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

Consider the set A = {$x\in \mathbb{Q}:0<(\sqrt{2}-1)x<\sqrt{2}+1$} as a subset of $\mathbb{R}$. Which of the following statements is true?

Let S = {$x\in \mathbb{R}:x>1$ and $(1-x^4)/(1-x^3)>22$}. Which of the following is true about S?

Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true?

Let $(a_n{)}_n\ge 1$ be a bounded sequence in $\mathbb{R}$. Which of the following statements is FALSE?

What is the cardinality of the set of real solutions of eˣ + x = 1?

Let S be a dense subset of $\mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}a$ given function. Define $g:S\to \mathbb{R}$ by g(x) = f(x). Which of the following statements is necessarily true?

For each n ≥ 1 define $f_n:\mathbb{R}\to \mathbb{R}$ by $f_n(x)=x^2/\sqrt{x^2+1/n},x\in \mathbb{R}$, where $\surd$ denotes the non-negative square root. Wherever $li{m}_n\to \infty f_n(x)$ exists, denote it by f(x). Which of the following statements is true?

Let $A:{\mathbb{R}}^m\to {\mathbb{R}}^n$ be a non-zero linear transformation. Which of the following statements is true?

Let A be a 10 × 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true?

Let ((2, a), (b, c)) be a 2 × 2 real matrix for which 6 is an eigenvalue. Which of the following statements is necessarily true?

Let V be the real vector space of 2 × 2 matrices with entries in $\mathbb{R}$. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where A = ((2, 0), (0, 1)). What is the characteristic polynomial of T?

Let A = ((0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 1, 1), (0, 0, 1, 1)), and consider the symmetric bilinear form on ${\mathbb{R}}^4$ given by ⟨v, w⟩ = vᵗAw, for $v,w\in {\mathbb{R}}^4$. Which of the following statements is true?

For a quadratic form $f(x,y,z)\in \mathbb{R}[x,y,z]$, we say that $(a,b,c)\in {\mathbb{R}}^3$ is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)?

Let f be an entire function. Which of the following statements is FALSE?

Consider the contour $\gamma$ given by $\gamma (\theta )=e^{(}2i\theta )$ for $\theta \in [0,\pi /2];\gamma (\theta )=1+2e^{(}2i\theta )$ for $\theta \in [\pi /2,3\pi /2];\gamma (\theta )=e^{(}2i\theta )$ for $\theta \in [3\pi /2,2\pi ]$. Then what is the value of ${\int }_{\gamma }$ dz / (z(z − 2))?

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define ${\gamma }_r(t)=$ re^(it) (where $t\in [0,2\pi ])$ and $I_r=(1/2\pi i){\int }_{(}{\gamma }_r)(z^2+1)/((z-a)(z-b))$ dz. Which of the following statements is necessarily true?

For a complex number a such that 0 < |a| < 1, which of the following statements is true?

How many arrangements of the digits of the number 1234567 are there, such that exactly three of them occur in their original position? (E.g., in the arrangement 5214763, exactly the digits 2, 4 and 6 are in their original positions. In the arrangement 1243576, exactly the digits 1, 2 and 5 are in their original positions.)

The number of group homomorphisms from $\mathbb{Z}/150\mathbb{Z}$ to $\mathbb{Z}/90\mathbb{Z}$ is

Consider the ring R = { ${\sum }_{(}n\in \mathbb{Z})a_n{X}^n$ | $a_n\in \mathbb{Z}$, and $a_n\ne 0$ only for finitely many $n\in \mathbb{Z}$ }, with the usual addition and multiplication of such sums. Which of the following statements is true?

