CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Linear Integral Equations

1. A

   $u(\pi/2) = 4e^(\pi/2)$

2. B

   $u(\pi) = 3e^\pi$

3. C

   $u(-\pi) = 4e^(-\pi)$

4. D

   $u(-\pi/2) = 4e^\pi$

Multi-select · full marks only for the exact set · Probability

1. A

   $Y_{2}$ follows gamma distribution with $E(Y_{2}) = 2$.

2. B

   E(Y(3,n)) → 3 as $n \to \infty$

3. C

   $Y_{3}$ follows beta distribution with $E(Y_{3}) = 1/3$.

4. D

   Y(2,n) follows beta distribution with parameters 2 and n − 1.

Multi-select · full marks only for the exact set · Probability

1. A

   $P(X_{1} > 1) \ge P(X_{2} > 1)$

2. B

   $P(X_{1} > 1) \le P(X_{2} > 1/2)$

3. C

   $E(X_{1}) \ge E(X_{2})$ provided both the expectations exist.

4. D

   $h(t) = h_{1}(t) + h_{2}(t), t \in \mathbb{R}$, is the hazard function of the random variable Y = min{$X_{1}, X_{2}$}.

Multi-select · full marks only for the exact set · Probability

1. A

   E(Zᵐ) = 1/(m + 1), for all $m \in \mathbb{N}$

2. B

   $E(\Phi$⁻$^{1}(Z)) = 0$, where $\Phi(\cdot)$ is the cumulative distribution function of standard normal random variable.

3. C

   Z is degenerate at 0.

4. D

   If $Z_{1}$ and $Z_{2}$ are independent and identically distributed (i.i.d.) random variables having distribution same as the distribution of Z, then Z has the same distribution as $(Z_{1} + Z_{2})/2$.

Multi-select · full marks only for the exact set · Estimation

1. A

   $(\sum(i=1..8)X_{i})(\sum(i=8..13)X_{i})$ is an unbiased estimator of $48\theta^{2}$.

2. B

   $(1/13)\sum(i=1..13)X_{i}$ is method of moments estimator of $\theta$.

3. C

   There does not exist any unbiased estimator of $e^(-7\theta)$.

4. D

   $(e^(X_{7}), \sum(i=1..6)X_{i}, \sum(i=8..13)X_{i})$ is a sufficient statistic for $\theta$.

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   The design is a balanced incomplete block design.

2. B

   The design is not connected.

3. C

   The design is binary.

4. D

   The design is symmetric.

Multi-select · full marks only for the exact set · Linear Programming

1. A

   The optimum value is 3

2. B

   The optimum value is 3/2

3. C

   (0, 2, 1) is an extreme point of the feasible region

4. D

   (1/2, 0, 1) is the optimal solution

Multi-select · full marks only for the exact set · The Real Line

1. A

   If $lim_{n}\to\infty b_{n}$ exists in $\mathbb{R}$, then {$a_{n} : n \ge 1$} is bounded

2. B

   If $lim_{n}\to\infty b_{n} = 1$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

3. C

   If $lim_{n}\to\infty b_{n} = 1/2$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

4. D

   If $lim_{n}\to\infty b_{n} = 0$, then $lim_{n}\to\infty a_{n} = 0$

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is strictly increasing

2. B

   f is strictly decreasing

3. C

   f is either strictly increasing or strictly decreasing

4. D

   f is onto

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f need not be surjective

2. B

   f must be surjective if K = [0, 1]

3. C

   f is injective and f⁻$^{1} : f(K) \to K$ is continuous

4. D

   f is injective, but f⁻$^{1} : f(K) \to K$ need not be continuous

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   f(x) is not uniformly continuous on [0, 1)

2. B

   The sequence $(p_{n}(x))$ converges to f(x) pointwise on [0, 1)

3. C

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, 1)

4. D

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, c] for every 0 < c < 1

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   (1, 1, 1, 1, 1)

2. B

   (0, 0, 1, 2, 4)

3. C

   (1, 0, 1, 0, 1)

4. D

   (1, 0, 1, 0, 2)

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   There exists an injective $\mathbb{Q}-$linear transformation $T : \mathbb{R} \to \mathbb{Q}[x]$

2. B

   There exists an injective $\mathbb{Q}-$linear transformation $T : \mathbb{Q}[x] \to \mathbb{R}$

3. C

   The $\mathbb{Q}-$vector spaces $\mathbb{Q}[x]$ and $\mathbb{R}$ are isomorphic

4. D

   There do not exist non-zero $\mathbb{Q}-$linear transformations $T : \mathbb{R} \to \mathbb{Q}[x]$

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   There exists a non-zero vector $v_{1} \in \mathbb{R}^{4}$ such that Tv$_{1} = 2v_{1}$

2. B

   There exists a non-zero vector $v_{2} \in \mathbb{R}^{4}$ such that Tv$_{2} = v_{2}$

3. C

   For every non-zero vector $v \in \mathbb{R}^{4}$, the set {2v, 3Tv} is linearly independent

4. D

   T is a one-one function

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   The minimal polynomial of B is $x^{2} + x + 1$

2. B

   The minimal polynomial of B is $x^{2} - x + 1$

3. C

   $B^{3} = I_{4}$

4. D

   $B^{3} + I_{4} = 0$

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   1

2. B

   2

3. C

   3

4. D

   5

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   The dimension of V is even

2. B

   The trace of T is zero

3. C

   The minimal polynomial of T cannot have two distinct roots

4. D

   The minimal polynomial of T is equal to its characteristic polynomial

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   q is a quadratic form in $x_{1}, x_{2}, y_{1}, y_{2}$

2. B

   There exists $(t_{1}, t_{2}) \in \mathbb{R}^{2}$ such that $q_{1}(t_{1}, t_{2}) = 5$

3. C

   There does not exist $(s_{1}, s_{2}) \in \mathbb{R}^{2}$ such that $q_{2}(s_{1}, s_{2}) = -5$

4. D

   Given $\alpha \in \mathbb{R}$, there exists a vector $w \in \mathbb{R}^{4}$ such that $q(w) = \alpha$

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f(z) = 2024 for all $z \in \mathbb{C}$

2. B

   f is a constant function

3. C

   f is an injective function

4. D

   f is a bijective function

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   There exists an infinite set X such that f(z) = 0 for all z ∈ X

2. B

   There exists a non-empty closed set X such that f(z) = 0 for all z ∈ X

3. C

   The set X_k is unbounded for each k ≥ 1

4. D

   If there exists a bounded sequence (z_k)_(k≥1) such that z_k ∈ X_k for each k ≥ 1, then f has a zero

Multi-select · full marks only for the exact set · Groups

1. A

   3

2. B

   4

3. C

   5

4. D

   6

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   If f(a) is a unit in S for every non-zero element a ∈ R, then S is a field

2. B

   If f(a) is a unit in S for every non-zero element a ∈ R, then f(R) is a field

3. C

   If R is a field, then f(a) is a unit in S for every non-zero element a ∈ R

4. D

   If a is a unit in R, then f(a) is a unit in S

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   1

2. B

   8

3. C

   16

4. D

   24

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   r(t) → 0 as $t \to +\infty$

2. B

   r(ln 2) = e⁻$^{1}$

3. C

   r(ln 2) = 2e⁻$^{1}$

4. D

   r(t)eᵗ → 0 as $t \to +\infty$

Multi-select · full marks only for the exact set · Partial Differential Equations

1. A

   $e^\pi$

2. B

   $e^(-\pi)$

3. C

   $-e^\pi$

4. D

   $-e^(-\pi)$