PYQ DEC 2019 - VedPrep

CSIR NET MATHEMATICS (DEC - 2019)
Previous Year Question Paper with Solution.

Then which of the following statements is true?

(a) det M(t) is a polynomial function of degree 3 in t.

(b)

(c)

(d)

Ans. (d)

Sol.

Therefore, the roots are real and distinct then

2. The ≤ be the usual order on the field R of real numbers. Define an order ≤ on R by (a, b) ≤ (c, d) if (a < c), or (a = c and b ≤ d).

Consider the subset E = .

With respect to ≤ which of the following statements is true?

(a) inf(E) = (0, 1) and sup(E) = (1, 0)

(b) inf(E) does not exist but sup(E) = (1, 0)

(d) Both inf(E) and sup(E) do not exist

Ans. (b)

Sol.

Thus, inf E does not exist.

Therefore, inf (E) = {0, 1} but sup (E) does not exist.

3. For a quadratic form in 3 variables over R, let r be the rank and s be the signature. The number of possible pairs (r, s) is

(a) 13

(b) 9

(d) 16

Ans. (c)

Sol. For a quadratic form in 3 variables over R which is given as

Q(x, y, z) = ax² + by² + cz² + 2dxy + 2exz + 2exz + 2fyz

Rank (Q) = r = p + n and signature s = p – n

Case 1: Rank(Q) = Rank(A) = r = 3

Possibility-1: p = 3, n = 0

s = p – n = 3 – 0 = 3

(r, s) = (3, 3)

Possibility-2: p = 2, n = 1

s = p – n = 2 – 1 = 1

Possibility-3: p = 2, n = 2

s = p – n = 1 – 2 = –1

(r, s) = (3, –1)

Possibility-4: p = 0, n = 3

s = p – n = 0 – 3 = –3

(r, s) = (3, –3)

Case-2: Rank(Q) = Rank(2) = r = 2

Possibility-1: p = 2, n = 0

s = p – n = 2 – 0 = 2

(r, s) = (2, 2)

Possibility-2: p = 1, n = 1

s = p – n = 1 – 1 = 0

Possibility-3: p = 0, n = 2

s = p – n = 0 – 2 = –2

(r, s) = (2, –2)

Case-3: Rank(Q) = Rank (A) = r = 1

Possibility-1: p = 1, n = 0

s = 1, n = 0

s = p – n = 1 – 0 = 1

(r, s) = (1, 1)

Possibility-2: p = 0, n = 1

s = p – n = 0 – 1 = –1

(r, s) = (1, –1)

Case-4: Rank(Q), Rank(A) = r = 0

Possibility-1: p = 0, n = 0

r = p – n = 0 – 0 = 0

(r, s) = (0, 0)

Thus, total possibility = 4 + 3 + 2 + 1 = 10

Thus, number is possibility pairs (r, s) is 10.

Then which of the following statements is true?

(a) No subsequence of (f_m)_{m > 1} converges at every point of E

(b) Every subsequence of (f_m)_{m > 1} converges at every point of E

(d) There exists a subsequence of (f_m)_{m > 1} which converges to 0 at every point of E.

Ans. (c)

Sol.

5. Let M₄(R) be the space of all (4 × 4) matrices over R.

Let .

Then dim(W) is

(a) 7

(b) 8

(d) 10

Ans. (c)

Sol.

Therefore, dim(W) is 9.

6. Let V be a vector space of dimension 3 over R. Let be a linear transformation, given by the matrix A = with respect to an ordered basis (v₁, v₂, v₃) of V. Then which of the following statements is true?

(a) T(v₃) = 0

(b) T(v₁ + v₂) = 0

(d) T(v₁ + v₃) = T(v₂)

Ans. (c)

Sol. Let V be a vector space of dimension 3 over R and {v₁, v₂, v₃} is an order of V.

So, T(v) = Av

T(v) = 0

then Av = 0

From Cramer's rule

Hence, V = (1)v₁ + 1(v₂) + 1(v₃)

= v₁ + v₂ + v₃

But T(v) = 0

T(v) = T(v₁ + v₂ + v₃) = 0

Thus, correct option is (C).

7. Let be an infinite countable bounded subset of R. Which of the following statements is true?

(a) X cannot be compact

(b) X contains an interior point

(d) closure of X is countable

Ans. (c)

Sol. Compact = closed + bounded

For option (A): x-closed set, bounded y 1 x-compact, then option (A) is false.

For option (B): A point x is said to be an interior point of a neighbourhood of x which is totally contained in a set. Hence option (B) is false.

For option (c): Since is compact i.e., closed.

Thus, option (C) is correct.

For option (D): Counter example

It is uncountable. This option (d) is wrong.

Thus, correct option is (c).

8. Which of the following sets is countable?

(a) The set of all functions Q to Q.

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

(d) The set of all subsets of N.

Ans. (c)

Sol. In the cardinal number theory,

X = set of all possible sets

A and B are equivalent, if is bijective function

The define relation P on x-axis,

and B are equivalent where ρ is an equivalent relation on X. It partitions x into disjoint equivalence classes.

All elements belong single equivalence classes are equipotent. No two elements from two different from two different equivalence classes are equipotent. So, the equivalence class is called the cardinal number.

For the finite sets, cardinal number is the number of elements of the set. When A contains three elements then |A| = 3. For the infinite case, there are two methods.

1. Countable infinite sets: The sets of equipotent to N.

2. Uncountable infinite sets: The sets of equipotent to R.

Thus, option (a) is wrong.

So, option (d) is wrong.

There exists an unique function i.e., vanishes outside A which is finite set.

Now, a₁, a₂, a₃, ..., a_n which is different function. Since a₁, a₂, a₃, ..., a_n are points of Q is countable. So, this would be given by countable variants for the set of A and thus, countable number of function vanish out.

Thus, correct option is (c).

9. Let (x_n)_{n ≥ 1} be a sequence of non-negative real numbers. Then, which of the following is true?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Let sequence of non-negative real numbers.

