List of top Questions asked in Andhra Pradesh Post Graduate Engineering Common Entrance Test

Suppose \( R_1 \) and \( R_2 \) are reflexive relations on a set \( A \). Which of the following statements is correct?

Determine the value of $\lambda$ and $\mu$ for which the system of equations
$x + 2y + z = 6$,
$x + 4y + 3z = 10$,
$2x + 4y + \lambda z = \mu$
has a unique solution.

Let \( z \) be a complex variable and \( C : |z| = 3 \) be a circle in the complex plane. Then,

\[ \oint_C \frac{z^2}{(z - 1)^2(z + 2)} \, dz = \]

For a stable closed loop system, the gain at phase crossover frequency should always be:

If the matrix \[ A = \begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix} \] has three distinct eigenvalues, and one of its eigenvectors is \[ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \] then which of the following can be another eigenvector of \( A \)?

Which of the options correctly represents the Laplace inverse of \( \dfrac{2}{s^3} \)?

For a first order reaction, the starting material reduces to \( \dfrac{1}{4} \) of its initial value after 20 min.
The rate constant of the reaction will be: (Given \( \ln 4 = 1.38 \))

1 mole of Argon gas is heated at constant pressure from 200 K to 600 K.
If \( C_p = 4\ \text{cal} \cdot \text{deg}^{-1} \cdot \text{mol}^{-1} \), then the change in entropy will be:
(Given \( \ln 3 = 1.09 \))

If the Laplace transform of a function \( f(t) \) is given by \[ \frac{2s + 1}{(s + 1)(s + 2)} \] then \( f(0) \) is equal to ______

The probability of a component being defective is 0.01. There are 100 such components in a machine. Then the probability of two or more defective components in the machine is ...........

The value of the integral \( \int_1^3 \frac{2}{x} \, dx \), when evaluated by using Simpson’s \( \frac{1}{3} \) rule on two equal subintervals each of length 1, is ...........

If $\vec{F} = x(x^2 + y^2 + z^2)\hat{i} + 2y(x^2 + y^2 + z^2)\hat{j} + 3z(x^2 + y^2 + z^2)\hat{k}$, then $\text{div} \, \vec{F}$ at $(1,1,1)$ is equal to ______

Consider the ordinary differential equation \[ x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0 \] with \( y(x) \) as a general solution. Given the values of \( y(1) = 1 \), \( y(2) = 5 \), the value of \( y(3) \) is equal to ______