Consider the initial value problem (IVP$)y^{\prime }(x)=\surd$|$y(x)+\epsilon$|, $x\in \mathbb{R},y(0)=y_0$. Consider the following statements: S1: There is an $\epsilon >0$ such that for all $y_0\in \mathbb{R}$, the IVP has more than one solution. S2: There is $a{y}_0\in \mathbb{R}$ such that for all $\epsilon >0$, the IVP has more than one solution. Then

Let $\phi$ denote the solution to the boundary value problem (BVP) (xy′)′ − 2y′ + y/x = 1 for $1<x<e^4$, with y(1) = 0 and $y(e^4)=4e^4$. Then the value of $\phi (e)$ is

Let u = u(x, t) be the solution of the following initial value problem: $u_t+2024u_x=0$ for $x\in \mathbb{R},t>0$, and $u(x,0)=u_0(x)$ for $x\in \mathbb{R}$, where $u_0:\mathbb{R}\to \mathbb{R}$ is an arbitrary $C^1$ function. Consider the following statements: $S_1$: If $A_t:=$ {$x\in \mathbb{R}:u(x,t)<1$} and |$A_t$| denotes the Lebesgue measure of $A_t$ for every t ≥ 0, then |$A_t$| = |$A_0$|, $\forall t>0.S_2$: If $u_0$ is Lebesgue integrable, then for every t > 0, the function x ↦ u(x, t) is Lebesgue integrable. Then

If u = u(x, t) is the solution of the initial value problem $u_t=u_{xx}$ for $x\in \mathbb{R},t>0$, with u(x, 0) = sin(4x) + x + 1 for $x\in \mathbb{R}$, satisfying |u(x, t)| $<3e^{(}{x}^2)$ for all $x\in \mathbb{R}$ and t > 0, then

If the value of the approximate solution of the initial value problem $y^{\prime }(x)=x(y(x)+1),x\in \mathbb{R},y(0)=\beta$ at x = 0.2 using the forward Euler method with step size 0.1 is 1.02, then the value of $\beta$ is

Let B(0,1) = {$(x,y)\in {\mathbb{R}}^2$ | $x^2+y^2<1$} be the open unit disc in ${\mathbb{R}}^2,\partial B(0,1)$ denote the boundary of B(0,1), and $\nu$ denote the unit outward normal to $\partial B(0,1)$. Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a given continuous function. The Euler–Lagrange equation of the minimization problem min { ½∬_B(0,1) |$\nabla u$|${}^2$ dxdy + ½∬${}_B(0,1)e^{(}{u}^2)$ dxdy $+{\int }_{\partial }B(0,1)fu$ ds } subject to $u\in C^1($closure of B(0,1)) is

Let u be the solution of the Volterra integral equation ${\int }_0$ᵗ [½ $+\sin (t-\tau )]u(\tau )d\tau =\sin t$. Then the value of u(1) is

Consider a solid circular cylinder of radius 2 meters and height 3 meters of uniform density. If the density of the cylinder is $\rho$ kg/meter${}^3$, then the moment of inertia (in kg meter${}^2)$ of the cylinder about a diameter of its base is

Let $A_1,A_2,A_3$ be events satisfying $0<P(A_i)<1$ for i = 1, 2, 3. Which of the following statements is true?

Let X be a random variable with cumulative distribution function given by F(x) = 0 if x < 0; F(x) = (x + 1)/3 if 0 ≤ x < 1; F(x) = 1 if x ≥ 1. Then the value of P(1/3 < X < 3/4) + P(X = 0) is equal to

Let {$X_n$ | n ≥ 0} be a homogeneous Markov chain with state space S = {0, 1, 2, 3, 4} and transition probability matrix

Let $\alpha$ denote the probability that starting with state 4 the chain will eventually get absorbed in closed class {0, 3}. Then the value of $\alpha$ is

Consider a petrol pump which has a single petrol dispensing unit. Customers arrive there in accordance with a Poisson process having rate $\lambda =1$ minutes. An arriving customer enters the petrol pump only if there are two or less customers in the petrol pump, otherwise he/she leaves the petrol pump without taking the petrol (at any point of time a maximum of three customers are present in the petrol pump). Successive service times of the petrol dispensing unit are independent exponential random variables having mean 1/2 minutes. Let X denote the average number of customers in the petrol pump in the long run. Then E(X) is equal to

Let a point P be chosen at random on the line segment AB of length $\alpha$. Let $Z_1$ and $Z_2$ denote the lengths of line segments AP and BP respectively. Then the value of E(|$Z_1-Z_2$|) is

Consider a distribution with probability mass function f(x | $\theta )=(1-\theta )/2$ if x = 0; 1/2 if $x=1;\theta /2$ if x = 2; and 0 otherwise, where $\theta \in (0,1)$ is an unknown parameter. In a random sample of size 100 from the above distribution, the observed counts of 0, 1 and 2 are 20, 30 and 50 respectively. Then, the maximum likelihood estimate of $\theta$ based on the observed data is

