CSIR NET Maths previous year question papers

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

1. A

   $(\sqrt{2}, 3)$

2. B

   $(\sqrt{2}, 6)$

3. C

   $(\sqrt{3}, 22/3)$

4. D

   $(\sqrt{3}, 23/3)$

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

1. A

   J[y] ≤ J[0] for all $y \in B_{0}(0, \varepsilon)$

2. B

   J[y] ≤ J[0] for all $y \in B_{1}(0, \varepsilon)$

3. C

   J[y] ≥ J[0] for all $y \in B_{0}(0, \varepsilon)$

4. D

   J[y] ≥ J[0] for all $y \in B_{1}(0, \varepsilon)$

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

1. A

   $y(1) = (1 + \pi/4)e$

2. B

   $y(1) = (1 + \pi/2)e$

3. C

   $y(\sqrt{3}) = (1 +$ $3\pi/4)e^{\sqrt{3}}$

4. D

   $y(\sqrt{3}) = (1 +$ $4\pi/3)e^{\sqrt{3}}$

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

1. A

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

2. B

   f(x) = eˣ

3. C

   f(x) = 2 − 3x

4. D

   f(x) = x − 1

Multi-select · full marks only for the exact set · Classical Mechanics

1. A

   $a^{2} + b^{2} = 2$

2. B

   a − b = 2

3. C

   a + b = 2

4. D

   a = 1, b = 1

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

1. A

   $E(X^{4}) = 3$

2. B

   P(X ∈ {1, 2, 5}) = 1/2

3. C

   E(eˣ) does not exist

4. D

   P(X ≤ 0) = 1/2

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

1. A

   E(X) ≤ 2/3

2. B

   $E(X^{2}) = 2/3$

3. C

   E(|X|) = 2/3

4. D

   Var(X) > 2/3

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

1. A

   E(Y) = 0

2. B

   P(Y > 0) < P(Y < 0)

3. C

   P(Y < −1) < P(Y > 1)

4. D

   $E(Y^{2}) = 1$

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

1. A

   The distributions of X − 12.5 and 12.5 − X are identical

2. B

   P(X ≤ 4) = P(X ≥ 22)

3. C

   The coefficient of variation (in percentage) of X is 20

4. D

   P(X ≤ 4.9) = P(X ≥ 20.1)

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

1. A

   P(X > 4) = P(X > 1)P(X > 2)

2. B

   P(X > 4 | X > 2) = P(X > 2)

3. C

   E(X) = 1/3

4. D

   6X ~ $\chi^{2}_{3}$

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $lim_{t\to\infty}$ P(N(t) = 1) = 2/9

2. B

   $P(T_{1} > 0) = 1/3$

3. C

   $E(T_{1}) = 1/90$

4. D

   $E(T_{2}) = 1/35$

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $P(W_{2}$ˣ $< W_{3}$ʸ) = 11/16

2. B

   $P(W_{1}$ˣ $< W_{1}$ʸ) = 1/2

3. C

   $P(W_{2}$ˣ $< W_{3}$ʸ) = 13/16

4. D

   $P(W_{1}$ˣ $< W_{1}$ʸ) = 1/4

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

1. A

   X̄ is the method of moments estimator of $\theta$

2. B

   $X_{(1)}$ is the maximum likelihood estimator of $\theta$

3. C

   $X_{(1)} - 1/n$ is the uniformly minimum variance unbiased estimator of $\theta$

4. D

   $X_{(1)}$ is a sufficient statistic for $\theta$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   If $\psi_{1}$ rejects $H_{0}$, then $\psi_{2}$ also rejects $H_{0}$

2. B

   If $\psi_{1}$ does not reject $H_{0}$, then $\psi_{2}$ also does not reject $H_{0}$

3. C

   If $\psi_{2}$ rejects $H_{0}$, then $\psi_{1}$ also rejects $H_{0}$

4. D

   If $\psi_{2}$ does not reject $H_{0}$, then $\psi_{1}$ also does not reject $H_{0}$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   The test in (A) is the uniformly most powerful test of size $\alpha$

