CSIR NET Maths previous year question papers

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 $z\in(-\varepsilon,\varepsilon)$.

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).

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

1. A

   {$x \in \mathbb{R}$ | [x] = 1}

2. B

   {$x \in \mathbb{R}$ | x − [x] = 1/2}

3. C

   {$x \in \mathbb{R}$ | $x \ge 0, \sqrt{x} \in \mathbb{Q}$}

4. D

   {$x \in \mathbb{R}$ | $x^{2} \in \mathbb{Q}$}

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

1. A

   sup A is finite.

2. B

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) < \sup A$

3. C

   $\lim \inf(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 2$

4. D

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 3$

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

1. A

   S is empty.

2. B

   S is non-empty.

3. C

   f must be uniformly continuous.

4. D

   f need not be uniformly continuous.

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

1. A

   $(g_{n})_{n}\ge_{1}$ converges uniformly on any compact subset of $\mathbb{R}$ but not on $\mathbb{R}$.

2. B

   $(g_{n})_{n}\ge_{1}$ converges uniformly on $\mathbb{R}$.

3. C

   There exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on $\mathbb{R}$.

4. D

   For every compact subset $K \subseteq \mathbb{R}$, there exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on K.

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 · Eigenvalues and Canonical Forms

1. A

   The set {$I, A, \dots, A^{n}$} is linearly independent but the set {$I, A, \dots, A^(n^{2})$} is not linearly independent.

2. B

   If A is a singular matrix, then the set {I, A, …, A^k} spans a (k + 1)-dimensional subspace of V for all k ≤ rank(A).

3. C

   The sets {$I, A, \dots, A^{n}$} and {$I, A, \dots, A^(n^{2})$} both span the same subspace of V.

4. D

   The set {$I, A, \dots, A^{n}$} is linearly dependent.

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

1. A

   $\varphi$ is one-to-one but not onto.

2. B

   $\varphi$ is onto but not one-to-one.

3. C

   There exists $f \in \mathbb{R}[x]$ such that deg f ≤ n and $\ker \varphi =$ {fg | $g \in \mathbb{R}[x]$}.

4. D

   deg h ≥ n for every nonzero $h \in \ker \varphi$.

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

1. A

   The characteristic polynomial of T can be $x^{2} + x + 1$.

2. B

   The characteristic polynomial of T can be x(x − 1).

3. C

   The minimal polynomial of T can be $x^{2}$.

4. D

   The characteristic polynomial of T can be $x(x - e^(2\pi{}i/3))$.

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

1. A

   There exists a nonzero $v \in \mathbb{R}^{4}$ such that $q_{1}(T)v = 0$.

2. B

   There exists a nonzero $v \in \mathbb{R}^{4}$ such that $q_{1}(T)v = 0$ and $q_{2}(T)v = 0$.

3. C

   For all nonzero $v \in \mathbb{R}^{4}, v$ and Tv are linearly independent.

4. D

   If $q_{1}(T)v = 0$ for some nonzero $v \in \mathbb{R}^{4}$, then {v, Tv, $T^{2}v, T^{3}v$} is a basis of $\mathbb{R}^{4}$.

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

1. A

   The set {$v_{1}, v_{2}$} is linearly independent.

2. B

   If $c_{1} \ne c_{2}$, then ⟨$v_{1}, v_{2}$⟩ = 0.

3. C

   If $c_{1} = c_{2}$, then $c_{3} = c_{2}$.

4. D

   If $c_{1} \ne c_{2}$, then $c_{3} = c_{2}$.

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

1. A

   $f(v_{2}, v_{1}) = -s$.

2. B

   If r = s = t, then either f(v, v) ≥ 0 for all v ∈ V or f(v, v) ≤ 0 for all v ∈ V.

3. C

   If f is positive definite, then r ≠ s, s ≠ t and r ≠ t.

4. D

   If f is positive definite, then $f(v_{1}, v_{2}) = 0$.