CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   A ⊊ C and B ⊊ C

2. B

   A ⊊ C ⊊ B

3. C

   A ⊊ B ⊊ C

4. D

   A ⊊ B = C

Single choice · The Real Line

1. A

   If A ⊊ B and B is finite, then there is a one-to-one map from $S_{2}$ to $S_{1}$.

2. B

   If $B = \mathbb{N}$, then there exists a one-to-one map from $S_{2}$ to B.

3. C

   If $B = \mathbb{N}$ and A is finite, then there exists a one-to-one map from B to $S_{1}$.

4. D

   If A is finite, then $S_{2}$ is finite for any B.

Single choice · Eigenvalues and Canonical Forms

1. A

   101

2. B

   67

3. C

   103

4. D

   113

Single choice · Inner Product Spaces and Forms

1. A

   $f_{4} \in V$

2. B

   {$f_{1}, f_{2}, f_{3}$} is an orthonormal basis of V.

3. C

   The dimension of V is 2.

4. D

   {$f_{1}, f_{2}, f_{3}$} is an orthogonal set but not orthonormal.

Single choice · Inner Product Spaces and Forms

1. A

   Q is a positive definite quadratic form on V.

2. B

   Q takes every positive real value.

3. C

   Q(x) = 2.

4. D

   For all f, g ∈ V, Q(f + g) = Q(f) + Q(g).

Single choice · The Real Line

1. A

   There is a bijective map from $\mathbb{N}$ to S.

2. B

   There is a surjective map from S onto (0,1).

3. C

   There is a bijective map from $\mathbb{N}$ to $(0,1)\S$.

4. D

   S is a countable set and $(0,1)\S$ is uncountable.

Single choice · Inner Product Spaces and Forms

1. A

   There exists $(a,b,c) \in \mathbb{R}^{3}${(0,0,0)} such that f(a,b,c) = 0.

2. B

   If there exists $(a,b,c) \in \mathbb{R}^{3}${(0,0,0)} such that f(a,b,c) = 0, then $(a,b,c) \in \mathbb{Q}^{3}$.

3. C

   {$(a,b,c) \in \mathbb{Z}^{3}$ ∣ f(a,b,c) = 0} is a finite set.

4. D

   If there exists $(a,b,c) \in \mathbb{Q}^{3}${(0,0,0)} such that f(a,b,c) = 0, then A has a positive eigenvalue and a negative eigenvalue.

Single choice · The Real Line

1. A

   The set of all finite subsets of $\mathbb{Z}$ is uncountable.

2. B

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function. If $S \subseteq \mathbb{R}$ is a countably infinite set, then f(S) is a countably infinite set.

3. C

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Then there exists an uncountable subset $T \subseteq \mathbb{R}$ such that $f(T) \subseteq \mathbb{R}$ is uncountable.

4. D

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. If R ⊆ image(f) is a countable set, then f⁻$^{1}(R)$ is countable.

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

   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 · 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

   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 · 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 · 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 · 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 · 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 $\ell^{2}$.

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 · 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 · 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.

Single choice · Vector Spaces and Linear Maps

1. A

   $(T^{3} - 3T^{2} + 3T)^2025(x) = x$

2. B

   $(T^{3} - 3T^{2} + 3T)^2025(x) = x + 1$

3. C

   $(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x$

4. D

   $(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x + 1$

Single choice · Determinants and Matrix Tricks

1. A

   W = ker(T)

2. B

   W ⊊ ker(T)

3. C

   W ∩ ker(T) ⊊ W

4. D

   W ∩ ker(T) ⊊ ker(T)

Single choice · The Real Line

1. A

   S is empty.

2. B

   There is a bijection between S and $\mathbb{N}$

3. C

   There is a bijection between S and $\mathbb{R}$

4. D

   There is a bijection between S and a non-empty finite set

Single choice · The Real Line

1. A

   There exists a non-empty finite set in C

2. B

   There exists a countably infinite set in C

3. C

   There exists an uncountable set in C

4. D

   C is empty