List of top Engineering Mathematics Questions

$X$ is a discrete random variable taking values $0,1,2$ with $P(X=0)=0.25$, $P(X=1)=0.5$, and $P(X=2)=0.25$. With $E[ . ]$ denoting expectation, compute $E[X]-E[X^2]$ (rounded off to one decimal place).

The solution $x(t)$, $t\ge 0$, to $\ddot{x}=-k\dot{x}$ ($k>0$) with $x(0)=1$ and $\dot{x}(0)=0$ is:

$F(z)=\dfrac{1}{1-z}$, when expanded as a power series around $z=2$, would result in $F(z)=\sum_{k=0}^{\infty} a_k (z-2)^k$ with ROC $|z-2|<1$. The coefficients $a_k,\;k\ge 0$, are given by the expression ________.

Choose solution set $S$ corresponding to the system of two equations \[ x-2y+z=0,\qquad x-z=0 \] (Note: $\mathbb{R}$ denotes the set of real numbers).

Which one among the following statements must be TRUE about the mean and the median of the scores of all candidates appearing for GATE 2023?

A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?

Given vectors \[ \vec{A} = 9\hat{i} - 5\hat{j} + 2\hat{k}, \vec{B} = 11\hat{i} + 4\hat{j} + \hat{k}, \vec{C} = -7\hat{i} + 14\hat{j} - 3\hat{k} \] which of the following statements are TRUE?

The sum of eigenvalues of the matrix \( \begin{bmatrix} 4 & 3 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & -3 \end{bmatrix} \) is ................... (in integer).

Order (O) and degree (D) of the differential equation \( \left(\frac{dy}{dx}\right)^3 = \sqrt{\frac{d^2y}{dx^2} + 10} \) are

The maximum value of function \(f(x) = 4x^3 - 24x^2 + 36\) in the domain [-1, 5] is ................... (round off to nearest integer).

A fluid flow field is given by the velocity vector \( \vec{V} = e^{xyz}(x\hat{i} + z\hat{k}) \). The curl of velocity at (1, 2, 3) is

Given, \( \vec{\phi} = xy\hat{i} + yz\hat{j} + xz\hat{k} \). S is a surface bounded by the planes x = 0, y = 0, z = 0, x = 3, y = 2, and z = 1. If \( \hat{n} \) is the unit vector normal to S, then \( \iint_S \vec{\phi} . \hat{n} dS \) is

The probability of setting an easy exam paper by three setters are \( \frac{1}{2}, \frac{1}{3}, \) and \( \frac{1}{4} \). If all three are setting one paper each, then the probability that at least one of the papers will be easy is ................... (round off to 2 decimal places).

The value of \( \lim_{x\to 1} \frac{7x^7-20x^5+13x}{3x^3+x-4} \) is

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?

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:

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

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).

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).

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: