CSIR NET Maths previous year question papers

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

1. A

   B(w, w) = 0

2. B

   There exists an $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

3. C

   There are infinitely many $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

4. D

   q is equivalent to the quadratic form $Q(x, y) = x^{2} - y^{2}$ for all $(x, y) \in \mathbb{R}^{2}$.

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

1. A

   If (a, b, c) = (−2, 1, −2), then every admissible extremal is a minimizer

2. B

   If (a, b, c) = (1, 0, 2), then every admissible extremal is a minimizer

3. C

   If (a, b, c) = (2, −1, 1), then every admissible extremal is a minimizer

4. D

   If (a, b, c) = (1, −2, 5), then every admissible extremal is a minimizer

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

1. A

   $E(X^{2}) = 4/3$

2. B

   P(|X − 4/5| ≥ 1) = 19/25

3. C

   The upper bound of P(|X − 4/5| ≥ 1), using Chebyshev's inequality, is 52/75

4. D

   The value of the moment generating function, M_X(t), at t = 1 is $(e^{2} - 1)/5$

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

1. A

   $(Y + (1/n)\ln(1 - \alpha/2), Y - (1/n)\ln(1 - \alpha/2))$

2. B

   $(Y + (1/2n)\ln(\alpha), Y)$

3. C

   $(Y + (2/n)\ln(\alpha/2), Y + (2/n)\ln(1 - \alpha/2))$

4. D

   $(Y + (1/n)\ln(\alpha/2), Y + (1/n)\ln(1 - \alpha/2))$

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

1. A

   The distribution of $(1/\sqrt{7})\sum_{i}$₌$_{1}^{10}(X_{i} - Y_{i} + 1)$ is N(0, 10)

2. B

   The distribution of $(1/19)\sum_{i}$₌$_{1}^{10}(X_{i} + Y_{i} - 11)^{2}$ is $\chi^{2}-$distribution with degrees of freedom 10

3. C

   The distribution of $2\sqrt{2}(X_{1} - 5)/\surd(\sum_{i}$₌$_{3}^{10}(X_{i} - 5)^{2})$ is t-distribution with degrees of freedom 9

4. D

   The distribution of $2\sum_{i}$₌$_{1}^{3}(Y_{i} - 6)^{2}/\sum_{i}$₌$_{4}^{9}(Y_{i} - 6)^{2}$ is F-distribution with degrees of freedom 3 and 6.

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

1. A

   $\rho$ may be −0.47

2. B

   The first principal component can only explain 32% of the total variation for some $\rho$

3. C

   The second principal component can explain more than 32% of the total variation for any $\rho$

4. D

   The variance of the first principal component is $1 + 2\rho$ for any $\rho$

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

1. A

   If f is Lebesgue integrable on $\mathbb{R}$, then $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite.

2. B

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, then f is Lebesgue integrable on $\mathbb{R}$.

3. C

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, and $f(x) \ge a_{1}$ for all $x \in \mathbb{R}$, then f is Lebesgue integrable on $\mathbb{R}$.

4. D

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite and f is bounded, then f is Lebesgue integrable on $\mathbb{R}$.

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖T(f) − T(g)‖ $\le \alpha$‖f − g‖.

2. B

   There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖$T^{2}(f) - T^{2}(g)$‖ $\le \alpha$‖f − g‖.

3. C

   The set {f ∈ C[0, 1] : T(f) = f} is a singleton set.

4. D

   ‖$T^{n}$‖ $\to \infty$ as $n \to \infty$.

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

1. A

   The method of moments estimator of $\theta$ is −2X̄

2. B

   The method of moments estimator of $\theta$ is 2X̄

3. C

   The maximum likelihood estimator of $\theta$ is min{$-2X_{(1)}/3, 2X_{(n)}$}

4. D

   The maximum likelihood estimator of $\theta$ is max{$-2X_{(1)}/3, 2X_{(n)}$}

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

1. A

   $W_{0} =$ {0}

2. B

   $W_{1} =$ {0}

3. C

   $W_{2} =$ {0}

4. D

   $W_{n} =$ {0}

Multi-select · full marks only for the exact set

1. A

   In every calendar month of 2024, at least one person was troublesome.

2. B

   In any two consecutive calendar months of 2024, at least one person was troublesome in at least one of the two calendar months.

