CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   There exist non-negative integers p, q such that both (A) and (B) are true.

2. B

   Both (A) and (B) are false if and only if p = q.

3. C

   For all non-negative integers p and q, (A) or (B) is true.

4. D

   There exists p ≠ q such that both (A) and (B) are false.

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 · The Real Line

1. A

   If $\ell$ is positive, then $\ell = \sqrt{3}/2$.

2. B

   If $\ell$ is positive, then $\ell = 1/\sqrt{2}$.

3. C

   If $\ell$ is negative, then $\ell = -\sqrt{3}/2$.

4. D

   If $\ell$ is negative, then $\ell = -1/2$.

Single choice · Continuity and Differentiation

1. A

   f is unbounded.

2. B

   f is increasing.

3. C

   $lim_{x\to\infty}$ f(x) = 2.

4. D

   f is decreasing.

Single choice · Sequences and Series of Functions

1. A

   $(f_{n})_{n}\ge_{1}$ converges uniformly on [0, 1] to a continuous function f.

2. B

   $(f_{n})_{n}\ge_{1}$ converges pointwise on [0, 1] to a discontinuous function f.

3. C

   $(f_{n})_{n}\ge_{1}$ converges pointwise on [0, 1] to a continuous function f.

4. D

   $(f_{n})_{n}\ge_{1}$ does not converge pointwise on [0, 1].

Single choice · Eigenvalues and Canonical Forms

1. A

   $t^{2} + 7t + 10$

2. B

   $t^{2} - 7t + 29$

3. C

   $t^{2} - 7t - 10$

4. D

   $t^{2} - 7t + 10$

Single choice · Eigenvalues and Canonical Forms

1. A

   101

2. B

   67

3. C

   103

4. D

   113

Single choice · Vector Spaces and Linear Maps

1. A

   125

2. B

   144

3. C

   145

4. D

   480

Single choice · Vector Spaces and Linear Maps

1. A

   The dimension of $X/\ker \varphi$ is 3.

2. B

   $\ker \varphi$ is finite dimensional.

3. C

   The dimension of $X/\ker \varphi$ is 1.

4. D

   X is finite dimensional.

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 · Eigenvalues and Canonical Forms

1. A

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = f(C) = 0 and f(D) = I.

2. B

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = 0, f(C) = I and f(D) = I.

3. C

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = 0, f(B) = f(C) = I and f(D) ≠ 0.

4. D

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = 0, f(C) = I and f(D) = 0.

Single choice · Eigenvalues and Canonical Forms

1. A

   $M^{2} = N^{2}$

2. B

   $M^{3} = N^{3}$

3. C

   The characteristic polynomials of M and N are the same.

4. D

   The minimal polynomials of M and N are the same.

Single choice · Vector Spaces and Linear Maps

1. A

   1

2. B

   (q^(2n) − q^n)(q^(2n) − q^(n+1))/q^n

3. C

   (q^(2n) − 1)(q^(2n) − q)⋯(q^(2n) − q^(n−1))/q^n

4. D

   $(q^n - 1)(q^n - q)/((q^{2} - 1)(q^{2} - q))$

Single choice · The Real Line

1. A

   $\lim (x\to\infty) f(x) = \infty$

2. B

   $\lim (x\to\infty) f(x) = 0$

3. C

   $\lim (x\to\infty) f(x) = 1$

4. D

   $\lim (x\to\infty) f(x) = 1/2$

Single choice · The Real Line

1. A

   $\limsup x_{n} = 1$

2. B

   $\limsup x_{n} = \sqrt{2}$

3. C

   $\limsup x_{n} = 2$

4. D

   $\limsup x_{n} = 1 + \sqrt{2}$

Single choice · The Real Line

1. A

   The series converges for all $\alpha \in (0,1)$ and for all $\beta \in (1,\infty)$.

2. B

   The series converges for $\alpha = 1$ and $\beta = 1$.

3. C

   The series converges for $\alpha = 1$ and for all $\beta \in (1,\infty)$.

4. D

   The series converges for all $\alpha \in (0,\infty)$ and for all $\beta \in (0,\infty)$.

Single choice · Inner Product Spaces and Forms

1. A

   If v and w are eigenvectors of B for distinct eigenvalues, then ⟨v,w⟩ = 0.

2. B

   If v is an eigenvector of B and w ≠ 0 is such that ⟨w,v⟩ = 0, then w is an eigenvector of B.

3. C

   For every subspace $W \subseteq \mathbb{R}^{4}, W + W\perp = \mathbb{R}^{4}$, where W⊥ = {$v \in \mathbb{R}^{4}$ ∣ ⟨v,w⟩ = 0 for all w ∈ W}.

4. D

   If $W \subseteq \mathbb{R}^{4}$ is an eigenspace of B, then W has an orthonormal basis.

Single choice · Continuity and Differentiation

1. A

   f∘g is uniformly continuous, but g∘f is not uniformly continuous.

2. B

   g∘f is uniformly continuous, but f∘g is not uniformly continuous.

3. C

   Neither f∘g nor g∘f is uniformly continuous.

4. D

   Both f∘g and g∘f are uniformly continuous.

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 · Eigenvalues and Canonical Forms

1. A

   The characteristic polynomial of A has repeated roots.

2. B

   Every eigenvalue of A has absolute value 1.

3. C

   A does not have any eigenvalue on the imaginary axis.

4. D

   $A^{2}$ is the identity matrix.

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 · Continuity and Differentiation

1. A

   f′(x) ≤ 0 for all x > 0.

2. B

   $lim_{x\to\infty}$ f′(x) = 0.

3. C

   $lim_{x\to\infty}$ x f′(x) need not exist.

4. D

   $lim_{x\to\infty}$ x f′(x) = 0.