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 · Vector Spaces and Linear Maps

1. A

   125

2. B

   144

3. C

   145

4. D

   480

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 $\beta \in (0,\infty)$.

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

1. A

   B(w, w) = 0

2. B

   There exists an $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

3. C

   There are infinitely many $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

4. D

   q is equivalent to the quadratic form $Q(x, y) = x^{2} - y^{2}$ for all $(x, y) \in \mathbb{R}^{2}$.

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

1. A

   If f is Lebesgue integrable on $\mathbb{R}$, then $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite.

2. B

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, then f is Lebesgue integrable on $\mathbb{R}$.

3. C

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, and $f(x) \ge a_{1}$ for all $x \in \mathbb{R}$, then f is Lebesgue integrable on $\mathbb{R}$.

4. D

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite and f is bounded, then f is Lebesgue integrable on $\mathbb{R}$.

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖T(f) − T(g)‖ $\le \alpha$‖f − g‖.

2. B

   There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖$T^{2}(f) - T^{2}(g)$‖ $\le \alpha$‖f − g‖.

3. C

   The set {f ∈ C[0, 1] : T(f) = f} is a singleton set.

4. D

   ‖$T^{n}$‖ $\to \infty$ as $n \to \infty$.

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

1. A

   $W_{0} =$ {0}

2. B

   $W_{1} =$ {0}

3. C

   $W_{2} =$ {0}

4. D

   $W_{n} =$ {0}

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

1. A

   f is a uniformly continuous function.

2. B

   f is a bounded function.

3. C

   There exists $x\in(0,\infty)$ such that f(x)=0.

4. D

   The derivative of f is continuous.

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

1. A

   F is non-decreasing on $\mathbb{R}$

2. B

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

3. C

   F is left-continuous on $\mathbb{R}$

4. D

   F is right-continuous on $\mathbb{R}$

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

1. A

   If T is not diagonalizable over $\mathbb{R}$, then $T^{2}$ has an eigenvalue ≤0.

2. B

   If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.

3. C

   If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$.

4. D

   For each non-zero v∈V, {v,Tv,$T^{2}v$} is linearly dependent.

Single choice · Continuity and Differentiation

1. A

   f is monotone.

2. B

   f is continuous but not strictly monotone.

3. C

   f is not continuous.

4. D

   f is continuous but not uniformly continuous.

Single choice · The Real Line

1. A

   $\sup A = 2 + 2\sqrt{3}$

2. B

   $\sup A = 3 + 2\sqrt{2}$

3. C

   $\inf A = 2 + 2\sqrt{3}$

4. D

   $\inf A = 3 + 2\sqrt{2}$

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

1. A

   $(x^{n}g(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

2. B

   $(x^{n}g(x))_{n}\ge_{1}$ does not converge uniformly on [0, 1].

3. C

   $(x^{n}h(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

4. D

   $(x^{n}h(x))_{n}\ge_{1}$ does not converge uniformly on [0, 1].

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

1. A

   f is differentiable at 0 when $\alpha > 1$.

2. B

   f is differentiable at 0 when $\alpha = 1/2$.

3. C

   f is continuous at 0 when $\alpha = 0$.

4. D

   f is continuous at 0 when $\alpha > 0$.

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

1. A

   If f ∈ B, then dim ker f = 3

2. B

   dim B = 3

3. C

   dim A = 4

4. D

   If f ∈ A, then the image of f is a one-dimensional $\mathbb{Q}-$vector space.

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

1. A

   det(B) > 0 for every nonzero $(a, b, c, d) \in \mathbb{R}^{4}$.

2. B

   det(B) > 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$.

3. C

   det(B) ≤ 0 for every $(a, b, c, d) \in \mathbb{R}^{4}$.

4. D

   det(B) ≤ 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$.

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

1. A

   S is an unbounded subset of $\mathbb{R}$.

2. B

   S is a bounded subset of $\mathbb{R}$ and sup(S) = 1.

3. C

   There exists u ∈ X such that ‖u‖ = 1 and T(u) = 1.

4. D

   S is a closed subset of $\mathbb{R}$.

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

1. A

   f is continuous on $\mathbb{R}$

2. B

   f is differentiable on $\mathbb{R}$

3. C

   f is differentiable only at 0

4. D

   f is not differentiable at 0

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

1. A

   The integral I is convergent

2. B

   The integral I is not convergent

3. C

   The integral $I_{a}$ converges for a = 1/2 but not for a = 0

4. D

   The integral $I_{a}$ converges for all a ≥ 0

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 · 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].