List of top Engineering Mathematics Questions

The value of \( \lim_{x \to 0} \left( \frac{\tan 11x}{\tan 5x} \right) \) is:
If \[ \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} \quad \text{is the eigenvector of the matrix} \quad A = \begin{bmatrix} 8 & 11 & 3 \\ 4 & -1 & 3 \\ -4 & 10 & 6 \end{bmatrix}, \quad \text{then the corresponding eigenvalue is:} \]
The solution of the following differential equation represents \[ \frac{dy}{dx} = \frac{y + 1}{x} \]
Let \( X \) be a Poisson distributed random variable with parameter \( \lambda (> 0) \) such that it satisfies the equation \[ P(X = 1) = 3P(X = 3) - P(X = 2) {. Then, the value of } \lambda { is:} \]
If \( u(x, y, z) = x^2y + y^2z + z^2x \), the value of \[ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} { at the point } (1,1,1) { is:} \]
The diameter of a fibre is assumed to be a continuous random variable \( X \) with probability density function \[ f(x) = 6x(1 - x), \quad 0<x \leq 1 \] If \( P(X<\beta) = P(X>\beta) \), then the value of \( \beta \) (rounded off to 1 decimal place) is _________.
Consider the partial differential equation: \[ \frac{\partial^2 u}{\partial x^2} = \frac{1}{k} \frac{\partial u}{\partial t} + \sin x, \quad k>0 \] Amongst the following, the correct statement(s) for the above equation is/are:
The value of \( \alpha \) for which the Euler method with step size \( h = 0.1 \) provides \( y(1.1) = 1.2 \) for the following initial value problem is \[ \frac{dy}{dx} = -2xy^\alpha, \quad y(1) = 2 \]
Consider the function \( f(x) = |x| - 1 \). Which of the following statements is/are true in the interval \([-10, 10]\)?
Let \( M = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \) and \( K = \begin{bmatrix} 2 & -1 \\ -1 & 1 \end{bmatrix} \) satisfy the eigenvalue problem given by:
\[ (M - \alpha K)\phi = 0. \] The lowest eigenvalue \( \alpha \) is ________ (rounded off to two decimal places).
For all real values of \( x \) and \( y \), the partial differential equation in terms of \( \psi(x, y) \) given by \[ \frac{\partial^2 \psi}{\partial x^2} + 2 \frac{\partial^2 \psi}{\partial x \partial y} - 3 \frac{\partial^2 \psi}{\partial y^2} = 0 \] is ________
The Fourier series expansion of \( f(x) = \sin^2 x \) in the interval \((- \pi, \pi)\) is ______.
The value of \( \lim_{t \to 1} \frac{\ln t}{t^2 - 1} \) is ________.
The gradient of \( y = 3x^2 \sin(2x) \) at \( (0.2, 1) \) is:
The value of \( \int_1^3 \int_2^5 x^2 y \, dx \, dy \) is ________(answer in integer).
Consider \( f(t) = \cos(at) \), where \( a \) is a real constant. The Laplace transform of \( f(t) \) is _________.
A box contains 12 red and 8 blue balls. Two balls are drawn randomly from the box without replacement. The probability of drawing a pair of balls having the same color is ________
Choose the correct statement(s) from the following options, regarding Cauchy's theorem on complex integration \( \oint_C f(z)\,dz \), where \( C \) is a simple closed path in a simply connected domain \( D \).
The value of the integral \( \int_{-\pi}^{\pi} (\cos^6 x + \cos^4 x) \, dx \) is:
Newton-Raphson method is used to compute the inverse of the number 1.6. Among the following options, the initial guess of the solution that results in non-convergence of the iterative process is:
Consider the function \( f(z) = \frac{2z + 1}{z^2 - 2z} \), where \( z \) is a complex variable. The sum of the residues at singular points of \( f(z) \) is:
If one of the eigenvectors of the matrix
\[ A = \begin{bmatrix} 1 & 1 \\ -4 & x \end{bmatrix} \] is along the direction of
\[ \begin{bmatrix} 2\alpha \\ \alpha \end{bmatrix} \] where \( \alpha \) is any non-zero real number, then the value of \( x \) is ________ (in integer).
The solution of the differential equation \(\dfrac{dy}{dx} = \dfrac{y}{x}\) represents:
A \( 2n \times 2n \) matrix \( A = [a_{ij}] \) has its elements defined as: \[ a_{ij} = \begin{cases} \beta(i + j), & \text{if } i + j \text{ is odd} \\ -\beta(i + j), & \text{if } i + j \text{ is even} \end{cases} \] where \( n \) is an integer greater than 2, and \( \beta \) is any non-zero real number. What is the rank of matrix \( A \)?
Choose the eigenfunction(s) of stable linear time-invariant continuous-time systems from the following options.