Let $X_1,\dots ,X_{10}$ be a random sample from a distribution with the probability density function f(x|$\theta )=\theta x^{(}\theta -1)$ if 0 < x < 1, and 0 otherwise, where $\theta >0$ is an unknown parameter. The prior distribution of $\theta$ is given by $\pi (\theta )=\theta e^{(}-\theta )$ if $\theta >0$, and 0 otherwise. The Bayes estimator of $\theta$ under squared error loss is

Let $X_1,X_2$ be a random sample from $N(0,{\sigma }^2)$ distribution, where $\sigma >0$ and $N(\mu ,{\sigma }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\sigma }^2$. Suppose, for some constant $c,(c(X_1^2+X_2^2),\infty )$ is a confidence interval for variance ${\sigma }^2$ with confidence coefficient 0.95. Then the value of c is equal to

Let $X_1,X_2$ be a random sample from a population having probability density function f ∈ {$f_0,f_1$} where $f_0(x)=1/2$ if 0 ≤ x ≤ 2 and 0 otherwise, and $f_1(x)=1/4$ if 0 ≤ x ≤ 4 and 0 otherwise. For testing the null hypothesis $H_0:f=f_0$ against the alternate hypothesis $H_1:f=f_1$, the power of a most powerful test of size $\alpha =0.05$ is equal to

An analyst considers standardized values of observations on three variables, consumption (C), saving (S) and total income (TI) so that they have zero means and unit variances. She further considers disposable income (DI) where DI = C + S. In the simple linear regressions of DI on TI, DI on C and S on TI, the regression coefficients are 0.8, 0.5 and 0.4, respectively. There are 21 sample observations. Sample covariances and variances are calculated with divisor 20. Then, the value of sum of squared residuals in the regression of DI on S is

Let $X_0,X_1,\dots ,X_p(p\ge 2)$ be independent and identically distributed random variables with mean 0 and variance 1. Suppose $Y_i=X_0+X_i,i=1,\dots ,p$. The first principal component based on the covariance matrix of $Y=(Y_1,\dots ,Y_p{)}^T$ is

The expected number of distinct units in a simple random sample of 3 units drawn with replacement from a population of 100 units is

Consider the linear programming problem: max{$x_1+x_2+x_3$} subject to constraints $x_1+x_2-x_3\le 1,x_1+x_3\le 2,0\le x_1\le 1/2,x_2\ge 0$, and $0\le x_3\le 1$. Which of the following statements are true?

Let $(a_n{)}_n{\ge }_1$ be a sequence of positive real numbers. Let $b_n=a_n/\max${$a_1,\dots ,a_n$}, n ≥ 1. Which of the following statements are necessarily true?

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

Let $K\subseteq \mathbb{R}$ be non-empty and f : K → K be continuous such that |x − y| ≤ |f(x) − f(y)| for all x, y ∈ K. Which of the following statements are true?

Let $f:[0,1)\to [1,\infty )$ be defined by f(x) = 1/(1 − x). For n ≥ 1, let $p_n(x)=1+x+\cdots +x^n$. Then which of the following statements are true?

Let V be the subspace spanned by the vectors $v_1=(1,0,2,3,1),v_2=(0,0,1,3,5),v_3=(0,0,0,0,1)$ in the real vector space ${\mathbb{R}}^5$. Which of the following vectors are in V?

Consider $\mathbb{R}$ and $\mathbb{Q}[x]$ as vector spaces over $\mathbb{Q}$. Which of the following statements are true?

Let $T:{\mathbb{R}}^4\to {\mathbb{R}}^4$ be a linear map with four distinct eigenvalues and satisfying $T^4-15T^2+10T+24I=0$. Which of the following statements are necessarily true?

Let A be a 4 × 4 real matrix whose minimal polynomial is $x^2+x+1$ and let $B=A+I_4$. Which of the following statements are necessarily true?

