List of top Mathematics 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,

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

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 \( \dfrac{2}{s^3} \)?

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 ...........

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 ______

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 ______

The solution of the differential equation
\[ (D^3 - 5D^2 + 7D - 3)y = e^{-2x} \] is:

The given differential equation
$(x y^2 + n x^2 y) dx + (x^3 + x^2 y) dy = 0$ is exact when $n =$

The number of accidents occurring in AU region in a month follows Poisson distribution with mean $\lambda = 5$. The probability of less than two accidents in a randomly selected month is ______

Let $X$ be an exponential random variable with mean parameter one. Then the conditional probability $P(X>10 | X>5)$ is equal to

If \( X \) is a continuous random variable with the probability density
\[ f(x) = \begin{cases} K(1 - x^3), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Then, the value of \( K \) is ............

The particular integral of the differential equation
\[ \frac{d^2y}{dx^2} - 6 \frac{dy}{dx} + 9y = e^{3x} \text{ is ............} \]

If $(a, b, c)$ is the unique solution of the system of linear equations
$x + y + z = 2, 2x + y - z = 3, 3x + 2y + z = 4$,
then $b^2 + c^2 = ........$

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 _______

A number is selected randomly from each of the following two sets:
{1, 2, 3, 4, 5, 6, 7, 8}, {2, 3, 4, 5, 6, 7, 8, 9}
What is the probability that the sum of the numbers is 9?

If the function \( f(z) = -x^2 + xy + y^2 + i(ax^2 + bxy + cy^2) \) of complex variable \(z = x + iy\) is analytic in the complex plane, then the values of \(a, b, c\) are _______

Let \( A \) be a \(3 \times 3\) matrix and \( B = 2A^2 + A^{-1} - I \), where \( I \) is a \(3 \times 3\) identity matrix. If the eigenvalues of \( A \) are 1, –1 and 2, then the trace of \( B \) is ________

Let \( A = \begin{bmatrix} a+1 & b & c \\ a & b+1 & c \\ a & b & c+1 \end{bmatrix} \). If determinant of the matrix \( A \) is zero, then \( (a + b + c)^3 = \_\_\_\_\_\_ \)

The value of the real variable \( x>0 \) that minimizes the function \( f(x) = x^x e^{-x} \) is ________

The general solution of the differential equation \( x^2 y'' - xy' + 5y = 0 \) is _______