List of top Mathematics Questions asked in TS PGECET

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

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

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

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

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 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

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

The general solution of one dimensional wave equation \[ \frac{\partial^2v}{\partial t^2} = c^2 \frac{\partial^2v}{\partial x^2} \] is \(u(x,t)=\)

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 \(x_0=1.2\) is the initial guess of the solution of \(x^3+2x-1=0\), then the \(1^{\text{st}}\) iteration solution \(x_1=\)

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

Fourier series of \(x^2,\;0<x<2\) is \[ f(x)=\frac{a_0}{2} +\sum_{n=1}^{\infty}a_n\cos(n\pi x) +\sum_{n=1}^{\infty}b_n\sin(n\pi x), \] then \(b_2=\)

A convergent series from the given series is

The complete solution of \[ y^3\frac{\partial u}{\partial x} + x^2\frac{\partial u}{\partial y} = 0 \] is \(u(x,y)=\)

The Laplace transform of \[ \int_{0}^{t} p\sin(t-p)\,dp \] is

The iterative formula for finding \(2^{\frac{1}{5}}\) using Newton-Raphson method is

A multistep method for solving a differential equation numerically among the following is

Let \[ f(x)=x^3-\frac32x^2, \] which of the following is not correct?

If \[ I=\int_C x^2y\,dx-xy^2\,dy, \] where \(C\) is the circle \[ x^2+y^2=16, \] then \(2I=\)

The solution of \[ \frac{dy}{dx} + \left[2(x-1)\sin x+x(x-2)\cos x\right]y = \frac{e^{2x\sin x}}{e^{x^2\sin x}}, \] at \(x=2\) if \(y(\pi)=0\) is

Evaluate \[ \lim_{x\to0}\frac{e^{\tan x}-e^{\sin x}}{(1-\cos x)\tan x} \]

Consider the system of equations \[ x+y+z=4\mu,\qquad x+2y+2z=10\mu,\qquad x+3y+4\lambda z=\mu^2+15, \] where \(\lambda\) and \(\mu\) are real numbers. Which of the following is correct?