List of top Engineering Mathematics Questions

Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$?

If the polynomial \[ P(x)=a_0+a_1x+a_2\,x(x-1)+a_3\,x(x-1)(x-2) \] interpolates the points $(0,2)$, $(1,3)$, $(2,2)$, and $(3,5)$, then the value of $P\!\left(\tfrac{5}{2}\right)$ is ____________ (round off to 2 decimal places).

The value of $m$ for which the vector field \[ \vec F(x,y)=\big(4x^{m}y^{2}-2xy^{m}\big)\,\mathbf{i}+\big(2x^{4}y-3x^{2}y^{2}\big)\,\mathbf{j} \] is a conservative vector field, is ____________________ (in integer).

The probability of a person telling the truth is $\dfrac{4}{6}$. An unbiased die is thrown by the same person twice and the person reports that the numbers appeared in both the throws are the same. Then the probability that actually the numbers appeared in both the throws are same is ______________ (rounded off to 2 decimal places).

If the quadrature formula \[ \int_{-1}^{1} f(x)\,dx \;\approx\; \frac{1}{9}\Big(c_1 f(-1) + c_2 f\!\left(\tfrac{1}{2}\right) + c_3 f(1)\Big) \] is exact for all polynomials of degree less than or equal to $2$, then

The second smallest eigenvalue of the eigenvalue problem \[ \frac{d^2 y}{dx^2} + (\lambda - 3)\,y = 0,\qquad y(0)=y(\pi)=0, \] is

Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,3$. If \[ B = A^2 + 2A + I, \] where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ are:

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by \[ f(x,y)= \begin{cases} \dfrac{xy}{|x|+y}, & y \neq -|x|, \\[6pt] 0, & \text{otherwise}. \end{cases} \] Then which one of the following statements is TRUE?

The value of the integral $\displaystyle \iint_R xy\,dx\,dy$ over the region $R$, given in the figure, is ___________ (rounded off to the nearest integer).

If $f(x)=\dfrac{\sin x+\cos x}{\sin x-\cos x}$, the value of $f'(x)$ at $x=0$ is ________________.

The position $x(t)$ of a particle, at constant $\omega$, is described by $\dfrac{d^{2}x}{dt^{2}}=-\omega^{2}x$ with initial conditions $x(0)=1$ and $\left.\dfrac{dx}{dt}\right|_{t=0}=0$. The position of the particle at $t=\dfrac{3\pi}{\omega}$ is (in integer).

If a matrix $M$ is defined as $M=\begin{bmatrix}10 & 6 \\ 6 & 10\end{bmatrix}$, the sum of all the eigenvalues of $M^3$ is equal to ________________ (in integer).

The first derivative of the function \[ U(r)=4\left[\left(\frac{1}{r}\right)^{12}-\left(\frac{1}{r}\right)^{6}\right] \] evaluated at $r=1$ is ______________ (in integer).

An exhibition was held in a hall on 15 August 2022 between 3 PM and 4 PM during which any person was allowed to enter only once. Visitors who entered before 3:40 PM exited the hall exactly after 20 minutes from their time of entry. Visitors who entered at or after 3:40 PM, exited exactly at 4 PM. The probability distribution of the arrival time of any visitor is uniform between 3 PM and 4 PM. Two persons $X$ and $Y$ entered the exhibition hall independent of each other. Which one of the following values is the probability that their visits to the exhibition overlapped with each other?

Simpson’s one-third rule is used to estimate the definite integral $I=\displaystyle\int_{-1}^{1}\sqrt{1-x^{2}}\,dx$ with an interval length of $0.5$. Which one of the following is the CORRECT estimate of $I$ obtained using this rule?

Which one of the following is the CORRECT value of $y$, as defined by the expression given below?
\[ y = \lim_{x \to 0} \frac{2x}{e^x - 1} \]

The vector $\vec{v}$ is defined as \[ \vec{v} = zx\,\hat{i} + 2xy\,\hat{j} + 3yz\,\hat{k}. \] Which one of the following is the CORRECT value of divergence of $\vec{v}$, evaluated at the point $(x,y,z) = (3,2,1)$?

For a two-dimensional plane, the unit vectors $(\hat{e}_r,\hat{e}_\theta)$ of the polar coordinate system and $(\hat{i},\hat{j})$ of the cartesian coordinate system are related by \[ \hat{e}_r=\cos\theta\,\hat{i}+\sin\theta\,\hat{j}, \qquad \hat{e}_\theta=-\sin\theta\,\hat{i}+\cos\theta\,\hat{j}. \] Which one of the following is the CORRECT value of $\dfrac{\partial(\hat{e}_r+\hat{e}_\theta){\partial\theta}$?}

Given that $ F=\dfrac{|z_1+z_2|}{|z_1|+|z_2|} $, where $z_1=2+3i$ and $z_2=-2+3i$ with $i=\sqrt{-1}$, which one of the following options is CORRECT?

Ten playing cards numbered $1,\ldots,10$ are drawn with replacement. What is the probability that the second number is greater than the first (rounded to two decimal places)?

If $A=\begin{pmatrix}1 & 2 \\ 3 & 5\end{pmatrix}$, the value of $\left|A^{4}+3A^{2}-5A+6I\right|$ is ________________.

If $f(2)=5$ and $f(x)\,f(x+1)=3$ for all real $x$, the value of $f(10)$ is ________________.

The distance between the two points of intersection of $x^{2}+y=7$ and $x+y=7$ (rounded off to two decimal places) is __________________.