CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Singularities and Residues

1. A

   f(0) = 0

2. B

   f(z) = 0 for all z ∈ 𝔻.

3. C

   There exists a nonzero constant c such that f(z) = ce^(−1/z) for all z ∈ 𝔻 \ {0}.

4. D

   There exists a nonzero constant c and a positive integer n such that f(z) = cz$^{n}e^(-1/z)$ for all z ∈ 𝔻 \ {0}.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

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

2. B

   f′(0) = f″(0) = f‴(0) = 0

3. C

   There is an entire function $g : \mathbb{C} \to \mathbb{C}$ such that $f(z) = g(z^{4})$ for all $z \in \mathbb{C}$.

4. D

   f is necessarily a constant function.

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

1. A

   The value of the Euler $\varphi-$function is even for all integers n ≥ 3.

2. B

   Let G be a finite group and S a subset of G with |S| > |G|/2. Then {ab : a, b ∈ S} = G.

3. C

   The polynomial ring $\mathbb{R}[x_{1}, \dots, x_{n}]$ is a Euclidean domain for all integers n ≥ 1.

4. D

   The subset {f ∈ C([0, 1]) : f(1/2) = 0} of the ring C([0, 1]) of continuous functions from [0, 1] to $\mathbb{R}$ is a prime ideal.

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

1. A

   $\mathbb{Q}$ with ordinary addition

2. B

   $\mathbb{C}$ \ {0} with ordinary multiplication

3. C

   The cyclic group of order 5

4. D

   The symmetric group $S_{5}$

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

1. A

   S is a cyclic group.

2. B

   S has $\varphi(2025)$ elements, where $\varphi$ denotes the Euler $\varphi-$function.

3. C

   S has $\varphi(2025)$ generators, where $\varphi$ denotes the Euler $\varphi-$function.

4. D

   S has 81 elements.

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

1. A

   The set of all irrational numbers is dense in $(\mathbb{R}, \tau)$.

2. B

   For each prime number p, the set {$0, \sqrt{p}$} is dense in $(\mathbb{R}, \tau)$.

3. C

   [0, 1] is compact in $(\mathbb{R}, \tau)$.

4. D

   $(\mathbb{R}, \tau)$ is Hausdorff.

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

1. A

   The spaces B, the closure of W, and C are homeomorphic.

2. B

   The spaces B, W, C are homeomorphic.

3. C

   If C is compact, then A, B, C are homeomorphic.

4. D

   If A is connected, then B and C are connected.

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

1. A

   There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution y_b is bounded above on I_b

2. B

   There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution y_b is bounded below on I_b

3. C

   There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution y_b is bounded above on I_b

4. D

   There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution y_b is bounded below on I_b

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   $2\lambda + 24p = 0$

2. B

   $7\lambda - 12p = 4$

3. C

   $2\lambda + 24p = -3$

4. D

   $7\lambda - 12p = 11$

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   f(5) + g(3) = 50

2. B

   2f(5) − g(3) = 4

3. C

   f(1) + g(3) = 50

4. D

   f(5) + g(3) = 74

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   S(2) has exactly two elements

2. B

   S(1/2) has exactly one element

3. C

   S(2) is empty

4. D

   S(1/2) has exactly two elements

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   $\varphi(x) = 2x + 1$ for every x ∈ [−1, 1]

2. B

   There exists $\varepsilon > 0$ such that $J[y] \ge J[\varphi]$ for every $y \in B_{0}(\varphi, \varepsilon)$

3. C

   There exists $\varepsilon > 0$ such that $J[y] \ge J[\varphi]$ for every $y \in B_{1}(\varphi, \varepsilon)$

4. D

   There exists $\varepsilon > 0$ such that $J[y] \le J[\varphi]$ for every $y \in B_{1}(\varphi, \varepsilon)$

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

1. A

   f(x) = cos x

2. B

   f(x) = cos 5x

3. C

   f(x) = sin x

4. D

   f(x) = sin 5x

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)