List of top Mathematics Questions asked in TS PGECET

If \(P\) is an orthogonal matrix of order \(3\times3\), \(A=\begin{bmatrix}1& 2& 2\\2& 1& 2\\2& 2& 1\end{bmatrix}\) and the eigenvalues of \(P^{T}AP\) are \(\alpha,\beta,\gamma\), then \(\alpha^2+\beta^2+\gamma^2=\)

If the mean and variance of a binomial variate \(X\) are \(2\) and \(1\) respectively, then the probability that \(X\) takes a value greater than one is

Suppose that \(X\) and \(Y\) are independent random variables with probability densities \[ f_X(x)= \begin{cases} \dfrac{8}{x^3}, & x>2,[2mm] 0, & \text{elsewhere}, \end{cases} \qquad f_Y(y)= \begin{cases} 2y, & 0 0, & \text{elsewhere}. \end{cases} \] Then the expected value of \(Z=XY\) is

The particular integral of the differential equation \[ (D^2+1)y = \sin x\,\sin2x \] is

Curl of \[ \vec{V}=e^{xyz}(\hat{i}+\hat{j}+\hat{k}) \] at the point \((1,1,1)\) is

The initial value problem \[ (x-x^2)\frac{dy}{dx} = (2x-1)y, \qquad y(x_0)=y_0, \] has a unique solution if \((x_0,y_0)\) equals to

Given \[ \int_{0}^{3} f(x)\,dx=\int_{0}^{3}(4x^2-1)\,dx \] is approximated using Simpson's \(\frac{1}{3}\) rule with \(3\) subintervals and gives \(f(0)+a+f(3)\), then \(a\) is

If \(f(x)=|\sin x|\) for \(-\pi<x<\pi\) and the Fourier series of \(f(x)\) is \[ f(x)=\sum_{n=0}^{\infty}(a_n\cos nx+b_n\sin nx), \] then the value of the Fourier coefficient \(a_0\) is

The eigenvalues of a \(3\times3\) real matrix \(A\) are \(0\), \(-2\), \(-2\) and it satisfies the equation \[ A^4+4A^3=KA^2, \] then the value of \(K\) is

Rank of the matrix \[ A= \begin{bmatrix} 1& 0& 1& 1 0& 1&-3&-1 3& 1& 0& 2 1& 1&-2& 0 \end{bmatrix} \] is

If \[ u_n=\frac{\sqrt{2n^2-5n+1}}{4n^3-7n^2+2} \] and \[ v_n=\sqrt{n^2+1}-\sqrt{n^2-1}, \] then

A continuous random variable \(X\) has the density function \[ f(x)= \begin{cases} e^{-x}, & x>0, 0, & \text{elsewhere}, \end{cases} \] then the expected value of \[ g(x)=e^{\frac{2x}{3}} \] is

For a binomial variate \(X\) if \(n=5\) and \[ P(X=1)=8P(X=3), \] then its variance is

How many minimum number of iterations are required to get an accuracy of \(0.001\) for the interval \([1,3]\) in the bisection method?

The value of \[ \int_{0}^{\infty}\frac{x^{2}(1-x^{2})}{(1+x)^{24}}\,dx \] is

The general solution of \[ \log\left(\frac{dy}{dx}\right)=ax+by \] is

For a given \[ u(x,y)=x^3-3xy^2+3x^2-3y^2+1, \] the analytic function \[ f(z)=u+iv \] is

The value of \[ \left| \begin{array}{ccc} \log x & \log y & \log z \log ax & \log ay & \log az \log bx & \log by & \log bz \end{array} \right| \] is

The divergence of \[ \left(2x^{2}z\,\hat{i}+xy^{2}z\,\hat{j}+3yz^{2}\,\hat{k}\right) \] at the point \((1,1,1)\) is

The differential equation \[ \frac{\partial^2u}{\partial x^2} +4\frac{\partial^2u}{\partial x\partial y} +4\frac{\partial^2u}{\partial y^2} -\frac{\partial u}{\partial x} -2\frac{\partial u}{\partial y} =0 \] is classified as

Let \(A=[a_{ij}]\) for \(i,j=1,2,3,\ldots,n\) be an \(n\times n\), \((n>1)\) matrix such that \[ a_{ij}= \begin{cases} i!, & \text{if } i=j, j, & \text{if } i>j, 0, & \text{if } i<j. \end{cases} \] Which one of the following is correct?

A multistep method used to solve differential equations is

If \(y_1,\;y_2\) are two dependent solutions of a second order linear homogeneous differential equation then \(y_1y_2'-y_2y_1'\) is

If \[ F(x)= \begin{cases} 0, & x[2mm] \left(\dfrac{x+7}{a}\right), & -7[2mm] 1, & x>7, \end{cases} \] represents the probability distribution function of a continuous random variable, then \(a=\)

Mean and variance pair is given below. A pair representing the data of a binomial distribution is