CSIR NET Maths previous year question papers

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

1. A

   If each $f_{n}$ is uniformly continuous and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is uniformly continuous.

2. B

   If each $f_{n}$ is bounded and $(f_{n})_{n}\ge_{1}$ converges to f pointwise, then f is bounded.

3. C

   If each $f_{n}$ is bounded and continuous, $(f_{n})_{n}\ge_{1}$ converges pointwise to a bounded and continuous function f, then the convergence is uniform.

4. D

   If each $f_{n}$ is differentiable and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is differentiable.

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

1. A

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges uniformly to 0 on the interval (−M, M) for some positive real number M.

2. B

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$.

3. C

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$ and $(f'_{n})_{n}\ge_{1}$ does not converge pointwise to 0 on $\mathbb{R}$.

4. D

   $(f_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$ but not uniformly on $\mathbb{R}$.

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

1. A

   If A is invertible, then AB and BA have the same minimal polynomial.

2. B

   If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.

3. C

   If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.

4. D

   If AB and BA have the same minimal polynomial, then either A or B is invertible.

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

1. A

   (ST − TS$)^{2} = \lambda{}I$ for some $\lambda \in \mathbb{C}$.

2. B

   The characteristic polynomial of (ST − TS$)^{2}$ is $(x - \lambda)^{2}$ for some $\lambda \in \mathbb{C}$.

3. C

   If ST − TS has only one eigenvalue, then ST − TS $= \lambda{}I$ for some $\lambda \in \mathbb{C}$.

4. D

   If ST − TS has only one eigenvalue, then (ST − TS$)^{2}$ is the zero transformation.

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

1. A

   {f′(x), f(x) − f(x − 1), 1}

2. B

   {f(x + 1) − f(x), f(x) − f(x − 1), 1}

3. C

   {f(x), f′(x), 1}

4. D

   {f(x + 1), f(x − 1), f(x)}

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

1. A

   The number of two dimensional subspaces of V is 13.

2. B

   The number of surjective linear transformations from V to W is 624.

3. C

   The number of one dimensional subspaces of V is 13.

4. D

   The number of linear transformations from V to W is $3^{6}$.

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

1. A

   If trace of TT* is zero, then T = 0.

2. B

   Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.

3. C

   Suppose T = T* and N > 1 be an integer. Let v ∈ V be such that $T^{2^N}(v)$ = 0. Then T(v) = 0.

4. D

   Suppose that TT* = T*T and N > 1 be an integer. Let v ∈ V be such that T^N(v) = 0. Then T(v) = 0.

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 · 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 · 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 · The Real Line

1. A

   There exists a positive integer N such that for all n≥N, $b_{n,l}$ $= a_{l}$ for all 1≤l≤2023.

2. B

   There exists a positive integer N such that for all n≥N, $b_{n,2025}$ $= a_{2025} - 1$.

3. C

   $\beta_{n}$ is rational for infinitely many n.

4. D

   There exists a positive integer N such that for all n≥N, $b_{n,2024}$ $= a_{2024} - 1$.

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

1. A

   If f∈F is a continuous function and f(n)=0 for all positive integers n, then $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$.

2. B

   If f∈F is uniformly continuous and $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$, then $\lim(x\to\infty)f(x)=0$.

3. C

   There exists a function f∈F such that $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$ but $\lim(x\to\infty)f(x)\ne0$.

4. D

   If f∈F is Lebesgue measurable and $\lim(x\to\infty)\int_{0}$ˣf(t)dt=0, then f=0 almost everywhere on $[0,\infty)$.

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

1. A

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB+BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

2. B

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB−BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

3. C

   There exists $B \in M_{4}(\mathbb{C})$ such that AB+BA$=0, B^{2}=I$ and Trace(B)=0.

4. D

   If $B \in M_{4}(\mathbb{C})$ is such that AB+BA=0, then B is diagonalisable.

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 · Vector Spaces and Linear Maps

1. A

   The dimension of V is 4 or 5.

2. B

   If 2u−3v+5w=0, then {v,w,x,y} is a basis of V.

3. C

   If u−6v+7w=0, then the span of {v,w,x} is a 2-dimensional vector space.

4. D

   If u+w=v+x, then {u,v,w,y} is a basis of V.

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

1. A

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $g:\Omega_\varepsilon\to(-\delta,\delta)$ such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all $(x,y)\in\Omega_\varepsilon$.

2. B

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $h:\Omega_\varepsilon\to(-\delta,\delta)$ such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all $(x,z)\in\Omega_\varepsilon$.

3. C

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $k_{1}(0)=k_{2}(0)=0$ and $F(x,k_{1}(x),k_{2}(x))=-1$ for all $x\in(-\varepsilon,\varepsilon)$.

4. D

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $j_{1}(0)=j_{2}(0)=0$ and $F(j_{1}(z),j_{2}(z),z)=-1$ for all $z\in(-\varepsilon,\varepsilon)$.