CSIR NET Maths previous year question papers

Single choice

1. A

   All Manuals are Booklets.

2. B

   All Catalogues are Booklets.

3. C

   All Booklets are Catalogues.

4. D

   All Catalogues are Manuals.

Single choice · Ordinary Differential Equations

1. A

   Both (P) and (Q) are true.

2. B

   (P) is true, (Q) is FALSE.

3. C

   (P) is FALSE, (Q) is true.

4. D

   Both (P) and (Q) are FALSE.

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.

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   $y(x) = (1/15)(x^{4} + 14)$ is an extremal for $J_{3}$

2. B

   $y(x) = (1/3)(x^{2} + 2)$ is an extremal for $J_{1}$

3. C

   y(x) = x is an extremal for $J_{0}$

4. D

   $y(x) = (1/2)(x^{2} - x + 2)$ is an extremal for $J_{1}$

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 · 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 $U_\alpha\cap{}U_\beta=\emptyset$.

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

   det A = 0

2. B

   det A = −1

3. C

   B(x, x) ≠ 0 for all nonzero $x \in \mathbb{R}^{4}$.

4. D

   If $x \in \mathbb{R}^{4}$ is nonzero, then there exists $y \in \mathbb{R}^{4}$ such that B(x, y) ≠ 0.

Single choice · Vector Spaces and Linear Maps

1. A

   If A is one-to-one but not onto, then m > n

2. B

   If A is onto but not one-to-one, then m < n

3. C

   If A is bijective, then m = n

4. D

   If A is one-to-one, then m = n

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

1. A

   F is Riemann integrable.

2. B

   If F(x) = 0 for all x ∈ [0, 1], then f(x) = 0 for all x ∈ [0, 1].

3. C

   F is uniformly continuous on [0, 1].

4. D

   F is differentiable on (0, 1).

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $X_{n}$ converges to X almost surely as $n \to \infty$ implies that $X_{n}$ converges to X in probability as $n \to \infty$.

2. B

   $X_{n}$ converges to X in probability as $n \to \infty$ implies that $X_{n}$ converges to X almost surely as $n \to \infty$.

3. C

   If $\sum(n=1..\infty)$ ℙ[|$X_{n} - X$| $> \delta] < \infty$ for all $\delta > 0$, then $X_{n}$ converges to X almost surely as $n \to \infty$.

4. D

   If $X_{n}$ converges to X in distribution as $n \to \infty$, and X is a constant with probability 1, then $X_{n}$ converges to X in probability as $n \to \infty$.

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

1. A

   f is strictly increasing

2. B

   f is strictly decreasing

3. C

   f is either strictly increasing or strictly decreasing

4. D

   f is onto

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

1. A

   f need not be surjective

2. B

   f must be surjective if K = [0, 1]

3. C

   f is injective and f⁻$^{1} : f(K) \to K$ is continuous

4. D

   f is injective, but f⁻$^{1} : f(K) \to K$ need not be continuous

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

1. A

   q is a quadratic form in $x_{1}, x_{2}, y_{1}, y_{2}$

2. B

   There exists $(t_{1}, t_{2}) \in \mathbb{R}^{2}$ such that $q_{1}(t_{1}, t_{2}) = 5$

3. C

   There does not exist $(s_{1}, s_{2}) \in \mathbb{R}^{2}$ such that $q_{2}(s_{1}, s_{2}) = -5$

4. D

   Given $\alpha \in \mathbb{R}$, there exists a vector $w \in \mathbb{R}^{4}$ such that $q(w) = \alpha$

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

1. A

   f is injective in a neighbourhood of (0, 0)

2. B

   There exists an open neighbourhood U of (0, 0) in $\mathbb{R}^{2}$ such that $f_{3}$ is a function of $f_{1}$ and $f_{2}$

3. C

   f maps an open neighbourhood of (0, 0) in $\mathbb{R}^{2}$ onto an open subset of $\mathbb{R}^{3}$

4. D

   (0, 0) is an isolated point of f⁻$^{1}(${f(0,0)})

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   [[5, 8], [1, 2]] ∈ S

2. B

   [[3, 2], [1, 2]] ∈ S

3. C

   [[−3, 1], [2, 3]] ∈ S

4. D

   [[2, 2], [4, 3]] ∈ S

Single choice · Vector Spaces and Linear Maps

1. A

   The product of two 2×2 real matrices of rank 2 is of rank 2.

2. B

   The product of two 3×3 real matrices of rank 2 is of rank at most 2.

3. C

   The product of two 3×3 real matrices of rank 2 is of rank at least 2.

4. D

   The product of two 2×2 real matrices of rank 1 can be the zero matrix.

Single choice · Inner Product Spaces and Forms

1. A

   $A^{2} = 0$.

2. B

   rank A = 1.

3. C

   rank A = 0.

4. D

   There exists $u \in \mathbb{R}^{2}, u \ne 0$ such that (u, u) = 0.

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

1. A

   f is necessarily one-one.

2. B

   f need not be one-one.

3. C

   f is necessarily onto.

4. D

   f need not be onto.

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

1. A

   F is injective.

2. B

   f is increasing.

3. C

   For every $(x', y') \in (0,1)^{2}$, there exists a unique $(x, y) \in (0,1)^{2}$ such that F(x, y) = (x′, y′).

4. D

   The total derivative DF(x, y) is invertible for all $(x, y) \in (0,1)^{2}$.