Let $M_5(\mathbb{C})$ be the complex vector space of 5 × 5 matrices with entries in $\mathbb{C}$. Let V be a non-zero subspace of $M_5(\mathbb{C})$ such that every non-zero A ∈ V is invertible. Which among the following are possible values for the dimension of V?

Let V (≠ {0}) be a finite dimensional vector space over $\mathbb{R}$ and T : V → V be a linear operator. Suppose that the kernel of T equals the image of T. Which of the following statements are necessarily true?

Let $q_1(x_1,x_2)$ and $q_2(y_1,y_2)$ be real quadratic forms such that there exist $(u_1,u_2),(v_1,v_2)\in {\mathbb{R}}^2$ such that $q_1(u_1,u_2)=1=q_2(v_1,v_2)$. Define $q(x_1,x_2,y_1,y_2)=q_1(x_1,x_2)-q_2(y_1,y_2)$. Which of the following statements are necessarily true?

Suppose that f is an entire function such that |f(z)| ≥ 2024 for all $z\in \mathbb{C}$. Which of the following statements are necessarily true?

Let f be an entire function such that for every integer k ≥ 1 there is an infinite set X_k such that f(z) = 1/k for all z ∈ X_k. Which of the following statements are necessarily true?

Which of the following numbers are order of some element of the symmetric group $S_5$?

Let R and S be non-zero commutative rings with multiplicative identities 1_R, 1_S, respectively. Let f : R → S be a ring homomorphism with f(1_R) = 1_S. Which of the following statements are true?

Let I be an ideal of the ring 𝔽${}_2[t]/(t^2(1-t{)}^2)$. Which of the following are the possible values for the cardinality of I?

If $x_1=x_1(t),x_2=x_2(t)$ is the solution of the initial value problem e^(−t) dx${}_1/$dt $=-x_1+x_2,e^{(}-t)$ dx${}_2/$dt $=-x_1-x_2,x_1(0)=1,x_2(0)=0$, and $r(t)=\sqrt{x_1^2(t)+x_2^2(t)}$, then which of the following statements are true?

Consider the initial boundary value problem (IBVP$)u_t+u_x=2u$ for x > 0, t > 0; u(0, t) = 1 + sin t for t > 0; u(x, 0) = eˣ cos x for x > 0. If u is the solution of the IBVP, then the value of $u(2\pi ,\pi )/u(\pi ,2\pi )$ is

Let B(0,2) = {$(x,y)\in {\mathbb{R}}^2:x^2+y^2<4$}, and $\partial B$ denote the boundary of B(0,2). Assume $(\alpha ,\beta )\ne (0,0),k\in \mathbb{R}$, and u is any solution to $-\Delta u=0$ in $B(0,2),\alpha u(x,y)+\beta (\partial u/\partial \nu )(x,y)=1+(x^2+y^2)k$ on $\partial B$, where $\nu (x,y)$ is the unit outward normal to B(0,2) at $(x,y)\in \partial B$. Consider the following statements: $S_1$: If $\beta =0$, then there exists $a(x_0,y_0)\in B(0,2)$ such that |$u(x_0,y_0)$| = |1 + 4k|/|$\alpha$|. $S_2$: If $\alpha =0$, then k = −1/4. Then

The infimum of the set {${\int }_a$ᵇ $\sqrt{1+(y^{\prime }(t){)}^2}$dt : $y\in C^1[a,b],y(a)=a^2,y(b)=b-5$} is

For $c\in \mathbb{R}$, consider the following Fredholm integral equation y(x) = 1 + x + cx${}^2+2{\int }_0^1(1-3$xt)y(t)dt. Then the values of c for which the integral equation admits a solution are

Consider a solid torus of constant density $\rho$, formed by revolving the disc $(y-b{)}^2+z^2\le a^2,x=0$ about the z-axis, where 0 < a < b. Then the moment of inertia of the solid torus about the z-axis is

Let X and Y be independent random variables with X ~ N(2, 4) and Y ~ N(−4, 9) where $N(\mu ,{\sigma }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\sigma }^2$. Given $\Phi (1)=0.8413,\Phi (2)=0.9772$ and $\Phi (3)=0.9987$ where $\Phi (\cdot )$ is the cumulative distribution function of a standard normal random variable. Which of the following statements are true?

