List of top Engineering Mathematics Questions

Solve differential equation: \[ x^2 \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} + 2y = 0, x \geq 1 \] with initial conditions $y=0, \; y'(1)=1$ at $x=1$. Find $y$ at $x=2$. (round off to two decimals)

Consider the equation $\dfrac{dy}{dx} + ay = \sin(\omega x)$, where $a$ and $\omega$ are constants. Given $y=1$ at $x=0$, select all correct statements as $x \to \infty$.

The system of equations \[ \begin{cases} x-2y+\alpha z=0,
2x+y-4z=0,
x-y+z=0 \end{cases} \] has a non-trivial solution for \(\alpha=\;\underline{\hspace{1cm}}\). \;(Answer in integer)

Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?

The direction in which a scalar field \(\phi(x,y,z)\) has the largest rate of change is along:

If a monotonic and continuous function \(y=f(x)\) has exactly one root in the interval \(x_1<x<x_2\), then:

Consider the one–dimensional wave (advection) equation \[ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0, -\infty<x<\infty,\ t\ge 0. \] For the initial condition \(u(x,0)=e^{-x^{2}}\), the solution at \(t=1\) is:

Given that \[ \frac{dy}{dx} = 2x + y, y(0) = 1 \] Using Runge-Kutta fourth order method, the value of \(y\) at \(x = 0.2\) is \underline{\hspace{2cm}} (rounded off to 3 decimal places).

In a locality 'A', the probability of a convective storm event is 0.7 with a density function \[ f_{X_1}(x_1) = e^{-x_1}, \; x_1 > 0 \] The probability of a tropical cyclone-induced storm in the same location is given by the density function \[ f_{X_2}(x_2) = 2 e^{-2x_2}, \; x_2 > 0 \] The probability of occurring more than 1 unit of storm event is \underline{\hspace{3cm}} (rounded off to 2 decimal places).

A vector \(\vec{F} = 5\hat{i} - 10\hat{j} + 8\hat{k}\) is passing through the origin of a 3-D frame. Considering the tendency of rotation in the counter clockwise direction as positive, the moment about a point \(A: (3, 4, 8)\) is:

If \(A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}, \; B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}\) and \((A + B)^2 = A^2 + B^2\), then the values of \(a\) and \(b\) are:

The probability that a storm event with a return period of 20 years will occur once in a 5-year period is \underline{\hspace{6cm}} (rounded off to 2 decimal places).

If \(A\) and \(B\) are square matrices of order \(3\) such that \(|A|=-1\) and \(|B|=3\), then \(|3AB|\) equals:

\(y=ae^{mx}+be^{-mx}\) is the solution of the differential equation:

The value of \[ I=\int_0^{\pi/2}\frac{(\sin x+\cos x)^2}{\sqrt{1+\sin 2x}}\,dx \] is:

If $\dfrac{dy}{dx} = 8y^2x^3$ and $y(2)=1$, then $\dfrac{1}{y(0)}$ (in integer) is __________________.

If the values of $x$ are 1, 2, and 3 and the corresponding values of $y$ are 9, 8, and 10 respectively, then the slope of the line of regression equation of $y$ on $x$ is (up to 1 decimal place) __________________.

Two eigenvalues of the following matrix are 3 and 6. The third eigenvalue is \[ \begin{bmatrix} -2 & -4 & 2 \\ -2 & 1 & 2 \\ 4 & 2 & 5 \end{bmatrix} \]

The Newton-Raphson method is being used for two iterations to find an approximate solution of the equation \( e^x - 1 = 0 \) with an initial guess of 1. The difference between the actual and approximate solutions (rounded off to 2 decimal places) is ____________________.

The probability of the standard normal variable taking values between 0 and 1 is 0.3413, between 0 and 2 is 0.4772, and between 0 and 3 is 0.4987. The average of marks in an examination is 68 and the standard deviation is 10. The percentage of examinees getting less than 48 marks is:

The value of \( y \) for which the following limit exists is \[ \lim_{x \to 1} \frac{2x^2 - yx - x + 3}{3x^2 - 5x + 2} \]

Consider the function \( z = \tan^{-1}\left(\frac{y}{x}\right) \), where \( x = u \sin v \) and \( y = u \cos v \). The partial derivative, \( \frac{\partial z}{\partial v} \) is

Consider the function \( z = x^3 - 2x^2y + xy^2 + 1 \). The directional derivative of z at the point (1, 2) along the direction \( 3\hat{i} + 4\hat{j} \) is

Two eigenvalues of the following matrix are 3 and 6. The third eigenvalue is \[ \begin{bmatrix} -2 & -4 & 2 \\ -2 & 1 & 2 \\ 4 & 2 & 5 \end{bmatrix} \]

The value of \( x \) for which the inverse of the following matrix does not exist is \[ \begin{bmatrix} 1 & 3 & 0 \\ 2 & x & 4 \\ -1 & 0 & 2 \end{bmatrix} \]