CSIR NET Maths previous year question papers

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

1. A

   $f \in L^{1}(\mathbb{R})$

2. B

   $f \in L^{2}(\mathbb{R})$

3. C

   $f \in L^{4}(\mathbb{R})$

4. D

   $f \in L^\infty(\mathbb{R})$

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

1. A

   (‖$f_{n}$‖$_{1})$ is a convergent sequence.

2. B

   (‖$f_{n}$‖$_{2})$ is a convergent sequence.

3. C

   Both (‖$f_{n}$‖$_{1})$ and (‖$f_{n}$‖$_{2})$ are convergent sequences.

4. D

   Neither (‖$f_{n}$‖$_{1})$ nor (‖$f_{n}$‖$_{2})$ is a convergent sequence.

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 · Functions of Several Variables

1. A

   The function h_(a,b) is differentiable on $\mathbb{R}$ for each $(a, b) \in \mathbb{R}^{2}$.

2. B

   There exists $(a, b) \in \mathbb{R}^{2}$ such that h_(a,b) is not differentiable at t = 0.

3. C

   The function f is differentiable at the point (0, 0).

4. D

   If the function f is differentiable at (0, 0) then g is identically zero.

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

1. A

   There exists a 5 × 5 real matrix whose minimal polynomial is $(x^{2} + x + 1)x^{3}$.

2. B

   There exists a 5 × 5 real matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}$.

3. C

   There exists a 5 × 5 complex matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}x$.

4. D

   There exists a 5 × 5 complex matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}$.

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

1. A

   n = 3 and the characteristic polynomials of A and B are the same.

2. B

   n = 4 and A has two distinct eigenvalues.

3. C

   n = 5 and A has some eigenvalue for which the dimensions of the eigenspaces of A and B are the same.

4. D

   n = 6 and A has only one eigenvalue and for this eigenvalue the dimensions of the eigenspaces of A and B are the same.

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

1. A

   $(f_{n})_{n}\ge_{1}$ does not converge pointwise on [0, 1].

2. B

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a continuous function on [0, 1].

3. C

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a discontinuous function on [0, 1].

4. D

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

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

1. A

   f(0) = 0

2. B

   f′(0) exists and f′(0) = 0.

3. C

   f″(0) exists and f″(0) = g(0).

4. D

   f″(0) exists but f″(0) ≠ g(0).

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

1. A

   f is Riemann integrable on [0, 1].

2. B

   The set of discontinuities of f cannot contain a non-empty open set.

3. C

   f is Lebesgue measurable function.

4. D

   f is Borel measurable function.

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 · Vector Spaces and Linear Maps

1. A

   DM ≠ MD.

2. B

   D + M is invertible.

3. C

   DM is invertible.

4. D

   rank(DM) = rank(MD).

Multi-select · full marks only for the exact set · Determinants and Matrix Tricks

1. A

   If A and B are upper triangular, then M is diagonalizable over $\mathbb{R}$.

2. B

   If A and B are diagonalizable over $\mathbb{R}$, then M is diagonalizable over $\mathbb{R}$.

3. C

   If A and B are diagonalizable over $\mathbb{R}$, then there exists $\lambda \in \mathbb{R}$ such that $M = \lambda{}I_{2}$.

4. D

   There exists $\lambda \in \mathbb{R}$ such that $M^{2} = \lambda{}I_{2}$.

Multi-select · full marks only for the exact set · Determinants and Matrix Tricks

1. A

   det(T) = 225

2. B

   det(T) = −225

3. C

   Trace(T) = 16

4. D

   Trace(T) = −16

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

   Let $x, y \in \mathbb{R}$ with x < y. Then there exists $r \in \mathbb{Q}$ such that x < (2^2024/e)r < y.

2. B

   Let $(a_{n})_{n}\ge_{2}$ be a sequence of positive real numbers. If there exists a positive real number L such that $\lim \sup(n\to\infty) a_{n}/\log n = L$, then $\lim \sup(n\to\infty) a_{n} < \infty$.

3. C

   The set of all finite subsets of $\mathbb{Q}$ is countably infinite.

4. D

   The set of continuous functions from $\mathbb{R}$ to the set {0, 1} is infinite.

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 · The Real Line

1. A

   $(a_{n})_{n}\ge_{1}$ does not converge.

2. B

   $\lim \sup(n\to\infty) a_{n} = \lim(n\to\infty) b_{n}$

3. C

   $\lim \inf(n\to\infty) a_{n} = \lim(n\to\infty) c_{n}$

4. D

   $\lim(n\to\infty) b_{n} = \lim(n\to\infty) c_{n}$

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

1. A

   $\lim(n\to\infty)$ |$X_{i}(n)$|/n exists for all 1 ≤ i ≤ 8.

2. B

   $\lim \inf(n\to\infty)$ |$X_{i}(n)$|/n ≥ 0 for all 1 ≤ i ≤ 8.

3. C

   $\lim \sup(n\to\infty)$ |$X_{i}(n)$|/n ≥ 1/8 for some 1 ≤ i ≤ 8.

4. D

   $\lim \sup(n\to\infty)$ |$X_{i}(n)$|/n < 1/8 for all 1 ≤ i ≤ 8.

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

1. A

   lim(x→0) f(x) exists.

2. B

   f is continuous at 0.

3. C

   f is differentiable at 0.

4. D

   lim(x→0) f′(x) does not exist.

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

1. A

   F is bounded.

2. B

   F is continuous on $\mathbb{R}$.

3. C

   F is uniformly continuous on $\mathbb{R}$.

4. D

   F is not uniformly continuous on $\mathbb{R}$.

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

1. A

   f is continuous at (0, 0).

2. B

   The partial derivatives $\partial{}f/\partial{}x$ and $\partial{}f/\partial{}y$ exist at (0, 0).

3. C

   f is differentiable at (0, 0).

4. D

   f is not differentiable at (0, 0).

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

1. A

   f is differentiable on $\mathbb{R}^{2}$ \ {(0, 0)}.

2. B

   All the directional derivatives of f exist at (0, 0).

3. C

   f is differentiable on $\mathbb{R}^{2}$.

4. D

   f is not continuous at (0, 0).

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

1. A

   There exists a subspace W of $\mathbb{C}^{n}$ such that $\mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

2. B

   There exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

3. C

   Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

4. D

   Suppose that there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}. Then there exists a positive integer k such that Tᵏ is the identity map.

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

1. A

   Both A and B are diagonalizable over $\mathbb{R}$.

2. B

   A is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$.

3. C

   Neither A nor B is diagonalizable over $\mathbb{R}$, but both A and B are diagonalizable over $\mathbb{C}$.

4. D

   Neither A nor B is diagonalizable over $\mathbb{C}$.

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

1. A

   S is a basis of W.

2. B

   S is an orthonormal basis of W.

3. C

   There exist f, g ∈ S such that ⟨f, g⟩ = 0.

4. D

   S contains an orthonormal basis of W.