Transition probability matrix of a homogeneous Markov chain with states 0, 1, 2, 3 is P = [[1/4, 3/4, 0, 0], [1, 0, 0, 0], [2/3, 0, 1/3, 0], [0, 0, 2/5, 3/5]], where rows and columns are indexed 0, 1, 2, 3. Which of the following statements are true?

Let X and Y be jointly distributed continuous random variables with joint probability density function f(x, y) = x/y if 0 < x < y < 2, and 0 otherwise. Which of the following statements are true?

Let $X_1,X_2$ denote lifetimes (in years) of 2 components of an electronic system. Let $Y_1=X_1+X_2,Y_2=\max${$X_1,X_2$} and $Y_3=\min${$X_1,X_2$}. Assume that $X_1$ and $X_2$ are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?

Let $X_1,X_2,X_3$ be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, then which of the following statements are true?

Let $X_1,\dots ,X_n(n\ge 2)$ be a random sample from $aU(-\theta ,2\theta )$ distribution, where $\theta >0$ is an unknown parameter. Let $\bar{X}=(1/n){\sum }_i$₌${}_1^n{X}_i,X_{(1)}=\min${$X_1,\dots ,X_n$} and $X_{(n)}=\max${$X_1,\dots ,X_n$}. Which of the following statements are true?

Let $Y_1,\dots ,Y_n(n\ge 2)$ be independent observations; $Y_i$ ~ $N(\beta x_i,{\sigma }^2),i=1,\dots ,n$; where $x_1,\dots ,x_n$ and ${\sigma }^2(>0)$ are known constants and $\beta \in \mathbb{R}$ is an unknown parameter. Consider $N({\beta }_0,{\tau }^2)$ prior for the parameter $\beta$, where ${\beta }_0$ and ${\tau }^2(>0)$ are known constants, and $N(\mu ,{\lambda }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\lambda }^2$. Suppose ȳ $=(1/n){\sum }_i$₌${}_1^n{y}_i$ and $\bar{x}=(1/n){\sum }_i$₌${}_1^n{x}_i$ are observed sample means. Under squared error loss function, which of the following statements are true?

Let $X_1,\dots ,X_{12}$ be a random sample from the N(2, 4) distribution and $Y_1,\dots ,Y_{15}$ be a random sample from the N(−2, 5) distribution, where $N(\mu ,{\sigma }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\sigma }^2$. Assume that the two random samples are mutually independent. Let $\bar{X}=(1/12){\sum }_i$₌${}_1^{12}{X}_i,S_1^2=(1/11){\sum }_i$₌${}_1^{12}(X_i-\bar{X}{)}^2$, Ȳ $=(1/15){\sum }_j$₌${}_1^{15}{Y}_j,S_2^2=(1/14){\sum }_j$₌${}_1^{15}(Y_j-$ Ȳ${)}^2$. Which of the following statements are true?

In a standard linear regression model, let $R^2$ and ${\bar{R}}^2$, respectively, denote the coefficient of determination and adjusted coefficient of determination. Which of the following statements are true?

Consider the two-way ANOVA model $Y_{ij}=\mu +{\alpha }_i+{\beta }_j+{\epsilon }_{ij},i=1,2;j=1,2$, where $\mu$ is the overall mean effect, ${\alpha }_i$ is the effect of the i-th level of factor $A,{\beta }_j$ is the effect of j-th level of factor $B,Y_{ij}$ is the response of the (i, j)-th experimental unit and ${\epsilon }_{ij}$ is the corresponding error with $E({\epsilon }_{ij})=0$ for i = 1, 2; j = 1, 2. Which of the following are estimable linear parametric functions?

Let $X=(X_1,X_2{)}^T$ be a bivariate random vector with covariance matrix $\sum =[[1,\sqrt{2}],[\sqrt{2},2]]$. Which of the following statements are true?

Let ${\sum }_n$₌${}_1^{\infty }{a}_n$ be a convergent series of real numbers. For n ≥ 1 define $A_n=a_n$ if $a_n>0$ and 0 otherwise; $B_n=a_n$ if $a_n<0$ and 0 otherwise. Which of the following statements are necessarily true?

Define $f:\mathbb{R}\to \mathbb{R}$ by f(x) = x|x|. Which of the following statements are true?