let Z[X] be the ring of polynomials over integers. Then the additive group Z[X] is

    (a) isomorphic to the multiplicative group Q⁺ of positive rational numbers.

    (b) isomorphic to the group of rational numbers Q under addition.

    (c) countable.

    (d) uncountable

Ans.    (a), (c)

Sol.    Suppose Z[x] be a ring polynomial i.e., polynomial algebra is a ring formed from the set of polynomials is one or more indeterminates with coefficient in another ring, often a field. Then, the additive group Z[x] are isomorphic to the multiplicative group Q⁺ of positive rational numbers and countable.

66. Let X = (0, 1) be the open unit interval and C(X, R) be the ring of continuous functions from X to R. For any Then which of the following are true?

    (a) I(X) is a prime ideal.

    (b) I(x) is a maximal ideal.

    (c) Every maximal ideal of C(X, R) is equal to I(X) for some

    (d) C(X, R) is an integral domain.

Ans.    (a), (b)

Sol.    Given, X (0, 1) be the open unit interval and C(X, R) be the ring of continuous functions from X to R. For any

    

    

    

    

    

    

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

67.

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (c)

Sol.    

    

    

    

    

    Hence, x³ + nx + 1 is reducible over Z if n {0, –2} and x³ + nx + 1 is irreducible over Z in n {0,–2}

    Options (b) and (c) are correct.

68. Let F₂₇ denote the finite field of size 27. For each we define

    

    Then which of the following are true?

    (a)

    (b)

    (c) |A₁| = 27

    (d)

Ans.    (a), (b), (d)

Sol.

69. Which of the following open covers of the open interval (0, 1) admit a finite subcover?

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (c)

Sol.

70. Let X be a topological space. Let A₁, A₂ be two dense subsets of X. If then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (d)

Sol.

71. Let y₁(x) be any non-trivial real valued solution of Let y₂(x) be the solution of Then

    (a) y₁(x) has infinitely many zeros

    (b) y₂(x) has infinitely many zeros

    (c) y₁(x) has finitely many zeros

    (d) y₂(x) has finitely many zeros

Ans.    (a), (d)

Sol.

72. Consider the equation a(x) is continuous function with period T. Let and be the basis for the solution satisfying

    Let denote the Wronskian of Then

    (a)

    (b)

    (c) if the given differential equation has a non-trivial periodic solution with period T.

    (d) if the given differential equation has a non-trivial periodic solution with period T.

Ans.    (a), (c)

Sol.

73. Let be a Lipschitz function such that f(x) = 0 if and only if x = ±n² where Consider the initial value problem

    Then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (b), (d)

Sol.

74. The general solution z = z(x, y) of (x + y)zz_x + (x – y)zz_y = x² + y² is

    (a) F(x² + y² + z², z² – xy) = 0 for arbitrary C¹ function F.

    (b) F(x² – y² – z², z² – 2xy) = 0 for arbitrary C¹ function F.

    (c) F(x + y + z, z – 2xy) = 0 for arbitrary C¹ function F.

    (d) F(x³ – y³ – z³, z – 2x²y²) = 0 for arbitrary C¹ function F.

Ans.    (b)

Sol.    Given PDE equation

    

    The Lagrange's auxiliary equations of (i) are

    

    Choosing x, –y, –2 as multiplication and from equation (ii)

    

    

    where C₁ is arbitrary constants.

    Again choosing multipliers s.t. y, x, –z in each function of (ii), we get

    

    

75. Let u be the solution of the problem

    

    Then

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (c)

Sol.    

    

    

    

    

    

    Belong to set such that every value of the set is less than or equal to that value

    

    

    Hence, option (a), (c) are correct.

76. The values of a, b, c so that the truncation error in the formula

    

    is minimum are

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

    

    Solving equation (i) and (ii), we get

    

    On solving, we get

    

77. Consider the equation x² + ax + b = 0 which has two real roots Then which of the following iteration scheme converges when x₀ is chosen sufficiently close to

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (c), (d)

Sol.    Let equation x² + ax + b = 0. The real roots of given equation are

    

    Hence, correct statements are (a), (c) and (d).

78. Let u be the solution of

    

    where f, g are in C²(R) and satisfy the following conditions.