Thus, option (a) and (c) will be discarded

Now, for option (d):

For option (b) as lim sup|x_n| = 0 iff lim x_n = 0.

Since, it is given that x_n > 0 therefore lim sup x_n = 0 iff lim x_n = 0 and lim x_n = 0

Thus, correct option is (b).

10. Let C[0, 1] be the space of continuous real valued functions on [0, 1].

Define = .

Then which of the following statements is true?

(a) is an inner product on C[0, 1]

(b) is a bilinear form on C[0, 1] but is not an inner product on C[0, 1].

(d)

Ans. (c)

Sol.

Bilinear product:

Inner product:

But it is not bilinear form, therefore it is not an inner product.

Thus, correct option is (c).

11. Let A = . The system of linear equations AX = Y has a solution

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

The system of linear equations Ax = y

Thus, correct option is (d).

12. What is the sum of

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Therefore, sum of I and II parts, we get

Hence, correct option is (b).

13. A permutation of [n] = {1, 2, ..., n} is called irreducible, if the restriction is not a permutation of [k] for any Let a_n be the number of irreducible permutations of [n]. Then a₁ = 1, a₂ = 1 and a₃ = 3. The value of a₄ is

(a) 12

(b) 13

(d) 15

Ans. (b)

Sol.

14. Let X be an infinite set. Consider the topology on X whose non-empty open sets are complements of finite sets. Then which of the following statements is true?

(a) X is disconnected

(b) X is compact

(d) Every sequence in X converges to a unique point in X

Ans. (b)

Sol.

(A) X is disconnected

So, condition (a) is false.

(B) X is compact

Let G be any arbitrary open curve of X. We take any arbitrary

Thus is a finite open subcover of X, X is compact.

Hence option (b) is true.

For option (c) we have

Thus, option (c) is false.

For option (d), every sequence converges to a unique point in X.

Let us take any arbitrary sequence {x_n} in X that does not take any values infinite number of times. Then {x_n} converges to every point of X. Let Then any open set containing x, say U, we get X/U = finite set.

U contains infinitely many points of the sequence [x_n] and excludes at most finitely many points of {x_n}.

This is true for neighbourhood of x. Thus {x_n} converges to x. Since x is arbitrary [x_n] converges every point of the set X. The sequence [x_n] does not converge to a unique point Houdsroff space then every sequence converge to a unique of x. is not a T-space.

15.

Then which of the following statements is false?

(a) T(z₁z₂) = T(z₁)T(z₂) for all

(b) T(z) is singular if and only if z = 0

(d)

Ans. (c)

Sol.

Option (c): T(z) is singular

Hence, option (b) is true.

Thus, correct option is (d).

16. Let S₅ be the symmetric group on five symbols. Then which of the following statements is false?

(a) S₅ contains a cyclic subgroup of order 6

(b) S₅ contains a non-abelian subgroup of order 8

(d) S₅ does not contain a subgroup of order 7

Ans. (c)

Sol. To find the solution, first we need to know the following results.

Result 1: A group G hs element of order d iff G has a cyclic subgroup of order d.

Result 2: If a group G has an element of order d, then G has a subgroup of order d. But converse need not be true.

Result 3: (i) If f =

(ii) If = f₁·f₂ .... where f₁, f₂ .... f_n are disjoint formulations. Then o(f) = lcm (of(f₁)·o(f₂) ... o(f_n)) where o(f₁) means order of f₁.

Result 4: If a positive number d + o(G), then the group G has no subgroup of order d.

Now, we have to find the order of elements of S₅ with the help of following table:

Table shows that S₅ has a element of order 6 which implies S₅ has a cyclic subgroup of order 6. Hence, option (A) is true. As, we know that, S₅ contains a subgroup of order 8 of type P₄ generated by <(1, 2, 3, 4), (1, 3)> which is non-abelian. Hence option (B) is true.

O(S₅) implies S₅ does not contain a subgroup of order 7. Hence, option (D) is true.

Thus, option (C) is false. In this question, we have to find the false statement. Hence, option (C) is correct.

17. Consider the polynomial f(z) = z² + ax + p¹¹ where is a prime. Suppose that a² ≤ 4p¹¹. Which of the following statements is true?

(a) f has a zero on the imaginary axis.

(b) f has a zero for which the real and imaginary parts are equal.

(d) f has exactly one real root.

Ans. (c)

Sol. Given, polynomial f(z) = z² + az + p¹¹ where a ∊ Z\{0}.

Suppose z = ki

For option (b), suppose z = k = ki

Hence, option (c), (d).

Hence, it has two real and equal roots.

Therefore, f distinct roots.

Thus, correct option is (c).

18. Let G be group of order pⁿ, p a prime number and n > 1. Then which of the following is true?

(a) Centre of G has a least two elements.

(b) G is always an abelian group.

(d) If H is any other group of order pⁿ then G is isomorphic of H.

Ans. (a)

Sol. (1) The number of normal subgroups of

where is known as positive divisors of n.

(2) Two groups G and are said to be isomorphic if there exist a mapping such that

(i) T is homomorphism

(ii) T is one-one

(iii) T is onto

For counter example, let us take Dihedral group D. As we know that o(D) = 8 which is of the form 2³. But D is non-abelian group. Hence option (B) is wrong.

Also, the number of normal subgroups of D is

t(4) + 3 = t(2²) + 3 = 3 + 3 = 6

Hence, option (C) is wrong.

Let us take Quaternion group Q₈ of order 8, which is also of the form 2³. But Q₈ D.

Hence, option (D) is wrong.

Hence, option (A) is correct.

19. Let be an entire function with for all Then which of the following statements is true?

(a) No such f exists

(b) such an f is not unique

(c)

(d) f need not be a polynomial function

Ans. (c)

Sol. Given that f is entire

Identity theorem:

According to identity theorem

Therefore, correct option is (c).

20.

Then which of the following statements is false?

(a) f(z) is continuous everywhere.

(b) f(z) is not analytic in any open neighbourhood of zero.

(d) f(z) is analytic in some open subset of C.

