CSIR NET Maths previous year question papers

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

1. A

   If A is invertible, then AB and BA have the same minimal polynomial.

2. B

   If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.

3. C

   If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.

4. D

   If AB and BA have the same minimal polynomial, then either A or B is invertible.

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

1. A

   (ST − TS$)^{2} = \lambda{}I$ for some $\lambda \in \mathbb{C}$.

2. B

   The characteristic polynomial of (ST − TS$)^{2}$ is $(x - \lambda)^{2}$ for some $\lambda \in \mathbb{C}$.

3. C

   If ST − TS has only one eigenvalue, then ST − TS $= \lambda{}I$ for some $\lambda \in \mathbb{C}$.

4. D

   If ST − TS has only one eigenvalue, then (ST − TS$)^{2}$ is the zero transformation.

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

1. A

   The number of two dimensional subspaces of V is 13.

2. B

   The number of surjective linear transformations from V to W is 624.

3. C

   The number of one dimensional subspaces of V is 13.

4. D

   The number of linear transformations from V to W is $3^{6}$.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   If f(z) = f(z + 1) for all $z \in \mathbb{C}$ then f is a constant function.

2. B

   If f(z) = f(z + 1) = f(z + i) for all $z \in \mathbb{C}$ then f is a constant function.

3. C

   If f(1/z) has a removable singularity at 0 then f is a constant function.

4. D

   If f is a non-constant function then f(1/z) has a pole at 0.

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   $lim_{z\to0}$ f(z) does not exist.

2. B

   $lim_{z\to0}$ f(z) exists and has absolute value ≤ 1.

3. C

   $lim_{z\to0}$ f(z) = 0.

4. D

   There exists $\theta \in \mathbb{R}$ such that f(z) = $e^{i\theta}z$ for all z ∈ 𝔻^×.

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

1. A

   A is nonempty.

2. B

   A is finite.

3. C

   A is infinite.

4. D

   A is uncountable.

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

1. A

   $\varphi_{1}(x + 2\pi)$ is also a solution of ODE

2. B

   $\varphi_{2}(x + 4\pi)$ is also a solution of ODE

3. C

   There are NO constants a, b such that $\varphi_{2}(x + 4\pi) = a\varphi_{1}(x) + b\varphi_{2}(x)$

4. D

   There exist $a, b \in \mathbb{R}$ such that $\varphi_{1}(x + 2\pi) = a\varphi_{1}(x) + b\varphi_{2}(x)$

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

1. A

   There exists a q such that cos(x) and e^x cos(x) are solutions of ODE

2. B

   There exists a q such that sin(x) and cos(x) are solutions of ODE

3. C

   There exists a q such that e^x sin(x) and e^x cos(2x) are solutions of ODE

4. D

   There exists a q such that xe^x and x(x − 1)e^x are solutions of ODE

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

1. A

   The probability density function of S is f_S(s) = $(1/2)e^{-s/2}$ if s > 0, and 0 otherwise.

2. B

   The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.

3. C

   Var(S) = 2.

4. D

   E(T) = 2/3.

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

1. A

   $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$

2. B

   $limsup_{n\to\infty}$ $t_{n} \le e$

3. C

   $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$

4. D

   $liminf_{n\to\infty}$ $t_{n} \ge e$

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

1. A

   The composition f ∘ g is Riemann integrable.

2. B

   If g(x) ≠ 0 for each x ∈ [a, b], then f/g is Riemann integrable.

3. C

   The positive square root $\sqrt{f^{2} + g^{2}}$ is Riemann integrable.

4. D

   The composition f ∘ g is Riemann integrable, if both f and g are continuous.

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

1. A

   G = GL$_{2}($𝔽_p) and H is a subgroup of order p.

2. B

   G = SL$_{2}($𝔽_p) and H is a subgroup of order p.

3. C

   p ≡ 3 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

4. D

   p ≡ 1 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   f(X) is irreducible in $\mathbb{Q}[X]$, but g(X) is not.

2. B

   g(X) is irreducible in $\mathbb{Q}[X]$, but f(X) is not.

