CSIR NET Maths previous year question papers

Single choice · Continuity and Differentiation

1. A

   f never vanishes in [0, 1].

2. B

   f has at most finitely many zeros in [0, 1].

3. C

   f has infinitely many zeros in [0, 1].

4. D

   f(1/2) = 0.

Single choice · Continuity and Differentiation

1. A

   S = ∅

2. B

   S ≠ ∅ and S is a proper subset of $(1, \infty)$

3. C

   $(2, \infty)$ is a proper subset of S

4. D

   S ∩ (0, 1) ≠ ∅

Single choice · Lebesgue Measure and Integration

1. A

   Y is an infinite set.

2. B

   Y has $2^{|X|}$ elements.

3. C

   There is a bijective function from X to Y.

4. D

   There is a surjective function from X to Y.

Single choice · Vector Spaces and Linear Maps

1. A

   A does not have any non-zero invariant subspace in $\mathbb{C}^{n}$.

2. B

   A has an invariant subspace in $\mathbb{C}^{n}$ of dimension n − 3.

3. C

   All eigenvalues of A are real.

4. D

   $A^{2}$ does not have any invariant subspace in $\mathbb{C}^{n}$ of dimension n − 1.

Single choice · Vector Spaces and Linear Maps

1. A

   The product of two 2×2 real matrices of rank 2 is of rank 2.

2. B

   The product of two 3×3 real matrices of rank 2 is of rank at most 2.

3. C

   The product of two 3×3 real matrices of rank 2 is of rank at least 2.

4. D

   The product of two 2×2 real matrices of rank 1 can be the zero matrix.

Single choice · Vector Spaces and Linear Maps

1. A

   If A is a real 3×2 matrix and B a real 2×3 matrix with BA $= I_{2}$, then AB $= I_{3}$.

2. B

   Let A = [[3, 3], [1, 2]]. Then there is a matrix B with integer entries such that AB $= I_{2}$.

3. C

   Let A = [[3, 1], [1, 2]] with entries in $\mathbb{Z}/6\mathbb{Z}$. Then there is a matrix B with entries in $\mathbb{Z}/6\mathbb{Z}$ such that AB $= I_{2}$.

4. D

   If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $= I_{3}$.

Single choice · Eigenvalues and Canonical Forms

1. A

   1

2. B

   n

3. C

   n(n+1)/2

4. D

   n(n+1)(2n+1)/6

Single choice · Inner Product Spaces and Forms

1. A

   $A^{2} = 0$.

2. B

   rank A = 1.

3. C

   rank A = 0.

4. D

   There exists $u \in \mathbb{R}^{2}, u \ne 0$ such that (u, u) = 0.

Single choice · Inner Product Spaces and Forms

1. A

   A_a is positive definite for all a < 3.

2. B

   A_a is positive definite for all a > 3.

3. C

   A_a is positive definite for all a ≥ −2.

4. D

   A_a is positive definite only for finitely many values of a.

Single choice · Groups

1. A

   G is a union of proper subgroups.

2. B

   G is a union of proper subgroups if |G| has at least two distinct prime divisors.

3. C

   If G is abelian, then G is a union of proper subgroups.

4. D

   G is a union of proper subgroups if and only if G is not cyclic.

Single choice · Rings and Fields

1. A

   88

2. B

   −88

3. C

   −2

4. D

   0

Single choice · Rings and Fields

1. A

   K is dense in $\mathbb{C}$.

2. B

   K is an algebraic extension of $\mathbb{Q}$.

3. C

   $\mathbb{C}$ is an algebraic extension of K.

4. D

   The smallest closed subset of $\mathbb{C}$ containing K is NOT a field.

Single choice · Analytic Functions

1. A

   For $c_{1}, c_{2} \in \mathbb{C}$, the level curves $u = c_{1}$ and $v = c_{2}$ are orthogonal wherever they intersect.

2. B

   $\nabla{}u \cdot \nabla{}v = 0$ at every point.

3. C

   If f is an entire function, then $\nabla{}u \cdot \nabla{}v = 0$ at every point.

4. D

   If $\nabla{}u \cdot \nabla{}v = 0$ at every point, then f is an entire function.

Single choice · Analytic Functions

1. A

   f(ℍ$) = \mathbb{C}$∖{0}.

2. B

   f(ℍ) ∩ ℍ is countable.

3. C

   f(ℍ) is bounded.

4. D

   f(ℍ) is a convex subset of $\mathbb{C}$.

Single choice · Zeros and Mappings

1. A

   100

2. B

   50

3. C

   30

4. D

   0

Single choice · Zeros and Mappings

1. A

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_{0} - n_p + 1$.

2. B

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_{0} - n_p - 1$.

3. C

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_{0} - n_p$.

4. D

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_p - n_{0}$.

Single choice · Ordinary Differential Equations

1. A

   3

2. B

   2

3. C

   1

4. D

   4

Single choice · Partial Differential Equations

1. A

   elliptic in {(x, y) : y > 0}

2. B

   parabolic in {(x, y) : x > 0, y > 0}

3. C

   hyperbolic in {(x, y) : xy ≠ 0}

4. D

   parabolic in {(x, y) : xy ≠ 0}

Single choice · Partial Differential Equations

1. A

   $lim_{t\to\infty}$ u(5, t) = 1

2. B

   $lim_{t\to\infty}$ u(5, t) = 2

3. C

   $lim_{t\to\infty}$ u(5, t) = 1/2

4. D

   $lim_{t\to\infty}$ u(5, t) = 0

Single choice · Numerical Analysis

1. A

   1.50

2. B

   1.65

3. C

   1.25

4. D

   1.15

Single choice · Calculus of Variations

1. A

   0

2. B

   1

3. C

   2

4. D

   countably infinite

Single choice · Linear Integral Equations

1. A

   $4/(1 + e^{2})$

2. B

   $2/(1 + e^{2})$

3. C

   $4/(e^{2} - 1)$

4. D

   $2/(e^{2} - 1)$

Single choice · Classical Mechanics

1. A

   $(1 - \cos \theta)\theta$̈ + ½ $\sin \theta (\theta$̇$)^{2} - (g/2a) \sin \theta = 0$

2. B

   $(1 - 2cos \theta)\theta$̈ $+ \sin \theta (\theta$̇$)^{2} - (g/a) \sin \theta = 0$

3. C

   $m(1 - 2cos \theta)\theta$̈ $+ \sin \theta (\theta$̇$)^{2} + (g/a) \sin \theta = 0$

4. D

   $m(1 - 2cos \theta)\theta$̈ $+ (m/2) \sin \theta (\theta$̇$)^{2} - (g/a) \sin \theta = 0$

Single choice · Probability

1. A

   P(X > 9) ≥ 11/12

2. B

   P(X < 9) ≥ 11/12

3. C

   $E(X - 3/4)^{2} \ge 11/12$

4. D

   (11/9)X ~ Poisson(11/12)

Single choice · Probability

1. A

   1581/3456

2. B

   1875/3456

3. C

   125/3456

4. D

   3331/3456