Ans. (d)

Sol. We have

In Cauchy-Riemann – f is analytic, so it is satisfies.

The sets of equations are

The function zf(z) is differentiable.

Cauchy-Riemann equation satisfies.

Thus, option (d) is true.

21.

(a) e^–1

(b) e^–2

(d) e²

Ans. (c)

Sol. Given solution

Hence, correct option is (c).

22.

The rate of convergence for iterative method is equal to

(a) 1

(b) 2

(d) 4

Ans. (a)

Sol.

Hence, the rate of convergence for given iterative method is 1.

Thus, option (a) is true.

23. For the following system of ordinary differentials equations

the critical point (0, 2) is

(a) a stable spiral

(b) an unstable spiral

(d) an unstable node

Ans. (c)

Sol. We have X = Ax.

All these properties of the critical point is dependent on the eigenvalue of A.

Linearization the equation at the critical point (0, 2).

F(x, y) = x(3 – 2x – 2y) = 3x – 2x² – 2xy

Linearization the equations at the critical point (0, 2).

The characteristic equation of A is

Both the eigen values of A are negative.

Therefore, (0, 2) is a stable node.

24. Let be the extremizing function for the functional I(y) = s.t. y(0) = 0, y(1) = 1. Then is equal to

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

such that y(0) = 0, y(1) = 1.

Here, F is an independent value of x, then

Putting these values in eqn. (i), we get

Thus, correct option is (b).

25. Consider the system of ordinary differential equations

Then for this system there exists

(a)

(b)

(c)

(d) no closed path R²

Ans. (d)

Sol. Bendixson' non-existence criterion

Suppose has the same sign throughout D.

There exists no closed path.

It never change in sign over R.

Therefore, it does not exist any closed path.

Thus, option (d) is correct.

26. Let u(x, y) be the solution of = 64 in the unit disc {(x, y)|x² + y² < 1} and such that u vanishes on the boundary of the disc.

Then is equal to

(a) 7

(b) 16

(d) 16

Ans. (c)

Sol. We have (In cartesian system)

In polar coordinate system

Using angular geometry

Integrating, we get

where c = constant.

Since = 0 at r = 0, c = 0.

Now, it is given as u vanishes on the boundary condition of the disc.

Now, on cartesian coordinate conditions

Hence, correct option is (c).

27. A consider a mass-less infinite straight wire with one end fixed at O. Assume that the wire is rotating in a plane about the point O with constant angular velocity Consider a bead of mass m sliding along the wire in the absence of external forces. Let r(t) denote the distance of the bead from O at time t ≥ 0 and = 0. Then which of the following statements is true?

(a)

(b)

(d) changes its sign for some t > 0

Ans. (a)

Sol. We have, Cauchy problem

Condition: x₀(s) = cos(s)

y₀(s) = sin(s), z₀(s) = 1, s > 0

Let the equation,

Solving, we get infinitely many solutions.

Hence, correct option is (d).

28. The Cauchy Problem and x₀(s) = cos(s), y₀(s) = sin(s), z₀(s) = 1, s > 0 has

(a) a unique

(b) no solution

(d) infinitely may solutions

Ans. (d)

Sol.

29. In a 2³ factorial design, the treatment combinations of three treatments A, B and C are allotted to 2 blocks of 4 plots each. Suppose the key block is as follows:

Key block: (1), a, bc, abc

Then the confounded treatment combination is

(a) AB

(b) AC

(d) ABC

Ans. (c)

Sol.

30. Suppose data (X₁, Y₁), (X₂, Y₂), ..., (X_n, Y_n) are generated as follows: Y₁, Y₂, ..., Y_n ~ i. i. d. Bernoulli and X_i |Y_i = y ~ Uniform (0, y + 1).

Define

Then, which of the following is a correct linear regression model for m(x) = E(Y_i|X_i= x) in the sense that the true m(x) is obtained for some values of the parameters

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

31. Let X₁ and X₂ be a random sample of size 2 from Uniform distribution Define M = max.{X₁, X₂}. What is the confidence coefficient of the confidence interval

(a) 0.6285

(b) 0.7585

(d) 0.95

Ans. (b)

Sol.

32. Let X be a real-valued random variable such that and E[e^x] = e^E[x]. Then which of the following is correct?

(a)

(b) E[X³] = (E[X])³

(c)

(d)

Ans. (b)

Sol.

33. Suppose are i.i.d. observations from the Uniform distribution on the unit square [0, 1] × [0, 1]. What is the probability that the rank correlation between the X_i and the Y_i values is 1?

(a) 0

(b)

(c)

(d)

Ans. (d)

Sol.

34. Let X and Y be independent Exponential random variables with means respectively with Let f_z(z) denote the density function of Z = X + Y. Then for z > 0

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Theorem: When X and Y are independent random variables having density functions f₁(x) and f₂(y) respectively then density function of Z = X + Y is

which is a convolution of f₁ and f₂.

Since, X and Y be independent exponential random variables with means respectively

Let Z = X + Y.

Now f₁(v) vanishes if v < 0 and f₂(z – v) vanishes if z – v < 0

35. Let X₁, X₂, ..., X_n (n < 2) be a random sample from a distribution with probability density function unknown where

Let be the sample mean and X_(n) = max.{X₁, X₂, ..., X_n}.

Then which of the following statements is correct?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

36. Two draw a sample of size n( > 5) using a without replacement scheme from a finite population {U₁, U₂, ..., U_N} of size N, the first unit is chosen using PPS (p₁, p₂, ..., p_N) scheme and the remaining (n – 1) units are drawn using SRSWOR. Then the probability that U₂ is included in the sample is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

37. Subject to the conditions the minimum value of the function 4x – 5y + 10 is

(a) 10

(b) 0

(d) –15

Ans. (d)

Sol. Here, minimize z = 4x – 5y + 10 subject to the conditions

Therefore, the minimum value i.e., Z_min. = –15.

Hence, correct option is (d).

