List of top Engineering Mathematics Questions asked in Telangana State Post Graduate Engineering Common Entrance Test

A and B are two nonsingular matrices. If the characteristic equation of A is $ a_0\lambda^3 + a_1\lambda^2 + a_2\lambda + a_3 = 0 $ and characteristic equation of $ B^{-1}AB $ is $ b_0\lambda^3 + b_1\lambda^2 + b_2\lambda + b_3 = 0 $, then

The order of convergence of the Newton's Raphson's method is

Which of the following numerical method is a multistep method to solve IVP or BVP's (IVP: Initial Value Problem, BVP: Boundary Value Problem. Multistep methods are typically for IVPs of ODEs.)

Which of the following pairs of numbers is possible to be the regression coefficients for a data of two variables

If $ x=\alpha, y=\beta, z=\gamma $ is a positive integral solution of the equation $ x+y+z=12 $, the probability that it is such that $ \alpha<\beta<\gamma $ is

If \( x + y + z = 45 \) and the maximum value of \( x^4 y^6 z^5 \) exists at \( x = \alpha,\, y = \beta,\, z = \gamma \), then the value of \[ \frac{\alpha + \beta}{\gamma} = ? \]

If it is known that \[ \int_0^{\infty} \frac{e^{ix}}{x} dx = \frac{\pi i}{2}, \] then compute: \[ \int_0^{\infty} \frac{\sin 5x}{x} dx = ? \]

Let \( \vec{F} \) be an irrotational vector function. If \( C \) is the closed curve which is the boundary of an open surface \( S \), then: \[ \oint_C \vec{F} \cdot d\vec{R} = ? \]

If \( |Z|>2 \) and the coefficients of \( Z^{12} \) and \( \frac{1}{Z^{12}} \) in the Laurent series expansion of \[ \frac{1}{(1 - Z)(2 - Z)} \] are \( A \) and \( B \) respectively, then compute \( 2024 - A \).

Given \[ \oint_C \frac{f(z)}{z - a} \, dz = 0, \] where \( C: |z| = K \). Consider the following conclusions:

• [I.] \( K<|a| \) and \( f(z) \) is analytic inside and on \( C \).
• [II.] \( f(z) = (z - a)^n g(z) \) for \( n \in \mathbb{N} \), and \( g(z) \) is analytic inside and on \( C \).

Which of the following is true?

Three students A, B and C write an entrance exam. The chance that at least one of them passes the exam is \( \frac{3}{4} \). If the probability that A and C pass the exam are respectively \( \frac{1}{2} \) and \( \frac{1}{4} \), then the probability that B does not pass that exam is:

The equations \[ a_1x + b_1y + c_1z = d_1,\quad a_2x + b_2y + c_2z = d_2,\quad a_3x + b_3y + c_3z = d_3 \] represent three planes. Given that \( d_1d_2d_3 \ne 0 \) and \[ \frac{a_i}{a_j} \ne \frac{b_i}{b_j} \ne \frac{c_i}{c_j} \quad \text{for } i, j = 1, 2, 3,\ i \ne j, \] Let \[ A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}, \quad B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}. \] If \( \text{rank}(A) = \text{rank}([A:B]) = 2 \), then the planes are such that

If the mean of a random variable which follows exponential distribution is \( \frac{1}{3} \), then the probability that the random variable \( X \) takes the values more than \( r \) is greater than \( \frac{1}{e^5} \), then

If \( \overline{X}_1 \) and \( \overline{X}_2 \) are any two eigenvectors corresponding to eigenvalues \( \lambda_1 \) and \( \lambda_2 \) respectively, then

Let \[ A = \begin{bmatrix} 1 & -5 & 2 \\ 7 & 0 & 6 \\ 5 & -3 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 6 & 7 \\ 3 & 8 & 4 \\ 7 & 6 & 1 \end{bmatrix}. \] If \( C \) and \( D \) are two \( 3 \times 3 \) matrices such that: - \( A + C \) is symmetric, - \( A - C \) is skew-symmetric, and - \( B = D^\top \), then find \( (C - D)^\top \).

A fair coin is tossed three times. Let \( X \) be the number of tails appearing and its distribution is: \[ \begin{array}{|c|c|c|c|c|} \hline X = x & 0 & 1 & 2 & 3 \\ \hline P(X = x) & \frac{1}{8} & \frac{3}{8} & \frac{3}{8} & \frac{1}{8} \\ \hline \end{array} \] Then the variance of \( X \) is:

If \( X \) is a continuous random variable whose probability density function is given by \[ f(x) = \begin{cases} k(4x - 2x^2), & 0<x<2 \\ 0, & \text{otherwise} \end{cases} \quad \text{then the value of } k \text{ is:} \]