CSIR NET Maths previous year question papers

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 · 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 $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{4})$.

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

1. A

   $A_{n}$ is finite for every integer n ≥ 1.

2. B

   $A_{n}$ is finite for some integer n ≥ 1.

3. C

   $A_{n}$ is infinite for some integer n ≥ 1.

4. D

   $A_{n}$ is countable (finite or infinite) for every integer n ≥ 1.

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

1. A

   The rank of T is equal to 2.

2. B

   Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$. Then $T^{2}$ must be zero.

3. C

   Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$. Then $T^{3}$ must be zero.

4. D

   (−2, −8, 3, 2, 0) is contained in the null space of T.

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

1. A

   V is the union of 2 proper subspaces.

2. B

   V is the union of 3 proper subspaces.

3. C

   V has a unique basis.

4. D

   V has precisely two bases.

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

1. A

   a + b > 8.

2. B

   a + b < 7.

3. C

   a + b is divisible by 2.

4. D

   a > b.

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

1. A

   X is invertible.

2. B

   X + I is invertible.

3. C

   XY = YX.

4. D

   Y is invertible.

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

1. A

   A is diagonalizable.

2. B

   If Aᵏ = I for some positive integer k, then $A^{2} = I$.

3. C

   If Aᵏ = 0 for some positive integer k, then $A^{2} = 0$.

4. D

   All eigenvalues of A are real.

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

1. A

   The characteristic polynomial of A is $x^{3}(x - 1)(x - 2\pi{}i)^{3}$.

2. B

   The minimal polynomial of A is $x^{2}(x - 1)(x - 2\pi{}i)^{3}$.

3. C

   The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 3 respectively.

4. D

   The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 2 respectively.

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

1. A

   A is positive definite.

2. B

   $A^{2}$ is positive definite.

3. C

   There exists a nonzero $v \in \mathbb{R}^{3}$ such that ⟨v, v⟩ = 0.

4. D

   rank A = 2.

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

1. A

   f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.

2. B

   $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.

3. C

   f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.

4. D

   If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.