38. There are three urns U₁, U₂, U₃ each with balls of two colours. U₁ contains 2 white balls and 3 black balls, U₂ contains 3 white balls and 2 black balls and U₃ contains 5 white balls and 5 black balls. An urn is chosen at random and a ball is drawn from the urn at random. What is the probability that U₂ was chosen given that the ball picked is black in colour?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

39. Let (X₁, Y₁), (X₂, Y₂) ... (X_n, Y_n), n ≥ 5 be a random sample from a Bivariate Normal distribution with all parameters unknown. For testing against if you use the usual t-test and your observed sample correlation coefficient is 0 then what is the p-value?

(a) 0

(b) 0.05

(d) 1

Ans. (d)

Sol.

40. Let {X_n : n > 0} be a two state Markov chain with state space S = {0, 1} and transition matrix

P = .

Assuming X₀ = 0 the expected return time to 0 is

(a)

(b)

(c)

(d) 3

Ans. (a)

Sol. We have {x_n : n > 10} be a two state Markov chain with state space S = {0, 1} and transition matrix

So, state transition diagram

When {X_n : n > 0} is an irreducible chain where all the states are +ve recurrent and a periodic.

Then

where U_i the mean recurrent time to return state (i).

Irreducible/Reducible chain.

Markov chain is said to be irreducible. If all state communicating with each other state.

Communicating State: If two states i and j are said to be communicating state if

Recurrent: The returning to that state is true.

Transient: The returning to that state is uncertain. After finite step the chain will not become to that state class of state.

Thus, the returning to 0 and 1 are sure i.e., recurrent.

Aperiodic: i is said to be a periodic, if If all the states are a periodic i.e., chain a periodic.

Hence, correct option is (a).

41. For each natural number n > 1 let a_n = where [x] = smallest integer greater than or equal to x. Which of the following statements are true?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

A function defined as f(x) = [x], the integral part of x which is the nearest and the greatest integer of x is called least integer function.

Limit superior is a sequence of real number which bounded above this

Limit inferior sequence of real number which is bounded below then

For option (a), we have

Thus, correct option is (d).

42. Let n be a fixed natural number. Then the series is

(a) absolutely

(b) divergent

(d) convergent

Ans. (d)

Sol.

Thus, correct options are A, B and D.

43. Let N > 5 be an integer. Then which of the following statements are true?

(a)

(b)

(c)

(d)

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

Sol.

44. Let and let X = Then

(a)

(b)

(c)

(d)

Ans. (c)

Sol. GL₃(R) is the set of 3 × 3 invertible matrices over R.

Triangular of matrix: Any matrix is said to be triangular if it is similar to diagonal matrix or upper triangular matrix of lower triangular matrix.

Jordan Normal Form: Jordan square matrix A can be similar to the triangular matrix (Jordan normal form) J (I = PAP^–1) over the field F, if the roots of its characteristic equation (eigenvalues of A) are in the field F.

Hence, it cannot be similar to triangular matrix over field R.

X = f

Option (b), (c) and (d).

A is 3 × 3 matrix, therefore its characteristic equation is cubic equation and that means it does not have real roots. But, it has three roots one is real and other two are complex conjugates.

Hence, it has three distinct eigenvalues in C. Thus it is diagonalizable over C.

So option (b) is false.

There, option (c) is true.

45. Let be an open subset of -function.

Suppose that for every the derivative at the derivative at x, df_x is non-singular. Then which of the following statements are true?

(a)

(b)

(d)

Ans. (a)

Sol.

46. Let X be a finite dimensional inner product space over C. Let T : be any linear transformation. Then which of the following statements are true?

(a)

(b)

(c)

(d)

Ans. (b), (c)

Sol. Let be any linear transformation and X be a finite dimensional inner product space over C.

A linear operator T on an IPS V is called unitary.

If TT^* = I = T^*T.

T is normal operator if TT^* = T^*T if A is real skew-symmetric matrix then A is normal.

Self-Adjoint: T is self-adjoint (Hermitian) if T = T^*

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

47. Let be a linear transformation n > 2. Suppose 1 is the only eigenvalue of T. Which of the following statements are true?

(a)

(b) (T – 1)^{n – 1} = 0

(d) (T – 1)^{n + 1} = 0

Ans. (c), (d)

Sol. Let us consider the matrix T = I all the natural number.

Then option (a) is false.

For option (b):

Here n = 3 eigenvalues of A is only.

Thus, in this case

Hence, option (b) is also false.

The fundamental theorem of algebra says that every n degree polynomial with complex coefficients exactly n roots in C. The only eigen value of T over C is 1. If f(x) is the characteristic polynomial of T, the f cannot have any other root than 1. Thus, it is of the form (x – 1)ⁿ over C. Since characteristic polynomial annihilates T. So (T – 1)ⁿ = 0.

Hence, option (C) is true.

Again (T – 1)ⁿ⁺¹ = (T – 1)ⁿ (T – 1)

= 0.(T – 1) = 0

Hence, option (D) is also true.

Thus, option (C) and (D) are true.

48. Let {a_n}_{n ≥ 1} be a bounded sequence of real numbers. Then,

(a) Every subsequence of {a_n}_{n ≥ 1} is convergent.

(b) There is exactly one subsequence of {a_n}_{n ≥ 1} which is convergent.

(d) There is a subsequence of {a_n}_{n ≥ 1} which is convergent.

Ans. (c), (d)

Sol.

49. The (X, d) be a compact metric space. Let be a continuous function satisfying Then which of the following statements are true?

(a) T is a one-one function.

(b) T is not a one-one function.

(d) If X is finite then T is onto.

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

Sol. Since (X, d) is a metric space is a function that satisfies the three conditions.

(X, d) is also compact. Hence X is closed and bounded. Converse may not be true always. Since for every we have

Since d is non-negative function (i),

Hence, T is one-one.

Thus, option (a) is true and option (b) is false.

Since (X, d) is compact and T is continuous T(x) is compact.

T(x) is closed and bounded.

Hence, option (c) is true.

X is a finite set and T is an one-one map from X to itself. So T onto.

Hence, option (d) is true.

