CSIR NET December 2024 Mathematical Sciences — Part B & C solved

Let A, B be two non-empty subsets of $\mathbb{R}$. Let S = {$f:A\to \mathbb{R}$ | f is continuous}, T = {$f:B\to \mathbb{R}$ | f is continuous}, U = {$f:A\times A\to \mathbb{R}$ | f is continuous}. Which of the following statements is true?

Let $(x_n{)}_n{\ge }_1$ be a sequence of real numbers that has a decreasing subsequence $(x_{(}{n}_k){)}_k{\ge }_1$. Assume that $\lim (k\to \infty )x_{(}{n}_k)=2025$. Which of the following statements is necessarily true?

Consider the sequences $(a_n{)}_n{\ge }_1$ and $(b_n{)}_n{\ge }_1$, defined by $a_n=-2\sqrt{n}+\sum (k=\mathrm{1..}n)1/\sqrt{k}$ and $b_n=-2\sqrt{n+1}+\sum (k=\mathrm{1..}n)1/\sqrt{k}$. Which of the following statements is true?

Let $f:\mathbb{R}\to \mathbb{R}$ be a non-constant continuous function. Which of the following statements is necessarily true?

Let $f:\mathbb{R}\to [0,\infty )$ be a bijective function. Which of the following statements is true?

Consider the sequences $(f_n{)}_n{\ge }_1$ and $(g_n{)}_n{\ge }_1$ of functions defined on the interval [−1, 1] by $f_n(x)=(-1{)}^n(x^2+n)/n^2$ and $g_n(x)=(-1{)}^n(x^2+n^2)/n^3$. Which of the following statements is true?

Let V be a vector space over $\mathbb{R}$. Let $T_1,T_2:V\to V$ be two $\mathbb{R}-$linear transformations such that $T_1+T_2$ and $T_1-T_2$ are linearly independent over $\mathbb{R}$. Consider the following statements: (A) The transformations $T_1$ and $T_2$ are linearly independent over $\mathbb{R}.(B)$ There exist $\mathbb{R}-$linear transformations $T_3,T_4:V\to V$ such that {$T_1+T_2,T_1-T_2,T_3,T_4$} is linearly independent over $\mathbb{R}$. Which of the following statements is true?

Let $T:{\mathbb{R}}^4\to {\mathbb{R}}^4$ be an $\mathbb{R}-$linear transformation such that $(T^2+T+I)(T-2I)=0$ and $(T^2+T+2I)(T^2-4I)=0$. Which of the following statements is FALSE?

Let A be a 3 × 3 complex matrix such that $A^3$ is the identity matrix. Which of the following statements is true?

For any matrix P, the transpose of P is denoted by Pᵗ. Consider the real matrix A = [[1, 1, 0], [1, 2, 1], [1, 1, 2]]. Which of the following statements is true?

Consider ${\mathbb{R}}^4$ with the standard inner product. Let V be the subspace of ${\mathbb{R}}^4$ spanned by the vectors (1, 0, 0, 1), (0, 1, 0, 1), and (0, 0, 1, 0). Which of the following is NOT an orthonormal basis of V?

Consider the real matrix A = [[0, 0, 1], [0, 1, 0], [1, 0, 0]]. Define $B:{\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R}$ by B(v, w) = vᵗAw. Which of the following statements is true?

For z = x + iy $\in \mathbb{C}$, let f(z) = u(x, y) + iv(x, y) define an entire function. Consider the function g(z) = u(x, −y) − iv(x, −y), for z = x + iy $\in \mathbb{C}$. Suppose that v(x, 0) = 0 for all $x\in \mathbb{R}$. Define E = {$z\in \mathbb{C}$ | f(z) = g(z)}. Which of the following statements is true?

Consider the half planes R₊ = {z = x + iy $\in \mathbb{C}:x>0$} and R₋ = {z = x + iy $\in \mathbb{C}:x<0$}, and the fractional linear transformations $T_1(z)=(z-1)/(z+1)$ and $T_2(z)=(z+1)/(z-1)$. Let disc 𝔻 = {$z\in \mathbb{C}$ : |z| < 1}. Which of the following statements is true?

Let $\gamma$ be the circle {$z\in \mathbb{C}$ : |z| = 3} oriented counterclockwise. Let f be an entire function. What is the value of A for which ${\int }_{\gamma }(A/(z-1)-f(z)/(z-2{)}^2)$dz = 0 holds?

Consider the entire function $f(z)=z^2(e$ᶻ − e⁻ᶻ) and the meromorphic function $g(z)=z^2/(e$ᶻ − e⁻ᶻ) on $\mathbb{C}$. Which of the following statements is true?

