List of top Mathematics Questions asked in TS PGECET

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 \(x_0=1.2\) is the initial guess of the solution of \(x^3+2x-1=0\), then the \(1^{\text{st}}\) iteration solution \(x_1=\)
The matrix \[ A= \begin{bmatrix} 1 & 1+i & 2i 1-i & 3 & 4 -2i & 4 & 5 \end{bmatrix} \] has
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 \(b_2=\)
An eigen vector of the matrix \[ A= \begin{bmatrix} 3 & 2 1 & 2 \end{bmatrix} \] is
A convergent series from the given series is
A multistep method for solving a differential equation numerically among the following is
The iterative formula for finding \(2^{\frac{1}{5}}\) using Newton-Raphson method is
The complete solution of \[ y^3\frac{\partial u}{\partial x} + x^2\frac{\partial u}{\partial y} = 0 \] is \(u(x,y)=\)
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
The Laplace transform of \[ \int_{0}^{t} p\sin(t-p)\,dp \] 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?
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 \(\alpha^2+\beta^2+\gamma^2=\)

Find the Laurent series of the function \[ f(z)=1+\frac{3}{z+2}-\frac{8}{z+3} \] in the region \[ |z|<2. \] 

Statement-I: 
The functions \[ u=x^2+y^2,\qquad v=\tan^{-1}\left(\frac{y}{x}\right) \] are functionally independent. 
Statement-II: 
The Jacobian \[ \frac{\partial(u,v)}{\partial(x,y)} \] is non-zero. 
The correct answer is:

The solution of $\frac{dy}{dx} - y - 2x + x^2 = 0$, $y(0) = 1$ at $x = 0.2$ using Euler's method is:
By assuming that the solution to the equation $x^4 - x - 10 = 0$ lies between $x = 1.8$ and $x = 2$, then the next approximate solution by Regula-falsi method is (Take $f(1.8) = -1.3024$):
The value of $\oint_C x \, dy - y \, dx$, where $C$ is the boundary of the rectangular region formed by $x = 5$, $x = 12$, $y = 0$, and $y = 8$ is:
Laplace transform of $\int_0^t u^2 \sin(t - u) \, du$ is:
If the integrating factor of $\frac{dy}{dx} + \left[(x^2 - 2x)\cos x + 2(x - 1)\sin x\right]y = x^2$ is $e^{f(x)}$, then $f(3) =$}
Two dice are thrown simultaneously. Let $X$ be the random variable representing the absolute difference of the numbers appeared on the dice. The mean of $X$ is:
If $f(x)$ is the probability density function of an exponential distribution and $\frac{1}{\lambda}$ is its mean, then $\int_0^\infty x^2 f(x) \, dx =$}