Thus, options (a), (c) and (d) are true.

50. Which of the following statements are true?

(a) Any two quadratic forms of same rank in n-variables over R are isomorphic.

(b) Any two quadratic forms of same rank in n-variables over C are isomorphic.

(d) A quadratic form in 4 variables may be isomorphic to a quadratic form in 10 variables.

Ans. (b), (d)

Sol. Isomorphic quadratic forms:

Result 1: Two quadratic forms each in n-variables are isomorphic over R. If they have the same rank and the same index on the same rank and the same signature.

We have given only equal ranks, not given anything about signature or index.

Hence, option (a) is false.

Option (b) and (c).

Result 2: Two quadratic forms each in n-variables are isomorphic over C iff they have same ran.

Hence, option (b) is true and option (c) is false.

Option (d): Matrix of quadratic form in 4 variables.

Matrix of quadratic for in 10 variables.

Quadratic forms associated with A in 4 variables and B is 10 variables are isomorphic to each other. Hence, option (D) is true.

Thus, options (B) and (D) are true.

51. Let be a monotonic function with

Which of the following statements are correct?

(a)

(b)

(c)

(d)

Ans. (c), (d)

Sol.

If possible let a neighbourhood bd such that

Similarly, we can show that

Thus, option (c) is correct.

Thus, option (b) is false and option (a) is also false.

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

52. If p(x) be a polynomial function in one variable of odd degree and g be a continuous function from R to R. Then which of the following statements are true?

(a)

(b) If g is a polynomial function then there exists such that p(x₀) = g(x₀).

(d) There is a unique point such that p(x₀) = g(x₀).

Ans. (c)

Sol.

53. Let n > 1 and Suppose is an n × n matrix such that a_ij = and Let d_n be the determinant of Which of the following statements are true?

(a)

(b)

(c)

(d)

Ans. (b), (d)

Sol.

Option (d) is true.

Option (c) is true.

Option (b) is true.

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

54. Let f(x) be a real polynomial of degree 4. Suppose f(–1) = 0, f(0) = 0, f(1) = 1 and f⁽¹⁾ = 0 where f^(k)(a) is the value of k^th derivative of f(x) at x = a. Which of the following statements are true?

(a)

(b)

(c)

(d)

Ans. (a), (d)

Sol. f(x) be a real polynomial of degree 4.

Let f(–1) = 0, f(0) = 0, f(1) = 1 and f⁽¹⁾(0) = 0

We have, f(x) = a₀ + a, x + a₂x² + a₃x⁴ + ...

f'(x) = a₁ + 2a₂x + 3a₃x² + 4a₄x³ + ...

If f(0) = 0 then a₀ = 0

f'(0) = 0, using equation (ii), a₁ = 0

f(x) = a₂x² + a₃x³ + a₄x⁴

f(1) = a₂ + a₃ + a₄ = 0

f(–1) = a₁ – a₃ + a₄ = 0.

From equations (iv) and (v), we get,

Equations (iv) and (vi), we get

From equation (iii),

55. Let be a linear transformation with characteristic polynomial (x – 2)⁴ and minimal polynomial (x – 2)². Jordan canonical form of T can be

(a)

(b)

(c)

(d)

Ans. (a), (b)

Sol. Jordan canonical form of a matrix A.

If A is matrix of order n × n then the Jordan canonical form of A is a matrix of order n × n expressed as

When J₁, J₂, ..., J_k are Jordan blocks of the form

Now, we can calculate the Jordan canonical form using characteristic polynomial and minimal polynomial characteristic polynomial minimal polynomial = The order of Jordan blocks for and d₁, d₂, ..., d_k with conditions d₁ + d₂ + ... + d_k = a_i and highest value of d_j is c_i.

Characteristic polynomial = (x – 2)⁴.

Minimal polynomial = (x – 2)².

The highest order of Jordan block for = 2 is c₁ = 2

The order of Jordan blocks for = 2 are d₁ = 2

d₂ = 2 with conditions.

d₁ + d₂ = 2 + 2 = a₁ = 4

or d₁ = 2, d₂ = 1, d₃ = 1, with conditions

d₁ + d₂ + d₃ = 2 + 1 + 1 = a₁ = 4

The Jordan blocks for = 2 are

56. Let be the metric space of Lebesgue square integrable functions on with a metric d given by

d(f, g) =

Consider the subset

Which of the following statements are true?

(a) A is bounded

(b) S is closed

(d) S is non-compact

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

Sol.

57. Let be a function defined by f(x, y) = if either if x = y = 0.

Then which of the following statements are true?

(a) f is continuous at (0, 0)

(b) f is a bounded function

(c)

(d) f is continuous at (1, 0)

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

Sol. be such that

which depends on the value of m.

The unit does not exist at (0, 0), f(x, y) does continuous at (0, 0).

We can write it in polar form:

which is finite.

which bounded.

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

58. Which of the following statements regarding quadratic forms in 3 variables are true?

(a) Any two quadratic forms of rank 3 are isomorphic over R.

(b) Any two quadratic forms of rank 3 are isomorphic over C.

(d) There are exactly three non zero quadratic forms of rank 2 upto isomorphism over R and C.

Ans. (b), (c)

Sol. Since, two real quadratic form in n-variable are real and equivalent iff they have the same rank and signature or index.

Hence, option (a) is false.

Now, two real quadratic form in n-variables are complex and equivalent, iff they have same rank.

So, option (B) is true.

For option (c):

For option (d): There are quadratic forms in option (a) and (b).

Then, option (d) is false.

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

59. Let C[0, 1] be the ring of all real valued continuous function on [0, 1].

Then which of the following statements are true?

(a) A is an ideal in C[0, 1] but is not a prime ideal in C[0, 1].

(b) A is a prime ideal in C[0, 1].

(d) A is a prime ideal in C[0, 1], but is not a maximal ideal in C[0, 1].

Ans. (a)

Sol.

An is an ideal of C[0, 1]

Now, consider the two function

Also consider another function

But, the function

So, A is not a prime ideal on C[0, 1] option (a) is true.

