CSIR NET Maths previous year question papers

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

1. A

   [0,1]

2. B

   (0,1]

3. C

   (10⁻$^{2025},1)$

4. D

   (−10⁻$^{2025},10$⁻$^{2025}]$

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

1. A

   For all x,y∈X and every non-empty subset A of X, we have d(x,A)−d(y,A) ≤ d(x,y).

2. B

   For every non-empty subset A of X, the function x↦d(x,A) is uniformly continuous on X.

3. C

   A non-empty subset A of X is closed if and only if d(x,A)>0 for all x in X outside A.

4. D

   If X is compact and $C_{1},\dots,C_{k}$ are non-empty closed sets of X such that $\cap{}C_{i}=\emptyset$, then the minimum value of $x\mapsto\sum{}d(x,C_{i})$ on X is 0.

Multi-select · full marks only for the exact set · The Real Line

1. A

   A is bounded below but not bounded above.

2. B

   A is bounded above but not bounded below.

3. C

   inf A = −1

4. D

   sup A = 1

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 · Sequences and Series of Functions

1. A

   The sequence of functions {$f_{n}$}$_{n}\ge1$ is uniformly bounded.

2. B

   The sequence of functions {$f_{n}$}$_{n}\ge1$ does not converge pointwise.

3. C

   The sequence of functions {$f_{n}$}$_{n}\ge1$ converges uniformly to the constant function 0.

4. D

   The set {$f_{n} : n\in\mathbb{N}$} is compact in C[0,1], where C[0,1] denotes the space of all real valued continuous functions on [0,1] equipped with the supremum norm.

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

1. A

   H contains an orthonormal basis of $\ell^{2}$.

2. B

   H is a linear subspace of $\ell^{2}$.

3. C

   H is a bounded subset of $\ell^{2}$.

4. D

   H is a convex subset of $\ell^{2}$.

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

1. A

   $L_\alpha$⁺ and $L_\alpha$⁻ exist for every $\alpha\in(0,1)$.

2. B

   If f is surjective, then f is continuous.

3. C

   If f is Riemann integrable, then f is continuous.

4. D

   If the left and right hand limits exist at $\alpha,\beta\in(0,1), \alpha\ne\beta$, then $U_\alpha\cap{}U_\beta=\emptyset$.

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

1. A

   There exists a cubic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.

2. B

   There exists a quadratic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.

3. C

   If $f(X)\in\mathbb{Q}[X]$ is such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

4. D

   If $f(X)\in\mathbb{Q}[X]$ is a cubic polynomial such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

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 · Eigenvalues and Canonical Forms

1. A

   There exist $\lambda\in\mathbb{C}$ and a non-zero v∈V such that Av$=\lambda{}v$.

2. B

   There exist $\lambda,\mu\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}w$.

3. C

   There exist $\lambda,\mu,\delta\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}v+\delta{}w$.

4. D

   There exists a three-dimensional subspace W such that Aw∈W for all w∈W.

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

1. A

   The set {$S\in{}W : S^{2}=S$} contains exactly 2 elements.

2. B

   $dim_\mathbb{R}(W) = 2$.

3. C

   If U∈L(V) is such that $U^{2}=U$ and $(T+U)^{2}=T+U$, then TU=0.

4. D

   If U∈L(V) is such that $U^{2}=U$ and $(T-U)^{2}=T-U$, then (TU$)^{2}=$TU.

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

1. A

   lim(x,y)→(0,0) f(x,y) does not exist.

2. B

   The partial derivatives of f at (0,0) do not exist.

3. C

   lim(x→0) f(x,0) = lim(y→0) f(0,y).

4. D

   f is differentiable at (0,0).

Single choice · The Real Line

1. A

   If A is finite, then there exists a bijection between S and U.

2. B

   If A is finite and B = [0, 1], then there is no bijection between S and T.

3. C

   There is no bijection between S and U for any choice of A.

4. D

   If A ≠ B, then there is no bijection between T and U.