Let p be a prime number. An element a of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})$ˣ is said to be a primitive root in $(\mathbb{Z}/p\mathbb{Z})$ˣ if the order of a in $(\mathbb{Z}/p\mathbb{Z})$ˣ is p − 1. Let S_p be the number of primitive roots in $(\mathbb{Z}/p\mathbb{Z})$ˣ and $\phi$ denote the Euler $\phi -$function. Which of the following statements is true?

Let G be a group of order n > 3 and H be a subgroup with 1 < |H| < n. Consider the set X = ⋃(g∈G) gHg⁻${}^1$. Which of the following statements is true?

Consider the ring homomorphism $\psi :\mathbb{C}[x,y]\to \mathbb{C}[t]$ defined by $\psi (f(x,y))=f(t^2,t^3)$. Which of the following statements is true?

Let $\mathbb{N}$ be the set of positive integers. Consider ${\mathbb{R}}^2$ with the Euclidean topology and the subsets A = {$(n,1/n):n\in \mathbb{N}$} and B = {$(n,1/m):n,m\in \mathbb{N}$}. Which of the following statements is true?

For any non-zero solution y = y(x) of the differential equation $(2x+3{)}^2(d^{2}y/$dx${}^2)+6(2x+3)($dy/dx) + 8y = 0, x > 0, denote S := {$x\in (0,\infty ):y(x)=0$}. Then

The initial value problem dy/dx $=\surd$|x − 1| sin y, y(0) = 1 has

The problem ${\partial }^{2}u/\partial x^2+{\partial }^{2}u/\partial y^2=0$ in {$(x,y)\in {\mathbb{R}}^2:x^2+y^2>1$}, u(x, y) = 1 on {$(x,y)\in {\mathbb{R}}^2:x^2+y^2=1$} has

The problem $\partial u/\partial x+\partial u/\partial y=u^2,u(x,0)=x^2,\forall x\in \mathbb{R}$ has a solution on an open set containing the line {$(x,y)\in {\mathbb{R}}^2$ : ax + by = 0} if

Consider the quadrature formula $\int$₋${}_1^1$ |x|f(x)dx ≈ (1/2)(f(−1) + f(1)). Then the degree of precision (also known as order of exactness) of the quadrature formula is

If $\phi \in C^4[0,1]$ is the extremal of the variational problem minimize $J[y]={\int }_0^1($yy${}^{\prime }+(y^{\prime \prime }{)}^2)$dx, subject to y(0) = 0, y′(0) = 1, y(1) = 2, y′(1) = 4, then $\phi (1/2)$ is equal to

If u is a solution of the integral equation $u(x)=\lambda {\int }_0^{1}K(x,t)u(t)$dt, where K(x, t) := x(1 − t) for 0 ≤ x ≤ t ≤ 1 and t(1 − x) for 0 ≤ t ≤ x ≤ 1, then

Consider a particle of mass $m_1$ moving along a horizontal line L that is perpendicular to a vertical wall. Let x denote the distance of the particle from the wall. Suppose a simple pendulum of length l having mass $m_2,m_2\ne m_1$, is attached to the particle, hanging below L with $\theta$ measured from the downward vertical. If the pendulum oscillates in a plane containing L, then the equations of motion in terms of the generalized coordinates x and $\theta$ are (g denotes the acceleration due to gravity)

Suppose that {$X_n$}$(n\in \mathbb{N})$ is a sequence of independent and identically distributed (i.i.d.) random variables with the common probability density function $f(x)=1/(\pi (1+x^2)),x\in \mathbb{R}$. Then which of the following statements is true?

Let $U_1,U_2,\dots ,U_6$ be 6 urns such that urn U_k contains $3k+k^2$ balls, out of which 3k are white balls and $k^2$ are black balls, k = 1, 2, …, 6. An urn is selected with the probability of selecting urn U_k being proportional to (k + 3). A ball is chosen randomly from the selected urn. Then the probability that urn $U_6$ was selected, given that the ball drawn is white, is equal to

Planes take off in a busy airport in accordance with the Poisson process with rate 60 planes per hour. 10% of these planes are cargo planes and 90% are passenger planes. Given that 10 cargo planes have taken off during one hour, what is the expected total number of planes that have taken off in that hour?