C[0, 1] is commutative ring with unity A is not prime ideal of C[0, 1].

Now, if A is a maximal ideal then C[0, 1]/A is a field. R is a commutative ring with unity.

Then I is

a prime ideal iff R/I is an integral domain.
a max. ideal iff R/I is a field.

R/I is a field R/I is an integral domain.

I is a prime ideal.

I is a max. ideal I also is prime ideal.

Thus, (c) and (d) are ruled.

Option (a) is correct.

60. Which of the following statements are true?

(a) There exist three mutually disjoint subsets of R each of which is countable and dense in R.

(b) For each there exists n mutually disjoint subsets of R each of which is countable and dense in R.

(d) There exist uncountable number of mutually disjoint subsets of R each of which is countable and dense in R.

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

Sol.

Hence, all options are effect.

61. Consider the power series

f(z) = .

Which of the following are true?

(a) Radius of convergence of f(z) is infinite

(b)

(d) f(z) has infinitely many zeros.

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

Sol.

Since, the product of two analytic functions is again analytic.

Then, analytic = ROC.

(A) Radius of convergence of entire function is infinite.

f(z) has infinitely may zeros.

Thus, case of option (a), (c) and (d).

62. Let F[X] be the polynomial ring in one variable over field F. Then which of the following statements are true?

(a) F[X] is UFD.

(b) F[X] is PID.

(d) F[X] is a PID but is not an Euclidean domain.

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

Sol. F[X] be the polynomial ring in one variable over a field F.

In order to solved the given question, we have learn the following results.

Result 1: F[X] is a principal ideal domain (PID) iff F is a field.

Result 2: F[X] is an Euclidean domain (ED), iff F is a field.

Result 3: If R is a unique factorisation domain (UFD), then R[X] is a unique factorisation domain (UFD).

Since F is a field implies F[X] is a principal ideal domain.

Again F is a field implies F[X] is an Euclidean domain (ED).

Thus, correct options are (B), (C) and (D) is false.

Also, F is a field implies F is a UFD.

Now, F is UFD implies F[X] is a UFD.

Thus, correct option is A.

Hence, options (A), (B) and (C) are true.

63. Let be a monic polynomial of degree n. Then which of the following are true?

(a)

(b)

(c)

(d)

Ans. (a), (b)

Sol. be a monic polynomial of degree n. Solving given question, some important points are:

Point 1: Let R be the commutative ring with unity. Let be a polynomial in one variable over, then f(x) is monic, iff f(x) is non-zero and its leading coefficient is 1.

Point 2: Let R is a integral domain. A non-zero, non unit polynomial is said to be irreducible polynomial over R if f(x) = g(x) h(x).

Where

Thein either g(x) is unit R[x] or h(x) is unit in R[x]

Point 3: If f(x) is irreducible over Z[r] then f(x) is irreducible over Q[x].

Point 4: If and f(x) is irreducible in Q[x] and where a is constant then f(x) is irreducible over Z[x].

Let f(x) is irreducible in Z[x], implies f(x) is irreducible in Q[x].

Hence, correct option is (A).

Let f(x) is irreducible in Q[x].

Also, f(x) is monic polynomial implies

Then, f(x) is irreducible in Z[x].

Hence, correct option is (B).

Let f(x) = x⁴ + x² + 1 = (x² + x + 1) (x² – x + 1) implies f(x) is irreducible in Z[r]. But polynomial f(x) = x⁴ + x² + 1 has no real root.

Hence, option (c) is false.

Let f(x) = x² – 2 has a real root. But f(x) = x² – 2 is not reducible in Z[x],

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

64. Let be an analytic function. For which of the following statements are true?

(a) f can take the value z₀ at finitely many points in .

(b)

(c)

(d)

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

Sol. Let be an analytic functions defined on D such that the set

has a limit point in D then

Theorem: The limit point of the set of zeros of a complex valued function is an isolated essential singularity.

For the option (a): f can take the value z₀ for finitely many points of

If f takes z₀ infinitely often in set then analytic function g(z) = f(z) – z₀ would have infinitely many zeros in the set.

. So, by BN theorem, the set of zeros of g(z) i.e., those infinitely many points from the set has a limit point in C which means that g has an isolated essential singularity in C which is a contradiction.

Thus, option (a) is true.

f is the constant function z₀ similarly in case of option (b).

So, the set E has limit point in C.

Since

Thus by limit uniqueness theorem

For option (c): f is the constant function z₀.

Thus it may not be true.

For example f(z) =

Hence, options (a), (b) and (d) are true.

65. Let I be an ideal of Z. Then which of the following statements are true?

(a) I is a principal ideal.

(b)

(c)

(d)

Ans. (a), (d)

Sol. In order to solve the question, first we required to know the given results:

Result 1: An ideal principal ideal.

Result 2: (i) I = {0} is prime ideal of Z.

(ii) I = pZ is prime ideal of Z where p is a prime number.

Result 3: When R is a continuous ring with unity then every maximal ideal of R is prime ideal of R.

Result 4: When R is a finite commutative ring with unity, then every prime ideal of R is maximal ideal of R.

We know that every ideal of Z is principal ideal.

F is principal idea of Z.

Thus, correct option is (A).

Also, {0} is prime ideal of Z. But {0} is not a maximal ideal of Z.

Thus, option (b) is false.

Z is a commutative ring with unity.

Every maximal ideal of Z is prime ideal of Z.

Thus, correct option is (d) and option (c) is false became Z is not a finite commutative ring.

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

66. Let U be an open subset of C and be an analytic function. Then which of the following are true?

(a) If f is one-one then f(U) is open in C.

(b) If f is onto then U = C.

(d) If f(U) is closed in C then f(U) is connected.

Ans. (a)

Sol.

67. Consider [n] = {1, 2, ..., n} with the discrete topology let X = be the product space with the product topology. For define T(x) = (1, a₁, a₂, ...). Then which of the following statements are true?

(a) Let for n = 1, 2, 3, ... be a sequence in X. Then it is convergent.

