CSIR NET Maths previous year question papers

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 · Cauchy Theory

1. A

   f has a pole at z = 0.

2. B

   There is an entire function g such that f = g on $\mathbb{C}$ \ {0}.

3. C

   f is a polynomial of degree at most 2.

4. D

   f has an essential singularity at z = 0.

Multi-select · full marks only for the exact set · Singularities and Residues

1. A

   f has a removable singularity at z = 0.

2. B

   f has a simple pole at z = 0.

3. C

   $\int_C (p(z)/z)$dz = 0, where C is any counterclockwise oriented smooth closed curve in $\mathbb{C}$.

4. D

   $\int_\gamma f(z)$dz $= 2\pi$ia$_{1}$, where $\gamma =$ {$z \in \mathbb{C}$ : |z| = 2025} is a counterclockwise oriented circle.

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

1. A

   If an abelian group G has property (P), every subgroup of G has property (P).

2. B

   If an abelian group G has property (P), then every element of G has finite order.

3. C

   The group $\mathbb{Z}$ has property (P).

4. D

   If an abelian group G has property (P), then no element has order $p^{2}$, where p is a prime number.

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

1. A

   The ring of polynomials in 3 variables over $\mathbb{R}$.

2. B

   The ring of complex analytic functions on the open unit disc in $\mathbb{C}$.

3. C

   The ring of 2 × 2 real matrices.

4. D

   The ring of continuous functions from [0, 1] to $\mathbb{R}$.

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

1. A

   Any group with p(p + 2) elements, where p and p + 2 are both prime numbers, has property (J).

2. B

   Any finite abelian group has property (J).

3. C

   The symmetric group $S_{3}$ has property (J).

4. D

   Any group with 77 elements has property (J).

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

1. A

   If K is the splitting field of a non-constant polynomial over $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

2. B

   If K is a normal extension of $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

3. C

   If K is the set of all the roots of the polynomial $x^{121} - x$ in an algebraic closure of 𝔽$_{11}$, then K is Galois over 𝔽$_{11}$, where 𝔽$_{11}$ is the field with 11 elements.

4. D

   If $K = \mathbb{Q}(2^(1/13), \zeta_{13})$, where $\zeta_{13}$ is a primitive 13th root of unity in $\mathbb{C}$, then K is Galois over $\mathbb{Q}$.

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

1. A

   If X and Y are compact, Y is Hausdorff and f is onto, then X is also Hausdorff.

2. B

   If X is an infinite compact set and f is a homeomorphism from X to f(X) (where f(X) is given the subspace topology), then Y is compact.

3. C

   If X is Hausdorff and f is onto, then Y is Hausdorff.

4. D

   If f is a homeomorphism, then X is second-countable if and only if Y is second-countable.

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

1. A

   $\partial^{2}w/\partial\xi^{2} + \partial^{2}w/\partial\xi\partial\eta + \partial^{2}w/\partial\eta^{2} = 0$

2. B

   $\partial^{2}w/\partial\xi^{2} + \partial^{2}w/\partial\eta^{2} = 0$

3. C

   $\partial^{2}w/\partial\xi^{2} + 4(\partial^{2}w/\partial\xi\partial\eta) + \partial^{2}w/\partial\eta^{2} = 0$

4. D

   $2(\partial^{2}w/\partial\xi^{2}) + 2(\partial^{2}w/\partial\xi\partial\eta) + \partial^{2}w/\partial\eta^{2} = 0$

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   $S_{1}$ and $S_{2}$ are true but NOT $S_{3}$

2. B

   $S_{2}$ and $S_{3}$ are true but NOT $S_{1}$

3. C

   $S_{1}$ and $S_{3}$ are true but NOT $S_{2}$

4. D

   Each of $S_{1}, S_{2}$, and $S_{3}$ is true

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   a + b = 1

2. B

   a + b = c

3. C

   a + b + c = 0

4. D

   $a^{2} + b^{2} = c^{2}$

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

1. A

   f(x) = cos x

2. B

   f(x) = cos 5x

3. C

   f(x) = sin x

4. D

   f(x) = sin 5x

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

1. A

   R(x, t) = 1

2. B

   R(x, t) = e^(t−x)

3. C

   u(ln 8) = 10

4. D

   u(ln 7) = 9

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

1. A

   $Y_{3}$ follows gamma distribution with $E(Y_{3}) = 3$.

2. B

   E(Y(2,n)) → 2 as $n \to \infty$

3. C

   $Y_{2}$ follows beta distribution with $E(Y_{2}) = 1/2$.

4. D

   Y(3,n) follows beta distribution with parameters 3 and n − 1.

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

1. A

   $(79X + 2024Y, 23Z^{2})$ is a sufficient statistic for $\theta$.

2. B

   $13(X + Y + Z)^{3}$ is a minimal sufficient statistic for $\theta$.

3. C

   2(X + Y + Z) is a complete sufficient statistic for $\theta$.

4. D

   (X − 2Y + Z) is an ancillary statistic.

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then E(r_S) = 0, ∀n ≥ 2.

2. B

   If n = 3, then P(r_S = 0) = 0.

3. C

   If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then Var(r_S) = 1/n, ∀n ≥ 2.

4. D

   If n = 4 and F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then P(r_S = 0) = 1/24.

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   If F_X(x) = F_Y(x) ∀x, then E(T_N) = m(N + 1)/2.

2. B

   If F_X(x) = F_Y(x) ∀x, then Var(T_N) = mn(N + 1)/24.

3. C

   If F_X(x) = F_Y(x) ∀x, then the distribution of T_N is symmetric about mn/2.

4. D

   The minimum and maximum possible values of T_N are m(m + 1)/2 and N(N + 1)/2 − m(m + 1)/2, respectively.

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   P(Unit 1 is included in the sample$) = p_{1}$

2. B

   P(Unit 1 and Unit 3 are included in the sample$) = p_{1}p_{3}[1/(1 - p_{1}) + 1/(1 - p_{3})]$

3. C

   P(Unit 1, Unit 3 and Unit 5 are included in the sample$) = p_{1}p_{3}p_{5}[1/((1 - p_{1})(1 - p_{3})) + 1/((1 - p_{3})(1 - p_{5})) + 1/((1 - p_{5})(1 - p_{1}))]$

4. D

   Expected number of distinct units in the sample is $2[1 - \sum(i=1..N)(1 - p_{i})^{2}]$

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

1. A

   The optimal solution is 4.

2. B

   An optimal point is (4, 1, 0).

3. C

   The optimal solution is 6.

4. D

   (1/2, 0, 0) is a corner point.

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.