Single choice · The Real Line

1. A

   $\lim \inf(n\to\infty) x_{n} \ge 2025$

2. B

   $\lim \sup(n\to\infty) x_{n} \le 2025$

3. C

   $\lim \inf(n\to\infty) x_{n} \le 2025$

4. D

   $\lim \inf(k\to\infty) x_(n_{k}) > 2025$

Single choice · The Real Line

1. A

   $(a_{n})_{n}\ge_{1}$ converges but $(b_{n})_{n}\ge_{1}$ does not converge.

2. B

   $(b_{n})_{n}\ge_{1}$ converges but $(a_{n})_{n}\ge_{1}$ does not converge.

3. C

   Both $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ converge.

4. D

   Neither $(a_{n})_{n}\ge_{1}$ nor $(b_{n})_{n}\ge_{1}$ converges.

Single choice · Continuity and Differentiation

1. A

   For every bounded subset $A \subseteq \mathbb{R}, f$⁻$^{1}(A)$ is a bounded subset of $\mathbb{R}$.

2. B

   For every Cauchy sequence $(x_{n})_{n}\ge_{1}$ in $\mathbb{R}, (f(x_{n}))_{n}\ge_{1}$ is a Cauchy sequence in $\mathbb{R}$.

3. C

   There exists $x \in \mathbb{R}$ such that f(x) = x.

4. D

   There exists $x \in \mathbb{R}$ such that f(x) = 0.

Single choice · Continuity and Differentiation

1. A

   f is monotone.

2. B

   f is continuous but not strictly monotone.

3. C

   f is not continuous.

4. D

   f is continuous but not uniformly continuous.

Single choice · Sequences and Series of Functions

1. A

   $\sum(n\ge1) f_{n}$ and $\sum(n\ge1) g_{n}$ are uniformly convergent on the interval [−1, 1].

2. B

   $\sum(n\ge1) f_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum(n\ge1) g_{n}$ is not.

3. C

   $\sum(n\ge1) g_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum(n\ge1) f_{n}$ is not.

4. D

   Neither $\sum(n\ge1) f_{n}$ nor $\sum(n\ge1) g_{n}$ is uniformly convergent on the interval [−1, 1].

Single choice · Vector Spaces and Linear Maps

1. A

   (A) is true but (B) is false.

2. B

   (B) is true but (A) is false.

3. C

   Both (A) and (B) are true.

4. D

   Both (A) and (B) are false.

Single choice · Eigenvalues and Canonical Forms

1. A

   T is diagonalizable over $\mathbb{R}$.

2. B

   The characteristic polynomial of T is $(x - 2)^{4}$.

3. C

   The characteristic polynomial of T is $(x^{2} + x + 2)(x^{2} - 4)$.

4. D

   For every $\mathbb{R}-$linear transformation $S : \mathbb{R}^{4} \to \mathbb{R}^{4}$, we have ST = TS.

Single choice · Eigenvalues and Canonical Forms

1. A

   A is diagonalizable.

2. B

   A has at least two distinct eigenvalues.

3. C

   The characteristic polynomial of A is $x^{3} - 1$.

4. D

   The minimal polynomial of A cannot have degree 2.

Single choice · Eigenvalues and Canonical Forms

1. A

   There exists a real invertible matrix P such that PAP⁻$^{1}$ is a diagonal matrix and Pᵗ$P = I_{3}$.

2. B

   There exists a real invertible matrix P such that PAP⁻$^{1}$ is a diagonal matrix and Pᵗ$P \ne I_{3}$.

3. C

   One of the eigenvalues of A is not real.

4. D

   A has only real eigenvalues and it is not diagonalizable over $\mathbb{R}$.

Single choice · Inner Product Spaces and Forms

1. A

   {$(-1/\sqrt{2}, 0, 0, -1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

2. B

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

3. C

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

4. D

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (1/\sqrt{6}, -\sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

Single choice · Inner Product Spaces and Forms

1. A

   B(v, v) = 0 if and only if v = 0.

2. B

   For every $\lambda \in \mathbb{R}$, there exists v such that $B(v, v) = \lambda$.

3. C

   There exists v ≠ 0 such that B(v, w) = 0 for all $w \in \mathbb{R}^{3}$.

4. D

   If B(v, w) = 0 then either v = 0 or w = 0.