(b) X is a compact, Hausdorff space.

(c)

(d)

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

Sol.

68. Let the be a field. Then which of the following statements are true?

(a) All extensions of degree 2 of F are isomorphic as fields.

(b) All finite extensions of F of same degree are isomorphic as fields if char (F) > 0.

(d) All finite normal extensions of F are isomorphic as fields if char (F) = 0.

Ans. (c)

Sol. The following results are required to know to find the solution:

Result 1: The degree of extension over field is equal to degree of irreducible polynomial over that field.

Result 2: The char (Z_n) = n where char (Z_n) means characteristic of ring Z_n.

Result 4: Let [K : F] be an algebraic field extension then [k : F] is normal if given any irreducible polynomial f(x) ∊ F[x], such that f(x) has at least one root in K.

The f(x) splits in K.

Result 5: An extension field K of F is said to be algebraic extension if every element of K is algebraic over F.

Thus, correct option is (c).

69. Let be an open connected set and be a non-constant analytic function.

Consider the following two sets X = : f vanishes on an open neighbourhood of z in U.

Then which of the following statements are true?

(a) X is closed in U

(b) Y is closed in U

(d) Y is open in U

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

Sol. Theorem 1: Let f and g be two continuous function on a set

Then

Theorem 2: Let f be an analytic function defined on a domain such that the set f(z) = 0} has a limit in U.

B has infinitely many limit points in U and

Thus, option (c) is true.

For options (a) and (d).

Hence, both are closed and open in U.

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

70. For a given integer k, which of the following statements are false?

(a)

(b)

(c)

(d)

Ans. (c), (d)

Sol. The following result is required to be remember in order to solve the problem.

Result 1: When f(x) is a polynomial with integer coefficient and it has a root.

Then that root persists mod m and mod n.

Thus, x be the unit in Z₈ and Z₉.

Option (a) and (b) are true and option (c) and (d) are false.

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

71. Consider the initial value problem

Then which of the following statements are true?

(a) There exists a unique solution in .

(b) Every solution is bounded in .

(d) The solution becomes unbounded in some sub-interval of .

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

Sol.

Boundary conditions,

If l = 0 then given ordinary equations

From partial fraction.

Solving (i), we get

Theorem: Let are continuous on [a, b] and suppose p(x) > 0 and r(x) > 0, then the S-L problem.

Hence, an increasing sequence of eigen values such that

and to each I_n there is an single eigen function y_n(x).

Then p(x), q(x) and r(nx) are continuous function and p(x) > 0 and r(x) >

Thus, since corresponds to trivial solution, therefore we have eigen values for I_n such that

Hence, all the eigen values of this SL problem are positive.

Hence, option (b) is correct.

72. Consider the eigen value problem

Then which of the following statements are true?

(a) all the eigenvalues are negative.

(b) all the eigenvalues are positive.

(d) there are no eigenvalues.

Ans. (b)

Sol.

73. A possible initial strip (x₀, y₀, z₀, p₀, q₀) for the Cauchy problem pq = 1 where z₀(s) = 1 for s > 1 is

(a)

(b)

(c)

(d)

Ans. (a), (b)

Sol. We have

Integrating p = a and q = b, pq = 1

Hence, option (c) is false.

Hence, option (d) is false.

Therefore, true options are (a) and (b).

74. Let u(x, t) be the solution

Then u(2, 3) is equal to

(a) 9

(b) 1

(d) 12

Ans. (a)

Sol.

Take initial condition u(x, 0) = f(x)

From Almbert solution,

Now, here c = 1, f(x) = 0 and g(x) = 0

u_CF = 0

So, the general solution for the given non-homogeneous wave equation

Hence, correct option is (a).

75. The values of a A, B, C for which the quadrature formula

is exact for polynomials of highest possible degree are

(a)

(b)

(c)

(d)

Ans. (a), (d)

Sol.

If n = 3

Solving equations (v) and (vi), we get

Now, from equations (vi), (v), we get

Now from equation (iv), we get

[From equation (vi)]

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

76. Assume that h₁, h₂, g₁ and

Let

be an integral equation. Consider the following statements:

S₁ : If the given integral equation has a solution for some then

S₂ : The given integral equation has a unique solution for every if l is not a characteristic number of the corresponding homogeneous equation.

Then

(a) Both S₁ and S₂ are true

(b) S₁ is true but S₂ is false

(d) Both S₁ and S₂ are false

Ans. (c)

Sol.

If l is not characteristic number of corresponding homogeneous equation.

Hence, option (c) is true.

77. The minimum value of the functional s.t. is equal to

(a)

(b) 1

(d)

Ans. (b)

Sol.

Also, given that

The boundary conditions are

where I is the Lagrange multiplier.

Now

Thus, the Euler equation for the functional F is

Case-I: If l = 0, y² = 0 solving, we get y(x) = c₁x + c₂

where c₁ and c₂ are arbitrary constants, using boundary condition we get c₁ = c₂ = 0 which gives to the trivial solution y(x) = 0.

where c₁ and c₂ are arbitrary constants.

Using boundary conditions,

leads, to trivial solution y(x) = 0.

Thus, we have

Hence, the minimum value of the functional is 1.

Thus, correct is (b).

78. Consider a mechanical system whose position is described using the generalized coordinates q₁, ..., q_n. Let be the kinetic energy of the system. If the generalized force Q_j, acting on the system is zero then the Lagrange equations of motion are

(a)

(b)

(c)

(d)

Ans. (a), (c)

Sol.

79. Let y be a solution of

y(0) = 0. Then y has

(a) infinitely many zeros in [0, 1]

(b) infinitely many zeros in

(c)

(d)

Ans. (b), (c)

Sol.

80. Let be an extremizing function for the functional 0 = y(1). Then an extremal y(x) satisfying the given conditions at 0 and 1 together with the natural boundary conditions is given by

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Euler's equation is given by

Hence, Euler's equation becomes

Hence, option (c) is true.

81. The integral equation has

(a) no solution

(b) unique solution

(d) infinitely many solutions

Ans. (a)

Sol.

