CSIR NET Maths previous year question papers

Let V be a real vector space and L(V) denote the space of linear operators on V. Let T∈L(V) be a non-zero operator such that $T^{2}=T$. Consider the subspace W of L(V) spanned by {$I, T^{n} : n$ is a positive integer}. Which of the following statements are necessarily true?

Let G be a finite non-abelian group. Which of the following statements are necessarily true?

Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate $\lambda(t)=1/(1+t), t>0$, and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are true?

Let a,b be distinct positive real numbers. Consider the function $f:\mathbb{R}^{2}\to\mathbb{R}$ given by f(x,y)=(ax+by$)^{2}/($ax$^{2}+$by$^{2})$ if (x,y)≠(0,0), and 0 if (x,y)=(0,0). Which of the following statements are necessarily true?

Let $X_{1},X_{2},\dots,X_{n} (n>5)$ be independent random variables such that $X_{t}=\alpha+\alpha^{2}t+\varepsilon_{t}$, for t=1,…,n, where $\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{n}$ are independent and identically distributed $N(0,\sigma^{2})$ random variables. Here $\alpha\in\mathbb{R}$ and $\sigma>0$ are unknown parameters. Which of the following statements are true?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} ((y')^{2}\sin x + (2cos x)y)$ dx subject to y(0)=0, y(1)=1. Then which of the following statements are true?

Let $Y=X\beta+\varepsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T}$ and the random error $\varepsilon$~$N_{n}(0,\sigma^{2}I), \sigma>0$ and n>p+1. The least squares method provides a unique estimator $\beta$̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

Let $\lambda\in\mathbb{R}$ be such that the integral equation $y(x) = \lambda\int$₋$_{1}^{1}(5$xt$^{3}+4x^{2}t+3$xt)y(t)dt admits a non-trivial solution y(x) such that y(1)=5/2. Then which of the following statements are true?

Let {$x_{n}$} be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_{n}\to\pi/2$ as $n\to\infty$. For $n\in\mathbb{N}$, let $e_{n}=x_{n}-\pi/2$. Let p>0 be such that $\lim(n\to\infty)$|$e_{n}$₊$_{1}$|/|$e_{n}$|^p exists and is non-zero. Then which of the following statements are true?

Consider the boundary value problem (BVP$) u_{xx}+u_yy=0$ in $\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1$}, u(x,y)=e^(x+y) on $\partial\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1$}. Then which of the following statements are true?

Consider the simple linear regression model $Y_{i}=\beta{}x_{i}+\varepsilon_{i}, i=1,2,\dots,n$, where $\beta>0, \sum{}x_{i}^{2}>0$, and the uncorrelated errors $\varepsilon_{i}$ have zero mean and finite variance $\sigma^{2}(>0)$. Let $\beta$̃$_{1}=\sum{}a_{i}$$Y_{i}$, where $a_{i}$'s minimize $E(\sum{}a_{i} Y_{i}-\beta)^{2}$ with respect to scalars $a_{1},a_{2},\dots,a_{n}$. Let $\beta$̃$_{2}$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=\sin^{2}z+\cos^{2}$|z|. Which of the following statements are true?

A monkey covers exactly 10 m on ground in each jump. What is the least number of jumps required to reach a distance 1 m away from where the monkey jumps first?

Cube root of 0.0125% is closest to

How many 5-digit numbers, using 0 to 9, can be generated such that '123' appears as a string and no digit appears more than once?

The diagrams show two orientations of a die having numbers 1 to 6 written on different faces. In the first orientation the visible faces are 3 on top, 4 on the front-left and 5 on the front-right; in the second, 4 on top, 2 on the front-left and 5 on the front-right. The number on the face opposite the face showing 3

There are four containers of equal height, whose bases are a circle, a square, a rectangle and an equilateral triangle having the same area. Which one of the following statements about these containers is true?

The diagrams show the rates of diamond and emerald in a range of sizes. Both graphs are straight lines through the origin: for diamond the price rises to 5 Lakh Rs/carat at a weight of 2.5 carat, and for emerald it rises to 2.5 Lakh Rs/carat at the same weight. A person wants to buy a diamond and an emerald of identical size for a total of Rs. 6,75,000/-. What is that size?

If A is B's daughter, B is C's brother and D is C's father, then what is A to D?

A block of marble 4 m × 3 m × 2 m in size is cut into square tiles of 1 m side having thickness of 10 cm. Assuming there is no wastage in cutting, how many tiles will be made?

A lady walks one-tenth of the total distance at 3 km/h, one-sixth she runs at 5 km/h, one-fifth at 6 km/h, and covers the remaining 16 km at 16 km/h by cycle. What is the total distance?

An OTP is made of six digits using 0 to 9. If three digits and their positions are known, what is the probability (in percentage) of discovering the full pin within 100 trials?

Ramesh is taller than Rajesh but not taller than Rupesh. Suresh's height is the average of the heights of Naresh and Rajesh. If Rajesh is taller than Naresh then who is the shortest among them?

The value of (1 − 1/2025)(1 − 1/2024)(1 − 1/2023)…(1 − 1/2001) is

One side and one diagonal of a rhombus are 13 cm and 24 cm, respectively. Then the area of the rhombus is