CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   $limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1$.

2. B

   $lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

3. C

   $liminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi$.

4. D

   $lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

Single choice · Continuity and Differentiation

1. A

   f(x) = sin(1/x)

2. B

   f(x) = $e^{-1/x^{2}}$

3. C

   f(x) = eˣ cos(1/x)

4. D

   $f(x) = \cos x \cdot \cos(\pi/x)$

Single choice · Eigenvalues and Canonical Forms

1. A

   The eigenspace of A for the eigenvalue 0 is two-dimensional.

2. B

   All the eigenvalues of A are real.

3. C

   $A^{3} = 0$.

4. D

   A is diagonalizable.

Single choice · Eigenvalues and Canonical Forms

1. A

   Both (a) and (b) are true.

2. B

   Both (a) and (b) are false.

3. C

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

4. D

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

Single choice · Inner Product Spaces and Forms

1. A

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $x_{i}y_{j}$

2. B

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $(x_{i}^{2} + y_{j}^{2})$

3. C

   ⟨x, y⟩ $= \sum_{j=1}^n$ $j^{3} x_{j}y_{j}$

4. D

   ⟨x, y⟩ $= \sum_{j=1}^n$ $x_{j} y_{n}$₋$_{j}$₊$_{1}$

Single choice · Groups

1. A

   If |G| $= p^{6}$, then G has a subgroup of index $p^{2}$.

2. B

   If |G| $= p^{6}$, then G has at least five normal subgroups.

3. C

   Center of G can be infinite.

4. D

   There exists G with |G| $= p^{6}$ such that G has exactly six normal subgroups.

Single choice · Rings and Fields

1. A

   0

2. B

   2

3. C

   4

4. D

   8

Single choice · Rings and Fields

1. A

   0

2. B

   1

3. C

   2

4. D

   3

Single choice · Lebesgue Measure and Integration

1. A

   S is in bijection with the set of rational numbers.

2. B

   S is in bijection with the set of real numbers.

3. C

   S is in bijection with S × S.

4. D

   S is in bijection with the power set of S.

Single choice · Singularities and Residues

1. A

   $\sum_{l=0}^\infty$ 1/(l+1)!

2. B

   $\sum_{l=0}^\infty$ 1/(l!(l+1))

3. C

   $\sum_{l=0}^\infty$ 1/(l!(l+1)!)

4. D

   $\sum_{l=0}^\infty$ $1/((l^{2}+l)!)$

Single choice · Analytic Functions

1. A

   f is an entire function.

2. B

   f has a simple pole at z = 0.

3. C

   f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$, where $a_{0} = 1, a_{1} = 0$ and $a_{n}$₊$_{2} = a_{n} + a_{n}$₊$_{1}$ for n ≥ 0.

4. D

   f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$, where $a_{0} = 1, a_{1} = 1$ and $a_{n}$₊$_{2} = a_{n} + a_{n}$₊$_{1}$ for n ≥ 0.

Single choice · Ordinary Differential Equations

1. A

   x(t) is a bounded solution for some $x_{0} \ne 0$.

2. B

   $e^{-6t}$|x(t)| → 0 as $t \to \infty$, for all $x_{0} \ne 0$.

3. C

   $e^{-t}$|x(t)| $\to \infty$ as $t \to \infty$, for all $x_{0} \ne 0$.

4. D

   $e^{-10t}$|x(t)| → 0 as $t \to \infty$, for all $x_{0} \ne 0$.

Single choice · Partial Differential Equations

1. A

   2

2. B

   3

3. C

   1/2

4. D

   1/3

Single choice · Calculus of Variations

1. A

   (P) has no stationary function (extremal).

2. B

   y ≡ 0 is the only stationary function (extremal) for (P).

3. C

   (P) has a unique stationary function (extremal) y not identically equal to 0.

4. D

   (P) has infinitely many stationary functions (extremals).

Single choice · Linear Integral Equations

1. A

   K(x,t) = t(1 − x) for t < x; x(1 − t) for t > x

2. B

   $K(x,t) = t^{2}(1 - x)$ for $t < x; x^{2}(1 - t)$ for t > x

3. C

   $K(x,t) = \sqrt{t}(1 - x)$ for $t < x; \sqrt{x}(1 - t)$ for t > x

4. D

   $K(x,t) = \sqrt{t^{3}}(1 - x)$ for $t < x; \sqrt{x^{3}}(1 - t)$ for t > x

Single choice · Probability

1. A

   $X_{(1)}$ and $X_{(4)}$ are independent.

2. B

   Expectation of $X_{(2)}$ is 1/2.

3. C

   Variance of $X_{(2)}$ is 8/81.

4. D

   $X_{(4)}$ is a degenerate random variable.

Single choice · Estimation

1. A

   $\lambda$⁻$^{1}$

2. B

   $\lambda$

3. C

   $\lambda^{2}$

4. D

   $e^{-\lambda}$

Single choice · Hypothesis Testing

1. A

   $P_{H_{0}}(R$ = 6) = 28/286, $P_{H_{0}}(R$ = 9) = 28/143.

2. B

   $P_{H_{0}}(R$ = 6) = 21/286, $P_{H_{0}}(R$ = 9) = 15/286.

3. C

   $P_{H_{0}}(R$ = 6) = 21/286, $P_{H_{0}}(R$ = 9) = 28/143.

4. D

   $P_{H_{0}}(R$ = 6) = 21/286, $P_{H_{0}}(R$ = 9) = 15/286.

Single choice · Linear Models and Multivariate

1. A

   $\sum Y_{i}x_{i} / \sum x_{i}^{2}$

2. B

   $\sum Y_{i} / \sum x_{i}$

3. C

   $(1/n) \sum Y_{i}/x_{i}$

4. D

   $(1/n) \sum Y_{i}x_{i} / x_{i}^{2}$

Single choice · Linear Models and Multivariate

1. A

   (0, 5)ᵀ, [[5, −3], [−3, 40]]

2. B

   (0, 5)ᵀ, [[5, −6], [−6, 20]]

3. C

   (0, 5)ᵀ, [[5, 3], [3, 20]]

4. D

   (0, 5)ᵀ, [[5, 6], [6, 40]]

Single choice · Linear Models and Multivariate

1. A

   $W_{1}(2, 2)$

2. B

   $W_{1}(1, 2)$

3. C

   $W_{1}(2, 1)$

4. D

   $2\chi^{2}_{4}$

Single choice · The Real Line

1. A

   inf U = 5.

2. B

   inf U = 4.

3. C

   inf U = 3.

4. D

   inf U = 2.

Single choice · The Real Line

1. A

   $\limsup a_{n} = \sqrt{3}/2$.

2. B

   $\limsup a_{2n} = 1$.

3. C

   $\limsup a_{2n} = 1/2$.

4. D

   $\limsup a_{3n} = 0$.

Single choice · The Real Line

1. A

   (a) is convergent, but (b) is not convergent.

2. B

   (a) is not convergent, but (b) is convergent.

3. C

   Both (a) and (b) are convergent.

4. D

   Neither (a) nor (b) is convergent.

Single choice · Continuity and Differentiation

1. A

   F(x) has exactly one real root.

2. B

   F(x) has exactly four real roots.

3. C

   F(x) has at most two real roots.

4. D

   F(x) has at most one real root.