CSIR NET Maths previous year question papers

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

1. A

   $Y_{3}$ follows gamma distribution with $E(Y_{3}) = 3$.

2. B

   E(Y(2,n)) → 2 as $n \to \infty$

3. C

   $Y_{2}$ follows beta distribution with $E(Y_{2}) = 1/2$.

4. D

   Y(3,n) follows beta distribution with parameters 3 and n − 1.

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

1. A

   $(79X + 2024Y, 23Z^{2})$ is a sufficient statistic for $\theta$.

2. B

   $13(X + Y + Z)^{3}$ is a minimal sufficient statistic for $\theta$.

3. C

   2(X + Y + Z) is a complete sufficient statistic for $\theta$.

4. D

   (X − 2Y + Z) is an ancillary statistic.

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

1. A

   If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then E(r_S) = 0, ∀n ≥ 2.

2. B

   If n = 3, then P(r_S = 0) = 0.

3. C

   If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then Var(r_S) = 1/n, ∀n ≥ 2.

4. D

   If n = 4 and F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then P(r_S = 0) = 1/24.

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

1. A

   If F_X(x) = F_Y(x) ∀x, then E(T_N) = m(N + 1)/2.

2. B

   If F_X(x) = F_Y(x) ∀x, then Var(T_N) = mn(N + 1)/24.

3. C

   If F_X(x) = F_Y(x) ∀x, then the distribution of T_N is symmetric about mn/2.

4. D

   The minimum and maximum possible values of T_N are m(m + 1)/2 and N(N + 1)/2 − m(m + 1)/2, respectively.

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

1. A

   P(Unit 1 is included in the sample$) = p_{1}$

2. B

   P(Unit 1 and Unit 3 are included in the sample$) = p_{1}p_{3}[1/(1 - p_{1}) + 1/(1 - p_{3})]$

3. C

   P(Unit 1, Unit 3 and Unit 5 are included in the sample$) = p_{1}p_{3}p_{5}[1/((1 - p_{1})(1 - p_{3})) + 1/((1 - p_{3})(1 - p_{5})) + 1/((1 - p_{5})(1 - p_{1}))]$

4. D

   Expected number of distinct units in the sample is $2[1 - \sum(i=1..N)(1 - p_{i})^{2}]$

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

1. A

   The optimal solution is 4.

2. B

   An optimal point is (4, 1, 0).

3. C

   The optimal solution is 6.

4. D

   (1/2, 0, 0) is a corner point.

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

1. A

   $(f_{n})_{n}\ge_{1}$ does not converge pointwise on [0, 1].

2. B

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a continuous function on [0, 1].

3. C

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a discontinuous function on [0, 1].

4. D

   $(f_{n})_{n}\ge_{1}$ does not converge uniformly on [0, 1].

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

1. A

   f(0) = 0

2. B

   f′(0) exists and f′(0) = 0.

3. C

   f″(0) exists and f″(0) = g(0).

4. D

   f″(0) exists but f″(0) ≠ g(0).

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

1. A

   f is Riemann integrable on [0, 1].

2. B

   The set of discontinuities of f cannot contain a non-empty open set.

3. C

   f is Lebesgue measurable function.

4. D

   f is Borel measurable function.

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

1. A

   If f ∈ B, then dim ker f = 3

2. B

   dim B = 3

3. C

   dim A = 4

4. D

   If f ∈ A, then the image of f is a one-dimensional $\mathbb{Q}-$vector space.

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

1. A

   DM ≠ MD.

2. B

   D + M is invertible.

3. C

   DM is invertible.

4. D

   rank(DM) = rank(MD).

Multi-select · full marks only for the exact set · Determinants and Matrix Tricks

1. A

   If A and B are upper triangular, then M is diagonalizable over $\mathbb{R}$.

2. B

   If A and B are diagonalizable over $\mathbb{R}$, then M is diagonalizable over $\mathbb{R}$.

3. C

   If A and B are diagonalizable over $\mathbb{R}$, then there exists $\lambda \in \mathbb{R}$ such that $M = \lambda{}I_{2}$.

4. D

   There exists $\lambda \in \mathbb{R}$ such that $M^{2} = \lambda{}I_{2}$.

