CSIR NET Maths previous year question papers

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

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

1. A

   The sequence $(y_{n})$ does NOT converge

2. B

   $y_{n} \to 0$ as $n \to \infty$

3. C

   $0 \le y_{n} \le 1$ for n = 0, 1, 2, …

4. D

   |y(nh$) - y_{n}$| → 0 as $n \to \infty$

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

1. A

   u(0) = 0

2. B

   $u(2\pi/\sqrt{3}) =$ $(1/6)(e^{4\pi/\sqrt{3}}$ − $e^{-2\pi/\sqrt{3}})$

3. C

   $u(\pi/(2\sqrt{3})) =$ $(1/6)(e^{\pi/\sqrt{3}}$ $- \sqrt{3}$ $e^{-\pi/(2\sqrt{3})})$

4. D

   $u(\pi/(2\sqrt{3})) =$ $(1/6)(e^{\pi/\sqrt{3}}$ $+ \sqrt{3}$ $e^{-\pi/\sqrt{3}})$

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

1. A

   $f'(\pi) = 3$

2. B

   $f(\pi) + f'(\pi) = 4 - \pi$

3. C

   $f(\pi) + f'(\pi) = 2 - \pi$

4. D

   f(0) + f′(0) = −4

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

1. A

   $\partial/\partial{}t [f, g] = [\partial{}f/\partial{}t, g] + [f, \partial{}g/\partial{}t]$

2. B

   If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion

3. C

   [[H, f], g] + [[g, H], f] + [[f, g], H] = 0

4. D

   If f and g are constants of motion, then [f, g] is a constant of motion

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

1. A

   $S_{n}/\sqrt{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

2. B

   $(T_{n} - n)/\sqrt{3n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

3. C

   $\sqrt{n} S_{n}/T_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

4. D

   $(T_{n} - n)/\sqrt{V}_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

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

1. A

   X̄ is an unbiased estimator of $\theta_{1}$

2. B

   $({\bar{X}} - X_{(1)})$ is an unbiased estimator of $\theta_{2}$

3. C

   $(X_{(1)} + X_{(n)})/2$ is the uniformly minimum variance unbiased estimator of $\theta_{1}$

4. D

   $(n + 1)(X_{(n)} - X_{(1)})/(2(n - 1))$ is the uniformly minimum variance unbiased estimator of $\theta_{2}$

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

1. A

   $U_{n}$ is an unbiased estimator of $\theta$

2. B

   $S_{n}^{2}$ is an unbiased estimator of $\theta^{2}$

3. C

   $(1/3)V_{n}$ is a consistent estimator of $\theta^{2}$

4. D

   The statistic $(U_{n}, V_{n})$ is complete

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

1. A

   The likelihood function is maximized at $\theta = 1/(x - 1)$

2. B

   The maximum value of the likelihood function is $e^{-1}/(x$ − 1)

3. C

   The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if (x − $1)e^{-x}$ < k, for some k > 0

4. D

   The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if x > c, for some c > 1