CSIR NET Maths previous year question papers

A Venn diagram shows the numbers of players playing one or more than one sport among Football, Cricket and Tennis: 18 play Football only, 24 play Cricket only, 20 play Tennis only, 7 play Football and Cricket but not Tennis, 11 play Football and Tennis but not Cricket, 8 play Cricket and Tennis but not Football, and 4 play all three sports. The percentage of players who play exactly two sports is closest to ____%.

A lady bought some apples, each costing Rs. 25, and some bananas each costing Rs. 6, for a total of Rs. 378. In how many ways could she have chosen the numbers of apples and bananas?

Let p, q be non-negative integers. Consider the following statements: (A) There is an integer k ≥ 1 such that p + k = q. (B) There is an integer k ≥ 1 such that q + k = p. Which of the following statements is true?

Let 𝔽$_{5}$ denote the field with 5 elements. How many 2 × 2 matrices with entries in 𝔽$_{5}$ have rank one?

Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map. For R > 0, let $\gamma_R : [0, 1] \to \mathbb{C}$ be the map t ↦ $Re^{2\pi{}it}$. Suppose that there exists $c \in \mathbb{R}$ such that $\int_{0}^{1}$ |(f ∘ $\gamma_R)(t) \gamma'_R(t)$| dt → c as $R \to \infty$. Which of the following statements is FALSE?

Let f be an entire function such that $f(\mathbb{C}) \subset$ {x + iy | y = x + 1}. Which of the following statements is true?

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

Suppose that we have a data set consisting of 2n + 1 observations for some $n \in \mathbb{N}$. Value of each observation is either x or x + r, where $x \in \mathbb{N}, r \ge 0$. Then, which of the following statements is always true?

A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is

Let $\alpha, \beta \in (0, \infty)$. Consider the infinite series $\sum_{n\ge4}$ $1/[n(log_{e} n)^\alpha (log_{e}(log_{e} n))^\beta]$. Which of the following statements is true?

Let {$Y_{n} : n \ge 1$} be a sequence of independent and identically distributed random variables, where $Y_{1}$ ~ Bernoulli(1/2). Define $Z = \sum_{n=1}^\infty$ $4Y_{n}/5^{n}$. Then, which of the following statements is true?

Let B(v, w) be a nondegenerate symmetric bilinear form on $\mathbb{R}^{2}$ and let q(v) = B(v, v) be the corresponding quadratic form. Suppose there exist vectors $v, w \in \mathbb{R}^{2}$ such that B(v, v) = 0 and B(v, w) ≠ 0. Which of the following statements are necessarily true?

For $a, b, c \in \mathbb{R}$, consider the variational problem: Minimize $J[y] = \int_{0}^{2} [a(y')^{2} + 2$byy′ + cy$^{2}]$ dx, subject to y(0) = 10, y(1) = 100. Then which of the following statements are true?

Let X be a random variable with the cumulative distribution function F_X(x) = 0 if x < 0, F_X(x) = (x + 2)/5 if 0 ≤ x < 2, and F_X(x) = 1 if x ≥ 2. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed random variables with the probability density function f(x|$\theta) =$ $e^{-(x-\theta)}$ if $x \ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is the unknown parameter. Further, let $Y = \min(X_{1}, X_{2}, \dots, X_{n})$. Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1 - \alpha)$, where $\alpha \in (0, 1)$?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{10}, Y_{10})$ be a random sample from a bivariate normal distribution BVN$(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho)$ with $\mu_{1} = 5, \mu_{2} = 6, \sigma_{1}^{2} = 4, \sigma_{2}^{2} = 9$ and $\rho = 1/2$. Then, which of the following statements are true?

Let the random vector $(X_{1}, X_{2}, X_{3})^{T}$ have the positive definite dispersion matrix with rows $(1, \rho, \rho), (\rho, 1, \rho)$ and $(\rho, \rho, 1)$. Then, which of the following statements are true?

Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$. Suppose that f is a non-negative Lebesgue measurable function on $\mathbb{R}$. Let $0 = a_{0} < a_{1} < a_{2} < \cdots$ be an unbounded sequence such that $a_{n+1}$ ≤ ca$_{n}$ for some real number c and for all n ≥ 1. Let A_k = {$x \in \mathbb{R}$ | a_k ≤ f(x) < $a_{k+1}$} for each k ≥ 0. Which of the following statements are true?

Let C[0, 1] be the $\mathbb{R}-$vector space of real valued continuous functions equipped with the norm ‖f‖ = $sup_{x\in[0,1]}$ |f(x)|. Let T : C[0, 1] → C[0, 1] be defined as $T(f)(x) = \int_{0}$ˣ f(t)dt, for x ∈ [0, 1]. Let $T^{n} = T$ ∘ T ∘ ⋯ ∘ T (n times). Which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from Uniform$[-3\theta/2, \theta/2]$ distribution, where $\theta>0$ is an unknown parameter. Let ${\bar{X}}, X_{(1)}$ and $X_{(n)}$ denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?

For n≥3, consider the space $M_{n}(\mathbb{C})$ of n×n complex matrices endowed with the inner product ⟨A,B⟩ = Trace(A*B). For 0≤k≤n, let $W_{k} =$ {$A \in M_{n}(\mathbb{C})$ : ⟨A,B⟩=0 for all $B \in M_{n}(\mathbb{C})$ with rank k}. Which of the following statements are necessarily true?

There are six persons. On each day of the year 2024 at least one of them borrowed money from another. A person P is labeled troublesome in a calendar month M of 2024 if P borrowed from the same person at least twice during month M. Which of the following statements are necessarily true?

Let $f : (0,\infty) \to (0,\infty)$ be the function defined by $f(x) = \int_{0}$ˣ $\sqrt{t}/(1+t^{2})$ dt, where $\sqrt{t}$ denotes the positive square root for t>0. Which of the following statements are true?

Consider the following real-valued functions $F_{1}$ and $F_{2}$ defined on $\mathbb{R}$, given by $F_{1}(x)=1$ if x≥1, 0 otherwise, and $F_{2}(x)=\int_{0}$ˣexp(−t)dt if x≥0, 0 otherwise. Define another function $F:\mathbb{R}\to[0,1]$ by $F(x)=(2/3)F_{1}(x)+(1/3)F_{2}(x)$ for all $x\in\mathbb{R}$. Which of the following statements are true?

Let V be a finite-dimensional $\mathbb{R}-$vector space and T:V→V a linear operator such that $T^{2}$ is diagonalizable over $\mathbb{R}$. Which of the following statements are necessarily true?