2. B

   The test in (B) is the uniformly most powerful test of size $\alpha$

3. C

   $P_\mu({\bar{X}}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $\mu > \mu_{0}$

4. D

   $P_{\mu_{0}}(Median${$X_{1}, \dots, X_{n}$} $> \mu_{0}) = 1/2$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   $({\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.05}$, ${\bar{X}} + (S/\sqrt{n})$ $t_{n-1,0.05})$

2. B

   $({\bar{X}} - (\sigma/\sqrt{n})$ $z_{0.05}$, ${\bar{X}} + (\sigma/\sqrt{n})$ $z_{0.05})$

3. C

   $[{\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9}$, $\infty)$

4. D

   $(-\infty, {\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9})$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   $T_{1,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$

2. B

   $T_{2,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$

3. C

   lim P_F($T_{2,n}$ > 1) = 1 for all F

4. D

   $T_{2,n}$ converges in distribution to a degenerate random variable under $H_{0}$

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   $(1/\sum{}n_{i}) \sum_{i}\sum_{j} Y_{ij}$ is an unbiased estimator of $\mu$

2. B

   $2\mu + \alpha_{1} + \alpha_{2}$ is an estimable linear parametric function

3. C

   $\mu + \alpha_{1} + \alpha_{2}$ is an estimable linear parametric function

4. D

   $(1/n_{2}) \sum_{j} (Y_{2j} -$ Ȳ$_{2})$ is an unbiased estimator of $\alpha_{2}$

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   $(1/\sigma^{2}) Y^{T}X\beta$̂ has a central $\chi^{2}_{p+1}$ distribution

2. B

   $(1/\sigma^{2})(Y - X\beta$̂$)^{T}(Y - X\beta$̂) has a central $\chi^{2}_{n-p-1}$ distribution

3. C

   $X\beta$̂ and $(Y - X\beta$̂$)^{T}(Y - X\beta$̂) are independently distributed

4. D

   $(1/\sigma^{2}) \sum(Y_{i} -$ Ȳ$)^{2}$ has a central $\chi^{2}_{n-1}$ distribution

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   $a_{22}$ ~ $\chi^{2}_{3}$

2. B

   ½ $a_{22}$ ~ $\chi^{2}_{5}$

3. C

   $(1/33)(a_{11} - 4a_{13} + 4a_{33})$ ~ $\chi^{2}_{3}$

4. D

   $(1/9)(a_{11} - 4a_{13} + 4a_{33})$ ~ $\chi^{2}_{5}$

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   If $X_{1}$ and $Y_{1}$ are independent, then $S_\rho = 0$

2. B

   $S_\rho = (6/\pi) \sin$⁻$^{1}(\rho/2)$

3. C

   If $S_\rho = 0$, then $X_{1}$ and $Y_{1}$ are independent

4. D

   If $X_{1}$ and $Y_{1}$ are independent, then $S_\rho = 1/2$

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

1. A

   E(M) = 4

2. B

   $E(M^{2}) = 17$

3. C

   $E(M^{3}) = 72$

4. D

   Var(M) = 1

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

1. A

   X̄ = 95

2. B

   $S^{2} = 77$

3. C

   C > 180/19

4. D

   R > 8

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

1. A

   f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.

2. B

   $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.

3. C

   f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.

4. D

   If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.

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

1. A

   Let $G_{1}$ and $G_{2}$ be finite groups such that |$G_{1}$| and |$G_{2}$| are coprime. Then any homomorphism from $G_{1}$ to $G_{2}$ is trivial.

2. B

   Let G be a finite group and f : G → G a homomorphism that fixes more than half of the elements of G. Then f(x) = x for all x ∈ G.

3. C

   Let G be a finite group having exactly 3 subgroups. Then G is of order $p^{2}$ for some prime p.

4. D

   Any finite abelian group G has at least d(|G|) subgroups, where d(m) is the number of positive divisors of m.