CSIR NET Maths previous year question papers

Single choice · Sampling and Design of Experiments

1. A

   33

2. B

   69

3. C

   89

4. D

   93

Single choice · Singularities and Residues

1. A

   $\int_{\gamma1}$ tan(z)dz $= 2\pi{}i$

2. B

   $\int_{\gamma1}$ tan(z)dz $= -2\pi{}i$

3. C

   $\int_{\gamma2}$ tan(z)dz $= 2\pi{}i$

4. D

   $\int_{\gamma2}$ tan(z)dz $= -2\pi{}i$

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 · Partial Differential Equations

1. A

   For each $x_{0} \in \mathbb{R}, e^t\cdot{}u(x_{0},t) \to 0$ as $t \to \infty$.

2. B

   For each $x_{0} \in \mathbb{R}$, |$u(x_{0},t)$| → 0 as $t \to \infty$.

3. C

   For each $x_{0} \in \mathbb{R}$, there exists $a t_{0} > 0$ such that $u(x_{0},t_{0}) \ge 4$.

4. D

   For each $x_{0} \in \mathbb{R}$, there exists $a t_{0} > 0$ such that $u(x_{0},t_{0}) \le -4$.

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 · Numerical Analysis

1. A

   −3/8

2. B

   −5/8

3. C

   5/8

4. D

   3/8

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 · Hypothesis Testing

1. A

   8/25

2. B

   13/25

3. C

   17/25

4. D

   22/25

Single choice · Probability

1. A

   P(Z ≥ 3/5) = 0.5, P(Z = 4/25) = 0

2. B

   P(Z ≥ 3/5) = 0.6, P(Z = 4/25) = 0

3. C

   P(Z ≥ 3/5) = 0.7, P(Z = 4/25) = 0.16

4. D

   P(Z ≥ 3/5) = 0.8, P(Z = 4/25) = 0.16

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 · Estimation

1. A

   $(X, X/(1-\sqrt{0.95}))$

2. B

   (X, X/0.95)

3. C

   $(X, X/\sqrt{0.95})$

4. D

   (0.95X, X)

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.

Single choice · Linear Models and Multivariate

1. A

   $(\sigma^{2}/3, \sigma^{2}/14)$

2. B

   $(3\sigma^{2}, 14\sigma^{2})$

3. C

   $(\sigma^{2}/14, \sigma^{2}/3)$

4. D

   $(14\sigma^{2}, 3\sigma^{2})$

Single choice · Linear Integral Equations

1. A

   $u(\pi) = 2e^\pi$.

2. B

   $u'(\pi) = e^\pi$.

3. C

   $u(\pi) + u'(\pi) = 3e^\pi$.

4. D

   $u(\pi) - u'(\pi) = e^\pi$.

Single choice · Limit Theorems and Markov Chains

1. A

   1/64

2. B

   37/64

3. C

   63/64

4. D

   27/32

Single choice · Groups

1. A

   There exists a finite group G such that |H_G| is even.

2. B

   There exists a finite group G such that |H_G| = 4n+1 for some n ≥ 3.

3. C

   For every finite group G, there exists a non-negative integer n such that |H_G| = 4n+1.

4. D

   For every finite group G, there exists a non-negative integer n such that |H_G| = 4n+3.

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.

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

1. A

   If each $f_{n}$ is uniformly continuous and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is uniformly continuous.

2. B

   If each $f_{n}$ is bounded and $(f_{n})_{n}\ge_{1}$ converges to f pointwise, then f is bounded.

3. C

   If each $f_{n}$ is bounded and continuous, $(f_{n})_{n}\ge_{1}$ converges pointwise to a bounded and continuous function f, then the convergence is uniform.

4. D

   If each $f_{n}$ is differentiable and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is differentiable.

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

1. A

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges uniformly to 0 on the interval (−M, M) for some positive real number M.

2. B

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$.

3. C

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$ and $(f'_{n})_{n}\ge_{1}$ does not converge pointwise to 0 on $\mathbb{R}$.

4. D

   $(f_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$ but not uniformly on $\mathbb{R}$.

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

   {f′(x), f(x) − f(x − 1), 1}

2. B

   {f(x + 1) − f(x), f(x) − f(x − 1), 1}

3. C

   {f(x), f′(x), 1}

4. D

   {f(x + 1), f(x − 1), f(x)}

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 · Inner Product Spaces and Forms

1. A

   If trace of TT* is zero, then T = 0.

2. B

   Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.

3. C

   Suppose T = T* and N > 1 be an integer. Let v ∈ V be such that $T^{2^N}(v)$ = 0. Then T(v) = 0.

4. D

   Suppose that TT* = T*T and N > 1 be an integer. Let v ∈ V be such that T^N(v) = 0. Then T(v) = 0.