CSIR NET Maths previous year question papers

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.

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

1. A

   If $lim_{n}\to\infty b_{n}$ exists in $\mathbb{R}$, then {$a_{n} : n \ge 1$} is bounded

2. B

   If $lim_{n}\to\infty b_{n} = 1$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

3. C

   If $lim_{n}\to\infty b_{n} = 1/2$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

4. D

   If $lim_{n}\to\infty b_{n} = 0$, then $lim_{n}\to\infty a_{n} = 0$

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

1. A

   f is strictly increasing

2. B

   f is strictly decreasing

3. C

   f is either strictly increasing or strictly decreasing

4. D

   f is onto

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

1. A

   f need not be surjective

2. B

   f must be surjective if K = [0, 1]

3. C

   f is injective and f⁻$^{1} : f(K) \to K$ is continuous

4. D

   f is injective, but f⁻$^{1} : f(K) \to K$ need not be continuous

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

1. A

   f(x) is not uniformly continuous on [0, 1)

2. B

   The sequence $(p_{n}(x))$ converges to f(x) pointwise on [0, 1)

3. C

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, 1)

4. D

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, c] for every 0 < c < 1

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

1. A

   (1, 1, 1, 1, 1)

2. B

   (0, 0, 1, 2, 4)

3. C

   (1, 0, 1, 0, 1)

4. D

   (1, 0, 1, 0, 2)

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

1. A

   There exists an injective $\mathbb{Q}-$linear transformation $T : \mathbb{R} \to \mathbb{Q}[x]$

2. B

   There exists an injective $\mathbb{Q}-$linear transformation $T : \mathbb{Q}[x] \to \mathbb{R}$

3. C

   The $\mathbb{Q}-$vector spaces $\mathbb{Q}[x]$ and $\mathbb{R}$ are isomorphic

4. D

   There do not exist non-zero $\mathbb{Q}-$linear transformations $T : \mathbb{R} \to \mathbb{Q}[x]$

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

1. A

   There exists a non-zero vector $v_{1} \in \mathbb{R}^{4}$ such that Tv$_{1} = 2v_{1}$

2. B

   There exists a non-zero vector $v_{2} \in \mathbb{R}^{4}$ such that Tv$_{2} = v_{2}$

3. C

   For every non-zero vector $v \in \mathbb{R}^{4}$, the set {2v, 3Tv} is linearly independent

4. D

   T is a one-one function

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

1. A

   The minimal polynomial of B is $x^{2} + x + 1$

2. B

   The minimal polynomial of B is $x^{2} - x + 1$

3. C

   $B^{3} = I_{4}$

4. D

   $B^{3} + I_{4} = 0$

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

1. A

   1

2. B

   2

3. C

   3

4. D

   5

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

1. A

   The dimension of V is even

2. B

   The trace of T is zero

3. C

   The minimal polynomial of T cannot have two distinct roots

4. D

   The minimal polynomial of T is equal to its characteristic polynomial

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

1. A

   q is a quadratic form in $x_{1}, x_{2}, y_{1}, y_{2}$

2. B

   There exists $(t_{1}, t_{2}) \in \mathbb{R}^{2}$ such that $q_{1}(t_{1}, t_{2}) = 5$

3. C

   There does not exist $(s_{1}, s_{2}) \in \mathbb{R}^{2}$ such that $q_{2}(s_{1}, s_{2}) = -5$

4. D

   Given $\alpha \in \mathbb{R}$, there exists a vector $w \in \mathbb{R}^{4}$ such that $q(w) = \alpha$

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

1. A

   $A_{n} \to 0$ and $B_{n} \to 0$ as $n \to \infty$

2. B

   If $\sum_{n}$₌$_{1}^\infty a_{n}$ is absolutely convergent, then both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are absolutely convergent

3. C

   Both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are convergent

4. D

   If $\sum_{n}$₌$_{1}^\infty a_{n}$ is not absolutely convergent, then both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are divergent