CSIR NET Maths previous year question papers

The following plot shows temperature as a function of x and y, brightest at the far corner (x, y) = (5, 5) and darkest at the origin, with the colour banding curving between them rather than running in straight stripes. Along which path is the temperature change minimum?

The minimum height of a plane vertical mirror that will allow a 6-feet tall person to see himself fully in it

Three periodic events repeat every 24 seconds, 54 seconds, and 56 seconds. If they coincide at 10:20:00, when will they next coincide?

The numbers (in millions) of coconut, guava, mango and apple trees in a region in 2010 and 2020 are shown in the following figure — coconut 15 → 12, guava 25 → 22, mango 20 → 18, apple 30 → 35. The maximum relative change in numbers was for

Some, but not all, faces of a six-faced cubical fair die are painted red (R) and the remaining green (G); and the die is thrown until red faces come up on top 4 times. Consider the following sequences of colours listed left to right as they appear on the top. A: GRRRR. B: GRGRRR. Which one of the following is true?

From a fully-filled vessel of 12-litre capacity, to bring 6 litres of water into an empty vessel of 8-litre capacity, what is the minimum number of times water must be poured, given that an empty 5-litre vessel is also available for use?

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?

Suppose $a_{1}, a_{2}, \dots, a_{300}$ are integers such that $a_{i}$₋$_{1} + a_{i} + a_{i}$₊$_{1} = 2025$ for all i = 2, 3, …, 299. If $a_{7} = -5, a_{9} = 37$, then the value of $a_{106}$ is

How many 5-digit numbers can be formed from the digits 0, 2, 3, 4, 6, 7 and 9, using each at most once, which are divisible by 5?

A recent survey suggests that the total fertility rate in a country has fallen below 2.1, the population replacement ratio. This necessarily implies that the

In a class, 40% and 20% students passed in Mathematics and Physics, respectively, and 10% students passed in both subjects. What is the probability of a randomly selected student to have passed in Physics if the student already passed in Mathematics?

The value of $1 + (1/2^{1} + 1/3) + (1/2^{2} + 1/5 + 1/6 + 1/7) + \dots + (1/2^{9} + \cdots + 1/1023)$ lies between

In a community, some artists are teachers, no teacher is a painter, all painters are artists, and all teachers are professionals. Then it can be definitely asserted that

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 A = {$p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}$ for some $n \in \mathbb{N} \cup$ {0}, gcd(p, q) = 1}, B = {$p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}5^{m}$ for some $n, m \in \mathbb{N} \cup$ {0}, gcd(p, q) = 1} and C = {p/q ∈ (0, 1) : p/q has terminating decimal expansion} be subsets of (0, 1). Which of the following statements is true?

Let A, B be non-empty subsets of $\mathbb{N}$ with cardinality |A| ≥ 2. Let $S_{1} =$ {f : A → B | f is one-to-one} and $S_{2} =$ {g : B → A | g is onto}. Which of the following statements is true?

Let $f : \mathbb{R}$ \ $\mathbb{Q} \to \mathbb{R}$ \ $\mathbb{Q}$ be the function defined as f(x) = (3x + 2)/(4x + 3). Let $x_{1} \in \mathbb{R}$ \ $\mathbb{Q}$. For n ≥ 1, define $x_{n}$₊$_{1} = f(x_{n})$. Suppose that the sequence $(x_{n})_{n}\ge_{1}$ converges to a real number $\ell$. Which of the following statements is true?

Let $f(x) = x log_{e}(1 + 1/x)$ for $x \in (0, \infty)$. Which of the following statements is true?

For each n ≥ 1, let $f_{n} : [0, 1] \to \mathbb{R}$ be defined as $f_{n}(x) =$ nx if $x \in [0, 1/n], f_{n}(x) = 2 -$ nx if x ∈ (1/n, 2/n), and $f_{n}(x) = 0$ if x ∈ [2/n, 1]. Which of the following statements is true?

Which of the following polynomials is the characteristic polynomial of a real 2 × 2 matrix A such that trace(A) = 7 and trace$(A^{2}) = 29$?

Consider the real matrix A whose rows are (29, 0, 55, 17), (1, 28, 46, 26), (17, 13, 33, 38) and (21, 67, 0, 13). What is the largest real eigenvalue of A?

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

Let X be the $\mathbb{R}-$vector space of all twice differentiable real valued functions on [0, 1]. Consider the linear map $\varphi : X \to \mathbb{R}^{3}$ defined by $\varphi(f) = (f(1), f'(1), f''(1))$. Which of the following statements is true?

Let $C[0, \pi]$ be the real vector space of real-valued continuous functions on the closed interval $[0, \pi]$. For positive integers n, define $f_{n} \in C[0, \pi]$ by $f_{n}(x) = \sin($nx)/sin x if $x \in (0, \pi), f_{n}(0) = n$, and $f_{n}(\pi) = (-1)^{n}$⁻$^{1}n$. Let V be the real subspace of $C[0, \pi]$ spanned by {$f_{1}, f_{2}, f_{3}$}. Consider the inner product on V given by ⟨f, g⟩ $= (2/\pi)\int_{0}^\pi f(x)g(x)\sin^{2}x$ dx. Which of the following statements is true?

Let V = {ax$^{3} +$ bx$^{2} +$ cx | $a, b, c \in \mathbb{R}$}. For f ∈ V, define $Q(f) = \int$₋$_{1}^{1} (f'(t))^{2}$ dt, where f′ denotes the derivative of f. Which of the following statements is FALSE?