Consider an M/M/3 queuing system with arrival rate $\lambda =2$ and service rate $\mu =3/2$. Define, for $i=1,2,\dots ,I_i=1$ if the first transition is from i to i + 1, and $I_i=0$ if the first transition from i is i − 1. Then, $Var(I_4)$ equals

Suppose X ~ Uniform(5, 10). Define Z = X + 3 if X ≤ 7, and Z = X − 3 otherwise. Then E(Z) is

Let X and Y be independent random variables such that X follows U(0, 1) distribution and Y follows Bernoulli distribution with success probability p ∈ (0, 1). Define Z = X + Y. Let $z_1=0.5,z_2=1.2,z_3=1.3,z_4=0.9,z_5=0.1,z_6=0.7$ be the observed values from the distribution of Z. Then the maximum likelihood estimate of p equals

Suppose that the probability density function of the random variable X is $f(x)=2(\theta -x)/{\theta }^2$ if $0\le x\le \theta$, and 0 otherwise, where $\theta >0$ is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for $\theta$ is

Let $X_1,X_2,\dots ,X_n$ be a random sample from distribution Poisson$(\theta ),\theta >0$. Let $\pi (\theta )=e^{(}-\theta ),\theta >0$, be the prior distribution of $\theta$. Under the squared error loss function, which of the following is the Bayes estimator of $e^{(}-2\theta )$?

Let X be a random variable with the probability density function $f(x)=2\theta x+2(1-\theta )(1-x)$ if 0 < x < 1, and 0 otherwise, where $\theta \in [0,1]$. Based on single observation x, the critical region of the most powerful test for testing null hypothesis $H_0:\theta =1/2$ against alternative hypothesis $H_1:\theta =1$, at level of significance $\alpha =0.25$, is

Consider a multiple linear regression model $Y_i={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_p{x}_{i}p+{\epsilon }_i,1\le i\le n,n>(p+1)$, where errors ${\epsilon }_i^{\prime }s$ are uncorrelated with zero mean and finite variance ${\sigma }^2>0$. Here, $Y_i$ is the i-th response. Let Ŷ${}_i$ be the i-th predicted response by the least squares estimation method, and let $\epsilon$̂${}_i=Y_i-$ Ŷ${}_i,1\le i\le n$. Then, which of the following statements is true?

Let X, Y and Z be independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is

Consider the following design where the columns represent blocks and the letters represent treatments — block 1: A, B; block 2: C, D; block 3: A, C; block 4: B, D; block 5: A, D; block 6: B, C; block 7: A, E; block 8: B, E; block 9: C, E; block 10: D, E. Then, which of the following statements is NOT true?

Let A, B, and C be sets. Which of the following sets is equal to A \ (B \ C)?

What is the number of injective functions from {1, 2, …, 7} to {1, 2, …, 10}?

For integers n ≥ 0, let $f_n:[-1,0]\to \mathbb{R}$ be defined by $f_n(x)=x/(1-x{)}^n$. Which of the following statements is true about the series ${\sum }_{(}n=0{)}^{\infty }{f}_n$?

Consider the sequences $(a_n{)}_n{\ge }_1$ and $(b_n{)}_n{\ge }_1$ defined by $a_n=(e^n+e$⁻${}^n)/2$ and $b_n=a_n$₊${}_1/a_n$. Which of the following statements is true?

Let $f:[0,1]\to \mathbb{R}$ be defined by $f(x)=\sin (x^2)$. Let $A=\lim (n\to \infty )({\sum }_{(}k=1{)}^{n}f(k/n)-n{\int }_0^{1}f(x)$dx). Which of the following statements is true?

Consider the power series ${\sum }_{(}n=1{)}^{\infty }[n^{(}{n}^2)/(n+1{)}^{(}{n}^2)]x^n$ with coefficients in real numbers $\mathbb{R}$. Which of the following statements is true?

Let U denote the span of {eᵗ, $e^2$ᵗ, $e^3$ᵗ} in the real vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Consider the $\mathbb{R}-$vector spaces V = {$f:U\to \mathbb{R}$ | f is an $\mathbb{R}-$linear transformation} and W = {f ∈ V | $f(e^3$ᵗ) = 0}. Which of the following statements is true?

Let $v=(a,b,c)\in {\mathbb{R}}^3$ be a nonzero vector that lies in the orthogonal complement (with respect to the standard inner product) of the row-space of the matrix A = [[2, 2, 7], [3, 1, 4]]. If a, b, c are all integers, then what is the smallest possible value of |a + b + c|?

Let A = [[0, a, 0], [0, 0, b], [c, 0, 0]], where a, b, c are real numbers with abc = 1. If $B=A+A^2+A^3$, then which of the following statements is true?

