CSIR NET Maths previous year question papers

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

1. A

   The subsequences {$x_{2n}$₊$_{1}$}, {$x_{2n}$} and {$x_{3n}$} are convergent and have the same limit.

2. B

   The subsequences {$x_{2n}$₊$_{1}$}, {$x_{2n}$} and {$x_{3n}$} are convergent.

3. C

   The subsequences {$x_{kn}$}$_{n}$ are convergent for every k ≥ 2.

4. D

   lim |$x_{n}$₊$_{1} - x_{n}$| = 0.

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

1. A

   If a = 0, then the sequence {$x_{n}$} converges to 1/2.

2. B

   If a = 0, then the sequence {$x_{n}$} converges to −1/2.

3. C

   The sequence {$x_{n}$} converges for every a ∈ (−1/2, 3/2), and it converges to 1/2.

4. D

   If a = 0, then the sequence {$x_{n}$} does not converge.

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

1. A

   There is no continuous real-valued function g defined on any interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

2. B

   There is exactly one continuous real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

3. C

   There is exactly one differentiable real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

4. D

   There are two distinct differentiable real-valued functions g on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

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

1. A

   Quadratic forms (a) and (b) are equivalent.

2. B

   Quadratic forms (a) and (c) are equivalent.

3. C

   Quadratic form (b) is positive definite.

4. D

   Quadratic form (c) is positive definite.

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   {$v \in \mathbb{Q}^{5}$ | Bv = 0} is a three-dimensional vector space over $\mathbb{Q}$.

2. B

   The linear transformation $T : \mathbb{Q}^{3} \to \mathbb{Q}^{5}$ given by T(v) = Bᵗv is injective.

3. C

   The column span of B is two-dimensional.

4. D

   The linear transformation $T : \mathbb{Q}^{3} \to \mathbb{Q}^{3}$ given by T(v) = BBᵗv is injective.

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

1. A

   If $T_{1}$ and $T_{2}$ are similar, then $W_{1}$ and $W_{2}$ are isomorphic vector spaces.

2. B

   If $W_{1}$ and $W_{2}$ are isomorphic vector spaces, then $T_{1}$ and $T_{2}$ have the same minimal polynomial.

3. C

   If $W_{1} = W_{2} = V$, then $T_{1}$ and $T_{2}$ are similar.

4. D

   If $W_{1}$ and $W_{2}$ are isomorphic, then $T_{1}$ and $T_{2}$ have the same characteristic polynomial.

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

1. A

   G is an Abelian group.

2. B

   G is a cyclic group.

3. C

   G is a simple group.

4. D

   G is not a simple group.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   f(z̄) is an entire function.

2. B

   conj(f(z)) is an entire function.

3. C

   conj(f(z̄)) is an entire function.

4. D

   conj(f(z̄)) + f(z̄) is an entire function.

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   (P) has solution(s) defined for all t > 0.

2. B

   (P) has a unique solution.

3. C

   (P) has infinitely many solutions.

4. D

   The solution(s) of (P) is/are Lipschitz.

Multi-select · full marks only for the exact set · Partial Differential Equations

1. A

   u changes sign in the interior of S.

2. B

   u(x, y) = v(x, y) along a line in S.

3. C

   v changes sign in the interior of S.

4. D

   v vanishes along a line in S.

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

1. A

   z(x) + 3y(x) = 2 for x ∈ [0,1].

2. B

   3z(x) + y(x) = 2 for x ∈ [0,1].

3. C

   y(x) + z(x) = 2x for x ∈ [0,1].

4. D

   y(x) + z(x) = x for x ∈ [0,1].

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

1. A

   If P(A | B) = 0 then P(B | A) = 0.

2. B

   If P(A | B) = 1 then P(B | A) = 1.

3. C

   If P(A | B) > P(A) then P(B | A) > P(B).

4. D

   If P(A | B) > P(B) then P(B | A) > P(A).

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

1. A

   P{$Y_{n} \in [0, 2n]$}

2. B

   P{$Y_{n} \in [n, 3n]$}

3. C

   P{$Y_{n} \in [2n, 4n]$}

4. D

   P{$Y_{n} \in [3n, 5n]$}

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

1. A

   q = 0.6

2. B

   E[Y] = 5.8

3. C

   Var(Y) = 3.04

4. D

   $Y = 0.4X_{1} +$ qX$_{2}$

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

1. A

   n⁻$^{1} \sum_{i=1}^{n}$ $X_{i}$

2. B

   n⁻$^{1} \sum_{i=1}^{2n}$ $X_{i}$

3. C

   n⁻$^{1} \sum_{i=1}^{n}$ $X_{2i}$₋$_{1}$

4. D

   n⁻$^{1}(\sum_{i=1}^{n}$ $X_{i} + W)$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   Size of the test $\varphi$ is 0.1.

2. B

   Size of the test $\psi$ is 0.05.

3. C

   (Power of the test $\psi) > 0.05$.

4. D

   (Power of the test $\psi) > ($Power of the test $\varphi)$.

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

1. A

   There exists an integer n ≥ 1 such that $n^{2} \sin(1/n) \ge x$.

2. B

   There exists an integer n ≥ 1 such that n cos(1/n) ≥ x.

3. C

   There exists an integer n ≥ 1 such that n $e^{-n}$ ≥ x.

4. D

   There exists an integer n ≥ 2 such that n (log $n)^{-1}$ ≥ x.

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 · Continuity and Differentiation

1. A

   f is continuous on $[0, \infty)$.

2. B

   f is uniformly continuous on $[0, \infty)$.

3. C

   g is continuous on $[0, \infty)$.

4. D

   g is uniformly continuous on $[0, \infty)$.

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

1. A

   There exists $\lambda$ such that $I_\lambda$ converges, but $K_\lambda$ does not converge.

2. B

   There exists $\lambda$ such that $K_\lambda$ converges, but $I_\lambda$ does not converge.

3. C

   There exists $\lambda$ such that $I_\lambda, K_\lambda$ both converge.

4. D

   There exists $\lambda$ such that neither $I_\lambda$ nor $K_\lambda$ converges.

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

1. A

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0.

2. B

   $\int$₋$_{1}^{1} f(x)p(x)$ dx = 0 for all real polynomials p(x).

3. C

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 odd.

4. D

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 even.

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

1. A

   $(f_{n})$ converges pointwise on [0, 1].

2. B

   $(f_{n})$ converges uniformly on compact subsets of [0, 1) but not on [0, 1).

3. C

   $(f_{n})$ converges uniformly on [0, 1) but not on [0, 1].

4. D

   $(f_{n})$ converges uniformly on [0, 1].

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

1. A

   F is continuous.

2. B

   F is uniformly continuous.

3. C

   F is differentiable.

4. D

   F has partial derivatives of all orders.

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 $(x, y) \in (0,1)^{2}$.

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

1. A

   Let $A_{1} =$ {$x^{2} + y^{2} < 1$}. For each $\gamma \in p(A_{1})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{1})$.

2. B

   Let $A_{2} =$ {$x^{2} + y^{2} \le 1$}. For each $\gamma \in p(A_{2})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{2})$.

3. C

   Let $A_{3} =$ {xy = 0}. For each $\gamma \in p(A_{3})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{3})$.

4. D

   Let $A_{4} =$ {xy = 1}. For each $\gamma \in p(A_{4})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{4})$.