Homogeneous equation.

homogeneous equation

Thus, eigen function y(x) = cos²x.

Thus, f(x) is not orthogonal to eigen function y(x).

No solution of the given integral equation.

Hence, option (a) is true.

82. Consider the ordinary differential equation

and the following numerical scheme to solve the ODE

If 0 < h < 1/2 then which of the following statements are true?

(a)

(b)

(d)

Ans. (b), (c)

Sol.

The option (a) is false and option (b) is true.

Hence, option (c) is also true.

Thus, option (d) is false.

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

83. Consider the random effect model 10 where b_i ~ i.i.d. N(0, are all independent of each other. The parameter space for the model is be the usual unbiased ANOVA estimators of and respectively and be the maximum likelihood estimators of respectively. Then, which of the following events can happen with positive probability for some parameter values?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

84. Let be a Markov chain with state space such that the transition probabilities are given by

for = 0, 1, 2, ... where 0 < q < 1. Then which of the following statements are correct?

(a) The Markov chain is irreducible

(b) The Markov chain is a periodic

(c)

(d) The Markov chain is positive recurrent

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

Sol.

85. A random variable T has a symmetric distribution if T and –T have the same distribution. Let X and Y be independent random variables. Then which of the following statements are correct?

(a) If X and Y have the same distribution then X – Y has a symmetric distribution.

(b) If X ~ N(3, 1) and Y ~ N(2, 2) then 2X – 3Y has a symmetric distribution.

(d) If X has a symmetric distribution then XY has a symmetric distribution.

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

Sol.

86. Let {(X_n, Y_n) : n ≥ 1} and (X, Y) be random variables on Then which of the following statements are correct?

(a)

(b)

(c)

(d)

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

Sol.

87. Consider a Balanced Incomplete Block Design of v treatments and b blocks of k plots each. Let N be the v × b incidence matrix of the design. Then which of the following statements are correct?

(a)

(b)

(c)

(d)

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

Sol.

88. Suppose in a single service queue, customers arrive at a Poisson rate of one per ten minutes, and the service time is Exponential at a rate of one service per five minutes. Let P_n be the probability that there are n customers in the system in steady state. Then which of the following statements are correct?

(a)

(b) The expected number of customers in the system is 1 in steady state.

(d) The expected amount of time a customer spends in the system in steady state is 10 minutes.

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

Sol.

89. Suppose the conditional pdf of a random variables X given is

where the prior distribution of is Uniform (0, 1). Based on a single observation x from X, which of the following statements are correct?

(a) The Bayes estimate for under squared error loss function is – x log_ex

(b) The Bayes estimate for under squared error loss function is –

(d) The Bayes estimate for under absolute error loss function is .

Ans. (b), (d)

Sol.

90. Let be i.i.d. observations with E(X_i) = 0 and Then which of the following are correct?

(a)

(b)

(c)

(d)

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

Sol.

91. Suppose that are independent where is positive definite.

Let be a fixed p-vector such that and define d = Z/||Z||. Then which of the following random variables has a distribution possibly after begin scaled by a constant factor?

(a)

(b)

(c)

(d)

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

Sol.

92. Let X₁, X₂ and X₃ be i.i.d. normal random variables with mean and variance q² where is unknown. Then which of the following statements are correct?

(a)

(b)

(c)

(d)

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

Sol.

93. Suppose X₁, X₂, ..., X_n and i.i.d. uniform Let X₍₁₎ = min.{X₁, ..., X_n} and X_(n) = max. {X₁, ..., X_n}. Then which of the following statements are correct?

(a)

(b)

(c)

(d)

Ans. (b), (c)

Sol.

94. Consider the following system with three independent u₁, u₂ and u₃:

Suppose that the failure probability of each component is p and let f(p) be the probability that the whole system is still functioning. Then which of the following statements are correct?

(a)

(b)

(c)

(d)

Ans. (a), (c)

Sol.

95. Given data {(x_i, y_i) : i = 1, 2, ..., n} where n > 2 and not all x_i's are identical, the simple linear regression model is fit. Let h_ii be the i^th diagonal element of the Hat matrix where X_{n × 2} is the corresponding model matrix. Then which of the following are possible for some choice of n and x₁, x₂, ..., x_n?

(a) h_ii = –1 for some i

(b) h_ii = 0 for some i

(d) All h_ii are equal

Ans. (c), (d)

Sol.

96. Consider a Markov chain with state space S. Let d(k) denote the period of state Which of the following statements are correct?

(a) and i is recurrent then j is recurrent.

(b)

(c)

(d) and i is null recurrent then j is positive recurrent.

Ans. (a), (b)

Sol.

97. Let X be a discrete random variable with sample space = {1, 2, ..., 10} and probability mass function p(x),

Consider testing the hypothesis

based on a single observation X. Then which of the following statements are correct?

(a) The test with critical region is most powerful of its size.

(b) The test with critical region {X < 2} is unbiased at level

(d) There exists a non-randomized test of size 0.05.

Ans. (a)

Sol.

98. Let X and Y be real-valued independent random variables on Then which of the following statements are correct?

(a) E[cos(tX + uY)] = E[cos(tX)] E[cos(uY)] – E[sin(tX)] E[sin(uY)] for all t,

(b) If X ~ N(2, 1) and Y ~ N(0, 2) then Var(X + Y) = 3 where represents Normal distributions with mean m and variance

(c)

(d)

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

Sol.

99. Let {X_n : n > 1} be i.i.d. with common unknown continuous distribution function where is the unique median of F. Define

For testing which of the following statements are correct?

(a)

(b) Test based on S_n is distribution-free under H₁.

(d) The sequence of right-tailed tests based on S_n, n > 1 is consistent.

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

Sol.

100. To estimate the population total where y₁, y₂, ..., y_N are study variables and N is the size of the finite population, a sample on size n is drawn using PPSWR(p₁, p₂, ..., p_N) scheme. If is the Hansen-Hurwitz estimator for Y then which of the following are correct?