For a variable x, consider the $\mathbb{R}-$vector space V = {$a_0+a_{1}x+a_2{x}^2$ | $a_1,a_2,a_3\in \mathbb{R}$}. Let T : V → V be the linear transformation defined by T(f) = f + df/dx, where df/dx denotes the derivative of f with respect to x. Which of the following statements is true?

Let V be the $\mathbb{R}-$vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let $T:V\to \mathbb{R}$ be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?

Consider the bilinear form $B:{\mathbb{R}}^4\times {\mathbb{R}}^4\to \mathbb{R}$ defined by $B(x,y)=x_1{y}_3+x_2{y}_4-x_3{y}_1-x_4{y}_2$, where $x=(x_1,x_2,x_3,x_4)$ and $y=(y_1,y_2,y_3,y_4)$ in ${\mathbb{R}}^4$. Let A denote the matrix of B with respect to the standard ordered basis of ${\mathbb{R}}^4$. Which of the following statements is true?

Let $f:\mathbb{C}\to \mathbb{C}$ be the function defined by f(z) = e^((cos(1+i)) sin z). For z = x + iy $\in \mathbb{C}$, write f(z) as u(x, y) + iv(x, y), where u, v are real-valued functions. Which of the following is the value of $(\partial u/\partial x)(0,0)$?

Let 𝔻 = {z = x + iy $\in \mathbb{C}$ : |z| < 1} be the open unit disc and f : 𝔻 $\to \mathbb{C}a$ holomorphic function such that f(0) = 0. Let $\psi (z)=$ |f(z)|${}^2$, and ${\partial }^2\psi /\partial x^2+{\partial }^2\psi /\partial y^2\equiv 0$. Which of the following statements is FALSE?

Let ℍ = {z = x + iy $\in \mathbb{C}$ | y > 0} and f : ℍ $\to \mathbb{C}$ be a non-constant holomorphic function satisfying |f(z)| < 1 for all z ∈ ℍ. Which of the following statements is true?

For integers m, n ≥ 1, let $I_{(}m,n)=(1/2\pi i){\int }_C{z}^m{\bar{z}}^n$ dz, where C is the circle {$z\in \mathbb{C}$ : |z| = 1} oriented counterclockwise. Which of the following statements is true?

For integers n > 1, let G(n) denote the number of groups of order n, up to isomorphism, i.e. G(n) is the number of isomorphism classes of groups of order n. Which of the following statements is true?

We say that a group G has property (A) if every non-trivial homomorphism from G to any group is injective. Which of the following groups has property (A)?

Let $\mathbb{C}[x,y]$ be the polynomial ring in two variables over $\mathbb{C}$. For which of the following ideals I, the quotient ring $\mathbb{C}[x,y]/I$ is NOT an integral domain?

Which of the following statements is true?

Suppose that the differential equation $d^{2}y/$dx${}^2+P(x)$dy/dx $+e^{(}2x)y=0,x\in \mathbb{R}$ transforms into a second order differential equation with constant coefficients under the change of independent variable given by s = s(x) satisfying (ds/dx)(0) = 1. Then which of the following statements is true?

Given that $y_1(x)=e^{(}2x)$ is a solution of the ordinary differential equation (ODE$)x{d}^{2}y/$dx${}^2-(3+4x)$dy/dx + (4x + 6)y = 0, x > 0. Let $y_2=y_2(x)$ be the solution of the ODE satisfying the conditions $y_2(1)=e^2/4$ and (dy${}_2/$dx$)(1)=3e^2/2$. Then which of the following statements is true?

Let u = u(x, y) be the solution of the Cauchy problem $x\partial u/\partial x+y\partial u/\partial y=u,(x,y)\ne (0,0)$, with $u(x,1)=\sqrt{1+x^2},x\in \mathbb{R}$. Then which of the following statements is true?

Let u = u(x, t) be a solution of the wave equation ${\partial }^{2}u/\partial t^2-{\partial }^{2}u/\partial x^2=0,x\in \mathbb{R},t>0$ satisfying the condition u(0, t) = 0, ∀t ≥ 0. Then which of the following statements is true?