    (i)    f(x) = g(x) = 0 for x <0    (ii)    0 < f(x) <1 for x > 0

    (iii)    g(x) > 0 for x > 0    (iv)    

    Then, which of the following statements are true?    

    (a) u(x, t) = 0 for all x < 0 and t > 0

    (b)

    (c) u(x, t) = 0 whenever x + t < 0

    (d) u(x, t) = 0 for some (x, t) satisfying x + t > 0

Ans.    (b), (c)

Sol.

79. Let and C = {(x, y)/x² + y² = 1} and let f and g be continuous functions. Let u be the minimizer of the functional

    

    Then u is a solution of    

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.

80. Consider the functional

    

    subject to y(0) = 1 and y(1) = 2. Then

    (a)

    (b)

    (c) every extremal y belongs to C¹([0, 1])

    (d) every extremal y belongs to C²([0, 1])

Ans.    (c), (d)

Sol.

81. Consider the integral equation

    

    (a)

    (b)

    (c) for f(x) = e^–x(1 – 3x²), a solution exists

    (d) for f(x) = e^–x(x + x³ + x⁵), a solution exists

Ans.    (c), (d)

Sol.

82. Let (q, p) be canonical variables. Consider the following transformations

    

    (ii) (Q, P) = (q tan p, log(sin p))

    

    Then

    (a) only the transformations given in (a) and (b) are canonical

    (b) only the transformations given in (b) and (c) are canonical

    (c) only the transformations given in (a) and (c) are canonical

    (d) all are canonical

Ans.    (d)

Sol.    Given (a, p) be canonical variables canonical condition

    pdg – pdq = exact

    

    

    Hence, option (a) is true.

    (B)            Q = q tan p

                 P = log sin p = pdq – pdq = p sin q log sin pd(q tan p)

                = p sin q – log sin p [q sec² pdq] + tan pda = (p – tan p log sin p)dq – q sec² p.log sin p.dg

                = d[(p – tan p log sin p)q].

    Hence, option (b) is true.

    

    Hence, option (c) is true. Thus, all are canonical variables.

    Option (d) is correct.

83. Suppose a normal Q – Q plot is drawn using a reasonably large sample x₁, ..., x_n from an unknown probability distribution. For which of the following distributions would you expect the Q – Q plot to be convex (J – shaped)?

    (a) Beta (5, 1)

    (b) Exponential (91)

    (c) Uniform (0, 1)

    (d) Geometric (1/2)

Ans.    (b), (d)

Sol.

84. (X_n, n > 1) and X are random variables on a probability space. Suppose that X_n converges to X in probability. Which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.

85. Consider a simple symmetric random walk on integers where from every state i you move to i – 1 and i + 1 with probability half each. Then which of the following are true?

    (a) The random walk is aperiodic

    (b) The random walk is irreducible

    (c) The random walk is null recurrent

    (d) The random walk is positive recurrent

Ans.    (b), (c)

Sol.

86. Consider a Markov Chain with state space {0, 1, 2} and transition matrix

    

    Then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (d)

Sol.

87. Suppose X, Y are i.i.d. Binomial (n, p) random variables. Which of the followings are true?

    (a) X +Y ~ Bin(2n, p)

    (b) (X, Y) ~ Multinomial (2n; p, p)

    (c) Var(X – Y) = E(X – Y)²

    (d) Cov(X + Y, X – Y) = 0

Ans.    (a), (c), (d)

Sol.

88. Let X₁, X₂, ... be i.i.d. is unknown.

Consider a class of estimators

for Then which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (b), (c), (d)

Sol.

89. Let X₁, X₂ be i.i.d. random variables with p.d.f.

    

    For testing the following tests are suggested.

      Test 1: Reject H₀ if and only if max{X₁, X₂} <1

      Test 2: Reject H₀ if and only if max{X₁, X₂) > 1

    Let denote the probability of Type I error for Test i, i = 1, 2. Which of the following are true?    

    (a) Test 1 is a level 0.05 test

    (b)

    (c) Test 2 is unbiased

    (d) Test 1 is unbiased

Ans.    (b), (c)

Sol.

90. Let be the square region with four vertices is unknown. Suppose that follows the uniform distribution on S. Which of the following statements are true?