Multi-select · full marks only for the exact set · Determinants and Matrix Tricks

1. A

   det(T) = 225

2. B

   det(T) = −225

3. C

   Trace(T) = 16

4. D

   Trace(T) = −16

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

1. A

   det(B) > 0 for every nonzero $(a, b, c, d) \in \mathbb{R}^{4}$.

2. B

   det(B) > 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$.

3. C

   det(B) ≤ 0 for every $(a, b, c, d) \in \mathbb{R}^{4}$.

4. D

   det(B) ≤ 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$.

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

1. A

   R is never an integral domain.

2. B

   r = −r for all r ∈ R.

3. C

   Every nonzero prime ideal of R is maximal.

4. D

   R must be a commutative ring.

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

1. A

   $\mathbb{R}[x]/(f)$ is a field.

2. B

   $\mathbb{R}[x]/(f)$ is a finite dimensional $\mathbb{R}-$vector space.

3. C

   $\mathbb{R}[x]/(f)$ is a direct sum of fields, each of which is isomorphic to $\mathbb{R}$ or $\mathbb{C}$.

4. D

   There are no non-zero elements $u \in \mathbb{R}[x]/(f)$ such that u^m = 0 for some m ≥ 1.

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

1. A

   Every solution of the ODE is bounded on $(0, \infty)$

2. B

   There exists a solution of the ODE which is unbounded on $(0, \infty)$

3. C

   Every solution of the ODE is unbounded on $(0, \infty)$

4. D

   Every solution of the ODE tends to zero as $x \to \infty$

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

1. A

   $\lim(t\to\infty) t$⁻$^{2}x(t)y(t) = 0$

2. B

   x(1) = 0, y(1/2) = 0

3. C

   x(1/2) = 0, y(1) = 0

4. D

   $\lim(t\to\infty) t$⁻$^{2}x(t)y(t) = 2$

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

1. A

   $u(x_{0}, y_{0}) = \sqrt{2} e^\pi$

2. B

   $u(x_{0}, y_{0}) = e^\pi$

3. C

   $u(x_{0}, y_{0}) = -1$

4. D

   $u(x_{0}, y_{0}) = -e^\pi$

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

1. A

   $\lim(t\to\infty) u(x, t) = 0$ for all x ∈ (0, 1)

2. B

   u(x, t) = u(1 − x, t) for all x ∈ (0, 1), t > 0

3. C

   $\int_{0}^{1}(u(x, t))^{2}$dx is a non-increasing function of t

4. D

   $\int_{0}^{1}(u(x, t))^{2}$dx is a non-decreasing function of t

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

1. A

   $(5/3)\sqrt{g/l}$

2. B

   $\sqrt{5g/(3l)}$

3. C

   $(5/2)\sqrt{g/l}$

4. D

   $\sqrt{5g/(2l)}$

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

1. A

   $X_{n}$ converges to X almost surely as $n \to \infty$ implies that $X_{n}$ converges to X in probability as $n \to \infty$.

2. B

   $X_{n}$ converges to X in probability as $n \to \infty$ implies that $X_{n}$ converges to X almost surely as $n \to \infty$.

3. C

   If $\sum(n=1..\infty)$ ℙ[|$X_{n} - X$| $> \delta] < \infty$ for all $\delta > 0$, then $X_{n}$ converges to X almost surely as $n \to \infty$.

4. D

   If $X_{n}$ converges to X in distribution as $n \to \infty$, and X is a constant with probability 1, then $X_{n}$ converges to X in probability as $n \to \infty$.

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

1. A

   $N_{n}/(3n)$ converges to e⁻$^{2}$ with probability one.

2. B

   $N_{n}$ converges to e⁻$^{2}$ in probability.

3. C

   $N_{n}/n$ converges to 3e⁻$^{2}$ in distribution.

4. D

   $(N_{n} - 3$ne⁻$^{2})/\sqrt{3n}$ converges in distribution to a normal random variable with mean zero and variance e⁻$^{2}(1 - 3e$⁻$^{2})$.

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

1. A

   P⁽$^{2}$⁾$_{00} = 3/4$

2. B

   P⁽$^{3}$⁾$_{10} = 39/64$

3. C

   The stationary probability that the chain is in state 2 is 2/7.

4. D

   State 1 is transient.

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

1. A

   e⁻$^{1}$

2. B

   e⁻$^{1}/2$

3. C

   e⁻$^{1}/4$

4. D

   e⁻$^{1}/12$