Let $f:\mathbb{R}\to \mathbb{R}$ be such that sup(x≠y) |f(x) − f(y)|/|x − y| = L, where $1<L<\infty$. Let $h:\mathbb{R}\to \mathbb{R}$ be a differentiable function satisfying |h′(x)| ≤ 3/4 for all $x\in \mathbb{R}$. For $\alpha >0$, define $g(x)=\alpha f(x)+h(x)$ for $x\in \mathbb{R}$. Consider the sequence {$x_k$}(k≥0) defined by $x_{(}k+1)=g(x_k),k=0,1,\dots$, where $x_0\in \mathbb{R}$. The sequence {$x_k$}(k≥0) converges to the solution of the equation x = g(x) if

Let S denote the set of all solutions of the Euler-Lagrange equation of the variational problem: minimize $J[y]={\int }_0^1(y^2+(y^{\prime }{)}^2)$dx, subject to $y(0)=0,y(1)=0,{\int }_0^1{y}^2$dx = 1. Then the set {$\phi (1/2):\phi \in S$} is equal to

Let $\lambda \in \mathbb{R}$, and $K:[0,1]\times [0,1]\to \mathbb{R}$ be a function such that every solution of the boundary value problem $(d^{2}u/$dx${}^2)(x)+\lambda u(x)=0;($du/dx)(0) = u(0), (du/dx)(1) = 0 satisfies the integral equation $u(x)+\lambda {\int }_0^{1}K(x,t)u(t)$dt = 0. Then

Two blocks of equal mass m are connected by a flexible inelastic cord of mass M. One block is placed on a smooth horizontal table, the other block hangs over the edge. The total potential energy of the entire cord is given by (−Mg$/2l)x^2$, where x is the distance of the hanging block from the edge of the table, l is the length of the cord, and g is the gravitational acceleration. Then

$\lim (n\to \infty ){\int }_0^1{\int }_0^1\cdots {\int }_0^1(x_1^2+\cdots +x_n^2)/(x_1+\cdots +x_n)$ dx${}_1\dots$ dx${}_n$ equals

Let $U_1,U_2,\dots ,U_5$ be 5 urns such that urn U_k contains $2k+k^2$ balls, out of which 2k are white balls and $k^2$ are black balls, k = 1, 2, …, 5. An urn is selected with probability of selecting urn U_k being proportional to (k + 2). A ball is chosen randomly from the selected urn. Then, the probability that the urn $U_5$ was selected, given that the ball drawn is white, is equal to

In an examination question paper, all questions are 'True' or 'False' type. These are arranged in such a way that three-fourth of times a question with answer 'True' is followed by a question with answer 'True'. Also two-third of times a question with answer 'False' is followed by a question with answer 'False'. If the question paper has 100 questions, the approximate probability that the correct answer of the 100-th question is 'True', is

Consider an M/G/1 queuing system with arrival rate $\lambda =1$ and independent and identically distributed successive service times having probability density function g(x) = xe⁻ˣ if x > 0, and 0 otherwise. Define, for $i=1,2,\dots ,I_i=1$ if the first transition is from i to i − 1, and $I_i=0$ if the first transition from i is i + 1. Then, $Var(I_2)$ equals

Suppose that X ~ binomial$(9,\theta ),0.7<\theta <1$, and Y ~ Poisson$(\lambda ),\lambda >0$. If 3E(Y) = E(X), then which of the following is true?

Let $X_1,X_2,\dots ,X_n(n\ge 2)$ be a random sample from a gamma distribution with shape parameter $\alpha >0$ and scale parameter $\beta =1$. For a suitable constant C, the rejection region of the most powerful test for testing $H_0:\alpha =1$ against $H_1:\alpha =2$ is of the form

Suppose $\phi$ is a most powerful test of size 0.05 for testing a simple null hypothesis $H_0$ against a simple alternative hypothesis $H_1$. If the power of the test is 0.4, then which of the following is true?

Let $X_1,X_2,\dots ,X_n(n\ge 2)$ be a random sample from Uniform$(\theta ,1-\theta )$ distribution, where $-\infty <\theta <1/2$. The maximum likelihood estimator of $\theta$ is

Let $X_1,X_2,\dots ,X_5$ be a random sample of size 5 from an absolutely continuous distribution having median M. Let S denote the number of $X_i^{\prime }s$ greater than 0. For testing $H_0:M=0$ against $H_1:M>0$, let $\phi (X)=1$ if $S>c,\nu$ if S = c, and 0 if S < c be a test of size $\alpha =0.05$, where $\nu \in [0,1]$ and c ∈ {−1, 0, …, 5} are fixed constants. Then $c+\nu$ equals

Consider a multiple linear regression model $Y_i={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_p{x}_{i}p+{\epsilon }_i,1\le i\le n,n>(p+1)$, where the errors ${\epsilon }_i^{\prime }s$ are uncorrelated with zero mean and finite variance ${\sigma }^2>0$. Here, $Y_i$ is the i-th response. Let Ŷ${}_i$ be the i-th predicted response by the least squares estimation method, and let $\epsilon$̂${}_i=Y_i-$ Ŷ${}_i,1\le i\le n$. Then, which of the following statements is true?