    (a) X and Y are uncorrelated

    (b) X and Y are independent

    (c)

    (d) If are two observations drawn independently then the maximum likelihood estimate of is max {|x₁| + |y₁|, |x₂| + |y₂|}

Ans.    (a), (c), (d)

Sol.

91. Let X_1,X₂, ..., X_{2n – 1} (n > 5) be i.i.d. with p.d.f. which is symmetric about θ having bounded support. Let X₍₁₎ < X₍₂₎ < ... < X_{(2n – 1)} be the order statistics of the random variables X₁, X₂, ..., X_{2n – 1}. Which of the following statements are corrects?

    (a)

    (b)

    (c)

    (d)

Ans.    (b), (c), (d)

Sol.

92. X follows a p-variate (p > 2) distribution with Which of the following are true?

    (a)

    (b)

    (c)

    (d)

Ans.    (a), (b), (c), (d)

Sol.

93. Consider a logistic regression model with a single repressor x, where the log odds ratio has intercept and slope Let T₁ be the ratio of the odds of success when x = 0 to that when x = –1. Similarly, let T₂ be the ratio of the odds of success when x = 1 to that when x = 0. Which of the following are true?

    (a) T₁ = T₂

    (b) T₁T₂ = 1

    (c)

    (d)

Ans.    (a), (d)

Sol.

94. Let b₁, b₂, ..., b₁₀ ~ i.i.d. for i = 1, 2, ..., 10; j = 1, 2, .., n, independently of b₁, b₂, ..., b₁₀. Define As an estimator of the statistic

is

    (a) unbiased

    (b)

    (c)

    (d)

Ans.    (c), (d)

Sol.

95. Consider a classification problem between two classes having densities f₁(x) = 1; 0 < x < 1 and f₂(x) = 1 + respectively. Assume that the prior probabilities of the two classes are equal. Which of the following are true?

    (a) The Bayes classifier classifies an observation to class-1 if

    (b) A randomly chosen observation from class-1 is misclassified with probability

    (c) A randomly chosen observation from class-2 is misclassified with probability

    (d) The average misclassification probability of the Bayes classifier is

Ans.    (a), (b), (c), (d)

Sol.

96.

    (a) The conditional variance of

    (b) The conditional variance of

    (c) The unconditional variance of

    (d) The unconditional variance of

Ans.    (b), (d)

Sol.

97. In a design with four treatments and three blocks of 2 plots each, treatments A and B are alloted to block 1 and 3 and treatments C and d are allotted to block 2. Hence the resulting design

    (a) is incomplete

    (b) is connected

    (c) is non-orthogonal

    (d) has all elementary treatment contrasts estimable

Ans.    (a), (c)

Sol.

98. Consider a system with two components whose lifetimes are i.i.d. exponential with hazards rate λ. Let h₁ and h₂ be the hazard functions of the system if the components are put in series and parallel respectively. Then which of the following are true?

    (a) h₂(t) < h₁(t) for all t > 0

    (b)

    (c)

    (d) h₂ is a strictly increasing function of t

Ans.    (a), (b), (d)

Sol.

99. You want to invest Rs. 1000 in companies I and II. If the market is good, company I will declare dividend of 50% while company II will declare 30%. If the market is bad, company I will declare dividend of 10% while company II will declare 20%. The prediction is that market will be good with probability 0.4 and bad with probability 0.6. The investment that maximizes expected dividend is

    (a) Rs. 1000 in company I and nil in company II

    (b) Nil in company I and Rs. 1000 in company II

    (c) Rs. 500 in each of the two companies

    (d) Rs. 600 in company I and Rs. 400 in company II

Ans.    (a)

Sol.

100. Consider as M/M/I Queue with arrival rate For t > 0, let N_t be the number of arrivals upto (and including) t. For k >1, let S_k be the arrival time of the k^th customer. Let A_t = t – S_Nt be the time elapsed after the last arrival and B_t = S_{Nt + 1} – t be the waiting time from t to the next arrival. Then which of the following are true?

    (a) A_t is an unbounded random variable

    (b)

    (c) B_t is an unbounded random variable

    (d)

Ans.    (c), (d)

Sol.