CSIR NET Maths previous year question papers

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 $\ell^{2}$.

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

1. A

   The transposition (1 2) belongs to G.

2. B

   |G| is divisible by p.

3. C

   A p-cycle belongs to G.

4. D

   $G = S_{p}$

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

1. A

   f is a real valued function.

2. B

   f(z) = 1 for all $z\in\mathbb{C}$.

3. C

   f is not an entire function.

4. D

   f has finitely many zeros on the imaginary axis.

Single choice · The Real Line

1. A

   If A is finite, then there exists a bijection between S and U.

2. B

   If A is finite and B = [0, 1], then there is no bijection between S and T.

3. C

   There is no bijection between S and U for any choice of A.

4. D

   If A ≠ B, then there is no bijection between T and U.

Single choice · Eigenvalues and Canonical Forms

1. A

   T is diagonalizable over $\mathbb{R}$.

2. B

   The characteristic polynomial of T is $(x - 2)^{4}$.

3. C

   The characteristic polynomial of T is $(x^{2} + x + 2)(x^{2} - 4)$.

4. D

   For every $\mathbb{R}-$linear transformation $S : \mathbb{R}^{4} \to \mathbb{R}^{4}$, we have ST = TS.

Single choice · Inner Product Spaces and Forms

1. A

   B(v, v) = 0 if and only if v = 0.

2. B

   For every $\lambda \in \mathbb{R}$, there exists v such that $B(v, v) = \lambda$.

3. C

   There exists v ≠ 0 such that B(v, w) = 0 for all $w \in \mathbb{R}^{3}$.

4. D

   If B(v, w) = 0 then either v = 0 or w = 0.

Single choice · Analytic Functions

1. A

   E is the real axis.

2. B

   E is the imaginary axis.

3. C

   E contains an open subset of $\mathbb{C}$, but $E \ne \mathbb{C}$.

4. D

   $E = \mathbb{C}$

Single choice · Groups

1. A

   For each $p < 100, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ converges.

2. B

   For each $100 \le p \le 200, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ diverges.

3. C

   For each $p > 200, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ converges.

4. D

   The element 4 mod 101 in $(\mathbb{Z}/101\mathbb{Z})$ˣ is a primitive root.

Single choice · Groups

1. A

   If G is abelian, then |X| = n.

2. B

   If |X| divides n, then G is abelian.

3. C

   |X| < n

4. D

   |X| divides n.

Single choice · Ordinary Differential Equations

1. A

   $d^{2}u/$dx$^{2} + \lambda{}u = 0, u(0) = 0 = u(1)$

2. B

   $d^{2}u/$dx$^{2} - \lambda{}u = 0, u(0) = 0 = u(1)$

3. C

   du/dx $+ \lambda{}u = 0, u(0) = 0 = u(1)$

4. D

   du/dx $- \lambda{}u = 0, u(0) = 0 = u(1)$

Single choice · Hypothesis Testing

1. A

   x < 1/4

2. B

   x > 3/4

3. C

   1/2 < x < 3/4

4. D

   1/4 < x < 1/2