CSIR NET Maths previous year question papers

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 · Functions of Several Variables

1. A

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $g:\Omega_\varepsilon\to(-\delta,\delta)$ such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all $(x,y)\in\Omega_\varepsilon$.

2. B

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $h:\Omega_\varepsilon\to(-\delta,\delta)$ such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all $(x,z)\in\Omega_\varepsilon$.

3. C

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $k_{1}(0)=k_{2}(0)=0$ and $F(x,k_{1}(x),k_{2}(x))=-1$ for all $x\in(-\varepsilon,\varepsilon)$.

4. D

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $j_{1}(0)=j_{2}(0)=0$ and $F(j_{1}(z),j_{2}(z),z)=-1$ for all $z\in(-\varepsilon,\varepsilon)$.

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

1. A

   [0,1]

2. B

   (0,1]

3. C

   (10⁻$^{2025},1)$

4. D

   (−10⁻$^{2025},10$⁻$^{2025}]$

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 · The Real Line

1. A

   A is bounded below but not bounded above.

2. B

   A is bounded above but not bounded below.

3. C

   inf A = −1

4. D

   sup A = 1

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

1. A

   f is a uniformly continuous function.

2. B

   f is a bounded function.

3. C

   There exists $x\in(0,\infty)$ such that f(x)=0.

4. D

   The derivative of f is continuous.

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

1. A

   F is non-decreasing on $\mathbb{R}$

2. B

   $\lim(x\to\infty) F(x) = 1$

3. C

   F is left-continuous on $\mathbb{R}$

4. D

   F is right-continuous on $\mathbb{R}$

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

1. A

   ABC is confounded with blocks

2. B

   BCD is confounded with blocks

3. C

   CDE is confounded with blocks

4. D

   ABDE is confounded with blocks

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

1. A

   The sequence of functions {$f_{n}$}$_{n}\ge1$ is uniformly bounded.

2. B

   The sequence of functions {$f_{n}$}$_{n}\ge1$ does not converge pointwise.

3. C

   The sequence of functions {$f_{n}$}$_{n}\ge1$ converges uniformly to the constant function 0.

4. D

   The set {$f_{n} : n\in\mathbb{N}$} is compact in C[0,1], where C[0,1] denotes the space of all real valued continuous functions on [0,1] equipped with the supremum norm.

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 · Probability

1. A

   Y has a Double Exponential distribution with location parameter ln(ln 2) and scale parameter 1

2. B

   Median of Y = ln(ln 2)

3. C

   P(Y≤0) = e⁻$^{1}$

4. D

   The derivative of the cumulative distribution function of Y at ln 3 is e^(−1/3)

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

1. A

   $L_\alpha$⁺ and $L_\alpha$⁻ exist for every $\alpha\in(0,1)$.

2. B

   If f is surjective, then f is continuous.

3. C

   If f is Riemann integrable, then f is continuous.

4. D

   If the left and right hand limits exist at $\alpha,\beta\in(0,1), \alpha\ne\beta$, then $U_\alpha\cap{}U_\beta=\emptyset$.

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   State 2 is a positive recurrent state

2. B

   Mean recurrence time of state 1 is 7/4

3. C

   State 4 is a transient state

4. D

   State 3 is aperiodic and ergodic

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

1. A

   P(XY=0) = (1+e⁻$^{3})/2$

2. B

   E(X+Y) = 3

3. C

   Var(X+Y) = 21

4. D

   Cov(X+Y,X−Y) = 0

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

1. A

   There exists a cubic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.

2. B

   There exists a quadratic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.

3. C

   If $f(X)\in\mathbb{Q}[X]$ is such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

4. D

   If $f(X)\in\mathbb{Q}[X]$ is a cubic polynomial such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

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

1. A

   $G(p,Q) = -Q[1 + log_{e}(p^{2}/(4Q))]$

2. B

   G(p,Q) = −pQ$[1 + log_{e}(Q^{2}/(4p))]$

3. C

   p = 2qe$^P, Q = q^{2}e^P$

4. D

   p = −2qe$^P, Q = -q^{2}e^P$

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

1. A

   |f(−1/5)| ≤ 1/5

2. B

   |f(−1/5)| ≤ 1/3

3. C

   |f′(−1/2)| ≤ 1/2

4. D

   |f′(−1/2)| ≤ 4/3

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

1. A

   $\int_\gamma1$ dz/(z sin z) = 0

2. B

   $\int_\gamma1$ dz$/(z \sin z) = 2\pi{}i$

3. C

   $\int_\gamma2$ dz$/(z \sin z) = 2\pi{}i$

4. D

   $\int_\gamma2$ dz/(z sin z) = 0

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

1. A

   If T is not diagonalizable over $\mathbb{R}$, then $T^{2}$ has an eigenvalue ≤0.

2. B

   If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.

3. C

   If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$.

4. D

   For each non-zero v∈V, {v,Tv,$T^{2}v$} is linearly dependent.

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

1. A

   $H_{0}$ is rejected

2. B

   The p-value of the test is greater than 0.01

3. C

   Under $H_{0}, E(K) = 3.5$

4. D

   If $X_{(i)}$ denotes the i-th smallest observation of the sample, then $[X_{(2)},X_{(6)}]$ is a confidence interval for M with confidence coefficient at least 0.95

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

1. A

   $\lim(x\to0+) y_{2}(x)/(x^{2}y_{1}(x))$ exists.

2. B

   $\lim(x\to0+) y_{2}(x)/($xy$_{1}(x))$ exists.

3. C

   lim(x→0+) xy$_{1}(x)/y_{2}(x)$ does NOT exist.

4. D

   $\lim(x\to0+) x^{2}y_{1}(x)/y_{2}(x)$ exists.

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

1. A

   P(X−Y=0 ∣ X+Y=2) = P(X−Y=0 ∣ X+Y=12)

2. B

   E((X−Y)/(X+Y)) = 0

3. C

   Cov(X+Y, X−Y) = 0

4. D

   (X+Y) and (X−Y) are independent

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

1. A

   There exist $\lambda\in\mathbb{C}$ and a non-zero v∈V such that Av$=\lambda{}v$.

2. B

   There exist $\lambda,\mu\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}w$.

3. C

   There exist $\lambda,\mu,\delta\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}v+\delta{}w$.

4. D

   There exists a three-dimensional subspace W such that Aw∈W for all w∈W.

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

   $G = S_{p}$