3. C

   Both f(X) and g(X) are irreducible in $\mathbb{Q}[X]$.

4. D

   Neither f(X) nor g(X) is irreducible in $\mathbb{Q}[X]$.

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

1. A

   S is dense in $\mathbb{R}$.

2. B

   S \ $\mathbb{Q}$ is dense in $\mathbb{R}$.

3. C

   S \ $\mathbb{Q}$ is discrete with subspace topology on S.

4. D

   S is connected.

Multi-select · full marks only for the exact set · Classical Mechanics

1. A

   $\partial/\partial{}t [f, g] = [\partial{}f/\partial{}t, g] + [f, \partial{}g/\partial{}t]$

2. B

   If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion

3. C

   [[H, f], g] + [[g, H], f] + [[f, g], H] = 0

4. D

   If f and g are constants of motion, then [f, g] is a constant of motion

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

1. A

   If there are no elements of order 4 in G, then G is abelian.

2. B

   If there are exactly two elements of order 4 in G, then G is abelian.

3. C

   If there are exactly six elements of order 4 in G, then G is abelian.

4. D

   There is an element of order 4 in G.

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

1. A

   There exists a positive integer N such that for all n≥N, $b_{n,l}$ $= a_{l}$ for all 1≤l≤2023.

2. B

   There exists a positive integer N such that for all n≥N, $b_{n,2025}$ $= a_{2025} - 1$.

3. C

   $\beta_{n}$ is rational for infinitely many n.

4. D

   There exists a positive integer N such that for all n≥N, $b_{n,2024}$ $= a_{2024} - 1$.

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

1. A

   $(\mathbb{R}, \tau_E)$ is a normal space.

2. B

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau_E)$ to $(\mathbb{R},\tau)$, then $(\mathbb{R},\tau)$ is a regular space.

3. C

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau)$ to $(\mathbb{R},\tau_E)$, then $(\mathbb{R},\tau)$ is a Hausdorff space.

4. D

   $\mathbb{R}$ is a regular space in the finite-complement topology.

Multi-select · full marks only for the exact set · Linear Integral Equations

1. A

   R(0,1,1/3) = 3/2

2. B

   R(1/2,1/2,1/3) = 2/3

3. C

   y(1) = 3e+1

4. D

   y(1) = e+3

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

1. A

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB+BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

2. B

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB−BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

3. C

   There exists $B \in M_{4}(\mathbb{C})$ such that AB+BA$=0, B^{2}=I$ and Trace(B)=0.

4. D

   If $B \in M_{4}(\mathbb{C})$ is such that AB+BA=0, then B is diagonalisable.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   R has a unique maximal ideal.

2. B

   Every finitely generated ideal of R is principal.

3. C

   R is an integral domain.

4. D

   R is a euclidean domain.

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

1. A

   (P) is true.

2. B

   (P) is FALSE.

3. C

   (Q) is true.

4. D

   (Q) is FALSE.

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

1. A

   For all x,y∈X and every non-empty subset A of X, we have d(x,A)−d(y,A) ≤ d(x,y).

2. B

   For every non-empty subset A of X, the function x↦d(x,A) is uniformly continuous on X.

3. C

   A non-empty subset A of X is closed if and only if d(x,A)>0 for all x in X outside A.

4. D

   If X is compact and $C_{1},\dots,C_{k}$ are non-empty closed sets of X such that $\cap{}C_{i}=\emptyset$, then the minimum value of $x\mapsto\sum{}d(x,C_{i})$ on X is 0.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   The function g has a pole at 0.

2. B

   If $g(\alpha)=0$, then |$\alpha$|≠1.

3. C

   The function g has only finitely many zeros.

4. D

   $max_{|z|=1}$ |g(z)| ≥ 1

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

1. A

   H contains an orthonormal basis of $\ell^{2}$.

2. B

   H is a linear subspace of $\ell^{2}$.

3. C

   H is a bounded subset of $\ell^{2}$.

4. D

   H is a convex subset of $\ell^{2}$.