3. C

   At least one person was troublesome in two calendar months of 2024.

4. D

   At least one person was troublesome in January 2024.

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

1. A

   f is a uniformly continuous function.

2. B

   f is a bounded function.

3. C

   There exists $x\in(0,\infty)$ such that f(x)=0.

4. D

   The derivative of f is continuous.

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

1. A

   F is non-decreasing on $\mathbb{R}$

2. B

   $\lim(x\to\infty) F(x) = 1$

3. C

   F is left-continuous on $\mathbb{R}$

4. D

   F is right-continuous on $\mathbb{R}$

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

1. A

   If T is not diagonalizable over $\mathbb{R}$, then $T^{2}$ has an eigenvalue ≤0.

2. B

   If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.

3. C

   If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$.

4. D

   For each non-zero v∈V, {v,Tv,$T^{2}v$} is linearly dependent.

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

1. A

   y(1/2) = 1

2. B

   y′(0) = 1

3. C

   y(1/4) = 2

4. D

   y′(1/2) = 1

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

1. A

   There exists a unique solution to BVP.

2. B

   The BVP does NOT have a solution.

3. C

   There exists a solution u to BVP such that u(x,y)=(1+e)/2 for some $(x,y)\in\Omega\cup\partial\Omega$.

4. D

   There exists a solution u to BVP such that $u(x,y)=(1+e^{3})/2$ for some $(x,y)\in\Omega\cup\partial\Omega$.

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

1. A

   $(x^{n}g(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

2. B

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

3. C

   $(x^{n}h(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

4. D

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

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

1. A

   For $\alpha = 1$, all the solutions of the ODE tend to zero as $x \to \infty$

2. B

   For $\alpha = 0$, there exists a solution of the ODE which tends to 1 as $x \to \infty$

3. C

   For $\alpha = -1$, there exists an unbounded solution of the ODE on $\mathbb{R}$

4. D

   For $\alpha = 2$, there exists an unbounded solution of the ODE on $\mathbb{R}$

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

1. A

   There are infinitely many extremals

2. B

   There are more than one but only finitely many extremals

3. C

   There is no extremal

4. D

   There is a unique extremal

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

1. A

   The BVP has a non-zero solution if $\lambda = 2$

2. B

   The BVP has a non-zero solution if $\lambda = 3$

3. C

   The BVP has a non-zero solution y = y(x) satisfying $y(k\pi/2) = 0$ for every integer k if $\lambda = 6$

4. D

   The BVP has a non-zero solution y = y(x) satisfying $y(k\pi/3) = 0$ for every integer k if $\lambda = 6$

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

1. A

   For M = 0, the maximum likelihood estimate of $\theta$ is equal to 0.

2. B

   For M = 10, the maximum likelihood estimate of $\theta$ does not exist.

3. C

   The maximum likelihood estimator of $\theta$ exists for M ∈ {1, 2, …, 9} and is equal to ln(10/M − 1).

4. D

   The method of moments estimator of $\theta$ exists and is equal to M/10.

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

1. A

   $T_{n}$ is a consistent estimator of $\theta$.

2. B

   $T_{n}$ is an unbiased estimator of $\theta$.

3. C

   $T_{n} - 1$ is a consistent estimator of $\theta$.

4. D

   $\lim(n\to\infty) E_\theta(T_{n}) = \theta + 1, \forall\theta \in \mathbb{R}$

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

1. A

   Every linear function of $\beta_{i}, 1 \le i \le n$, is not estimable.

2. B

   Every $\beta_{i}, 1 \le i \le n$, has infinitely many linear unbiased estimators, but a unique best linear unbiased estimator.

3. C

   Each $\beta_{i}, 1 \le i \le n$, has only one linear unbiased estimator.

4. D

   $Y_{2} - Y_{1}$ is the best linear unbiased estimator of $\beta_{2}$.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is differentiable at 0 when $\alpha > 1$.

2. B

   f is differentiable at 0 when $\alpha = 1/2$.

3. C

   f is continuous at 0 when $\alpha = 0$.

4. D

   f is continuous at 0 when $\alpha > 0$.

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

1. A

   f has a pole at z = 0.

2. B

   There is an entire function g such that f = g on $\mathbb{C}$ \ {0}.

3. C

   f is a polynomial of degree at most 2.

4. D

   f has an essential singularity at z = 0.