List of top Questions asked in TS PGECET

If X and Y are independent random variables with variances \(\sigma_x^2 = 5\) and \(\sigma_y^2 = 3\) then the variance of the random variable \(Z = 3X + 2Y - 5\) is

The value of the integral \(\displaystyle\oint_{C}\frac{z e^{-z}}{(z-1)(z-2)}\,dz\) where \(C:|z|=\dfrac{3}{2}\) is

For which values of \(k\) does the function \(f(z)=\left(x^3-3xy^2\right)+i\left(kx^2y-y^3\right)\) become analytic?

If \(u=\sin^{-1}\left(\dfrac{x^{1/3}+y^{1/3}}{x^{1/2}-y^{1/2}}\right)^{1/2}\), then \(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}=\)

For the function \(f(x)=\log x\), the number \(c\) strictly between \(e^2\) and \(e^3\) that satisfies \(f'(c)=\dfrac{f(e^3)-f(e^2)}{e^3-e^2}\) is

Let \(a,b\) and \(c\) be real numbers. Suppose there exist real numbers \(x,y,z\) which are not all zero such that the system of equations \(x = cy + bz\), \(y = cx + az\) and \(z = bx + ay\) has a non-zero solution then \(\left(a+b+c\right)^2 =\)

Let \(A=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\). If \(u_1\) and \(u_2\) are column matrices such that \(Au_1 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(Au_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}\), then \(u_1 - u_2\) is

The Eigenvalues of \(3\times 3\) real matrix A are 1, 2, 3 then \(A^{-1} =\)

In the earth's atmosphere, the lowest layer where all weather occurs is called

The directional derivative of \(f(x,y,z)=4e^{2x-y+z}\) at the point \((1,1,-1)\) in the direction of the vector \(\vec{a}=-4\hat{i}+4\hat{j}+7\hat{k}\) is