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

If \( A \) and \( B \) are two events having probabilities, \( P(A) = 0.6 \), \( P(B) = 0.3 \), and \( P(A \cap B) = 0.2 \), then the probability that neither \( A \) nor \( B \) occurs is ______

Let a random variable \( X \) follow Poisson distribution such that \( P(X = 0) = 2P(X = 1) \). Then, P(X = 3) = ______ 
 

Simpson's \( \frac{1}{3} \) rule is applied when

The probability distribution of a random variable \( X \) is given as follows. Then, \( P(X = 50) - \frac{P(X \leq 30)}{P(X \geq 20)} \) = 

The inverse Laplace transformation of \( \frac{s+5}{s^2 + 4s + 4} \) for \( t \geq 0 \), is
The value of the real variable \( x > 0 \) that minimizes the function \( f(x) = x^{-x} e^x \) is:
Let \( A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \) be a 3 × 3 matrix. If \( \alpha \) and \( \beta \) are the largest and smallest eigenvalues of \( A \), respectively, then \( \alpha - \beta = \_\_\_\_\_\_ \)
If the determinant of the 3 × 3 matrix \( A = \begin{pmatrix} a & 1 & 2 \\ b & 0 & -2 \\ 1 & -3 & 1 \end{pmatrix} \) is zero, then the values of \( a \) and \( b \) are:
If \(f(x) = |x - 1|\), then
The solution of the differential equation \(\frac{d^2y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0\) satisfying \(y(0) = 0\), \(y'(0) = 1\), is
The probability distribution of a random variable \( X \) is given below:
\[ \begin{array}{|c|c|c|c|c|c|} \hline X = x & 10 & 20 & 30 & 40 & 50 \\ \hline P(X = x) & k & 2k & 3k & 4k & 5k \\ \hline \end{array} \] Then, \( P(X = 50) - \dfrac{P(X < 30)}{P(X > 20)} \) = ...........
If \( X \) is a continuous random variable with the probability density function \[ f(x) = \begin{cases} K(1 - x^3), & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Then, the value of \( K \) is ..........
The complex-valued function \( f(z) = iz - |z|^2 \) is analytic at ...........
Convert the non-linear equation \( xy' + y = x^4 y^3 \) into a linear one using the transformation \( z = y^{-2} \).
The linear equation is ...........
If \( A = \begin{pmatrix} -3 & 2 \\ 1 & 0 \end{pmatrix} \) is a \( 2 \times 2 \) matrix, then \( A \) satisfies the relation ...........
Let \( i \) be an imaginary number such that \( i = \sqrt{-1} \). Let \( a \) and \( b \) be real numbers satisfying \( a^2 + b^2 = 1 \).
Then, the eigenvalues of the matrix \[ \begin{bmatrix} -a & b \\ b & a \end{bmatrix} \] are ...........
If \( \sin x \) is a solution of the differential equation \[ \dfrac{d^4 y}{dx^4} + 2 \dfrac{d^3 y}{dx^3} + 6 \dfrac{d^2 y}{dx^2} + 2 \dfrac{dy}{dx} + 5y = 0, \] then the general solution is ...........
The directional derivative of \( f(x, y, z) = xyz \) at the point \( (1, 2, 3) \) in the direction of the vector \( 2\hat{i} + \hat{j} - 2\hat{k} \) is ...........
Suppose \( R_1 \) and \( R_2 \) are reflexive relations on a set \( A \). Which of the following statements is correct?
In how many ways can a chairperson, a secretary, and a treasurer be chosen from a group of 10 people, if each person can hold only one position?
The ratio of the ages of A and B is 5:3. After 6 years, their ages will be in the ratio 6:4. What is the current age of A?
A project requires 12 workers to complete in 20 days. If 4 workers leave after 8 days, how many additional days will the remaining workers take to finish the project?
The integration factor of \( \frac{dy}{dx} + 2xy = e^{-x^2} \)

The rank of matrix \(\begin{bmatrix} k & -1 & 0 \\[0.3em]  0 & k & -1 \\[0.3em] -1 & 0 & k \end{bmatrix}\) is 2, for \( k = \)

If \(A = \begin{bmatrix} 4 & 2 \\[0.3em] -3 & 3 \end{bmatrix}\), then \(A^{-1} =\)