CSIR NET Maths previous year question papers

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

1. A

   The probability density function of S is f_S(s) = $(1/2)e^{-s/2}$ if s > 0, and 0 otherwise.

2. B

   The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.

3. C

   Var(S) = 2.

4. D

   E(T) = 2/3.

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 two fitted lines have the same slope

2. B

   The two fitted models have the same intercept

3. C

   The model with intercept passes through at least one of the observed data points

4. D

   The model without intercept passes through at lesast one of the observed data points

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

1. A

   A possible choice of L is the 5 × 2 matrix with rows (0, 0), (1, 0), (0, 1), (−1, 0), (0, 1)

2. B

   The test statistic for testing $H_{0}$ against H_A follows an F-distribution with (2, n − 4) degrees of freedom under $H_{0}$

3. C

   Sum of squares residuals under the restrictions $L^{T}\beta = 0$ follows $\sigma^{2}\chi^{2}_{n-3}$ distribution

4. D

   Sum of squares residuals without the restrictions $L^{T}\beta = 0$ follows a non-central $\sigma^{2}\chi^{2}_{n-4}$ distribution

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 · The Real Line

1. A

   $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$

2. B

   $limsup_{n\to\infty}$ $t_{n} \le e$

3. C

   $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$

4. D

   $liminf_{n\to\infty}$ $t_{n} \ge e$

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

1. A

   2/e

2. B

   4/e

3. C

   $log_{e}2 - 1$

4. D

   $2 log_{e}2 - 1$

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

1. A

   The composition f ∘ g is Riemann integrable.

2. B

   If g(x) ≠ 0 for each x ∈ [a, b], then f/g is Riemann integrable.

3. C

   The positive square root $\sqrt{f^{2} + g^{2}}$ is Riemann integrable.

4. D

   The composition f ∘ g is Riemann integrable, if both f and g are continuous.

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 · Functions of Several Variables

1. A

   $3 \partial{}f/\partial{}x_{1}(1, 1) + 3 \partial{}f/\partial{}x_{2}(1, 1) = f(1, 1)$

2. B

   $\partial{}f/\partial{}x_{1}(1, -1) - \partial{}f/\partial{}x_{2}(1, -1) = 3f(1, -1)$

3. C

   $x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 6f(x_{1}, x_{2})$

4. D

   $x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 9f(x_{1}, x_{2})$

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

1. A

   If AAᵀ = I, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

2. B

   If AAᵀ = I, then DF$_A : M_{2}(\mathbb{R}) \to S$ need not be surjective.

3. C

   If A is invertible, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

4. D

   If A is not invertible, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

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 · Eigenvalues and Canonical Forms

1. A

   T has at least four linearly independent eigenvectors.

2. B

   dim W ≥ 2.

3. C

   $\ker((T - 2I)^{2025}) = \ker((T - 2I)^{2026})$

4. D

   (T − 2I)(T − 3I) is a nilpotent operator.

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

1. A

   The matrix with rows (0, 0, 0, −1), (1, 0, 0, 0), (0, 1, 0, 2), (0, 0, 1, 0)

2. B

   The matrix with rows (0, 0, 0, 1), (1, 0, 0, 0), (0, 1, 0, −2), (0, 0, 1, 0)

3. C

   The matrix with rows (0, 1, 1, 0), (1, 0, 0, 1), (0, 0, 0, 1), (0, 0, 1, 0)

4. D

   The matrix with rows (0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0)

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

1. A

   I = 0

2. B

   $(1/2\pi{}i)I \in$ {$4n : n \in \mathbb{Z}, n \ge 1$}

3. C

   $I = 2\pi{}i \sum_{n=0}^\infty$ 1/n!

4. D

   $I = 2\pi{}i \sum_{n=0}^\infty$ 1/(n!(n + 1)!)

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

1. A

   G = GL$_{2}($𝔽_p) and H is a subgroup of order p.

2. B

   G = SL$_{2}($𝔽_p) and H is a subgroup of order p.

3. C

   p ≡ 3 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

4. D

   p ≡ 1 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

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

1. A

   If $G_{1}, G_{2}$ are two groups such that Aut$(G_{1})$ is isomorphic to Aut$(G_{2})$, then $G_{1}$ is isomorphic to $G_{2}$.

2. B

   If |G| = 2, then Aut(G × G) is abelian.

3. C

   If G is the group of complex numbers under addition, then Aut(G) is abelian.

4. D

   If G is finite, then Aut(G) is finite.

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

1. A

   If $G_{1}$ is normal in G, then $(G_{2}G_{1})/G_{1} \cong G_{2}/(G_{2} \cap G_{1})$.

2. B

   If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$, respectively, then $(G_{1} \times G_{2})/(H_{1} \times H_{2}) \cong (G_{1}/H_{1}) \times (G_{2}/H_{2})$.

3. C

   If $G_{1}$ is normal in $G_{2}$ and $G_{2}$ is normal in G, then $G_{1}$ is normal in G.

4. D

   Every subgroup of prime index in G is normal.

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

1. A

   If f is onto, then so is $f^{n}$ for all n ≥ 1.

2. B

   ker $f^{n+1}$ $= \ker f^{n}$ for some n ≥ 1.

3. C

   If f is onto, then f is one-to-one.

4. D

   If f is one-to-one, then f is onto.

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

1. A

   f(X) is irreducible in $\mathbb{Q}[X]$, but g(X) is not.

2. B

   g(X) is irreducible in $\mathbb{Q}[X]$, but f(X) is not.

3. C

   Both f(X) and g(X) are irreducible in $\mathbb{Q}[X]$.

4. D

   Neither f(X) nor g(X) is irreducible in $\mathbb{Q}[X]$.

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

1. A

   Let f(X) ∈ 𝔽_p[X] and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of f in 𝔽̄_p.

2. B

   Let f, g ∈ 𝔽_p[X] be irreducible polynomials of same degree and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of g in 𝔽̄_p.

3. C

   𝔽_p[X] has infinitely many irreducible polynomials.

4. D

   The set {a + b | a, b ∈ 𝔽_p} is contained in {$a^{2} + b^{2}$ | a, b ∈ 𝔽_p}.

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

1. A

   The preimage of a compact set under p is a compact set.

2. B

   The preimage of a connected set under p is a connected set.

3. C

   Every point $x \in \mathbb{R}$ has an open neighbourhood U_x such that the restriction $p|_{U_x}$ is a homeomorphism onto an open set in $\mathbb{R}$.

4. D

   The image of a bounded set under p is a bounded set.

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

1. A

   S is dense in $\mathbb{R}$.

2. B

   S \ $\mathbb{Q}$ is dense in $\mathbb{R}$.

3. C

   S \ $\mathbb{Q}$ is discrete with subspace topology on S.

4. D

   S is connected.

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

1. A

   f is a Lipschitz continuous function on D

2. B

   f is NOT a Lipschitz continuous function on D

3. C

   IVP has at least one solution

4. D

   IVP has NO solution