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 $\mathbb{C}$?

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 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).

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?

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:

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).

The state equation of a second order system is $\dot{x}(t)=Ax(t)$, with $x(0)$ as the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $A$ and $v_1$ and $v_2$ are the corresponding eigenvectors. For constants $\alpha_1$ and $\alpha_2$, the solution $x(t)$ of the state equation is _____________

Let $\mathbf{x}$ be an $n \times 1$ real column vector with length $l=\sqrt{\mathbf{x}^T\mathbf{x}}$. The trace of the matrix $P=\mathbf{x}\mathbf{x}^T$ is _____________

The value of the line integral ∫ from P to Q [ z² dx + 3y² dy + 2xz dz ] along the straight line joining the points P(1,1,2) and Q(2,3,1) is _____________

A random variable $X$, distributed normally as $N(0,1)$, undergoes the transformation $Y = h(X)$, given in the figure. The form of the probability density function of $Y$ is (In the options given below, $a, b, c$ are non-zero constants and $g(y)$ is a piece-wise continuous function)

Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\{\lambda_k \mid 1 \le k \le n\}$ and $\{\mathbf{v}_k \mid 1 \le k \le n\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1}AP$, respectively, are

The value of the contour integral, $\displaystyle \oint_{C}\left(\frac{z+2}{z^{2}+2z+2}\right)dz$, where the contour $C$ is $\left\{ z:\; \left| z+1-\tfrac{3}{2}j \right| = 1 \right\}$, taken in the counter clockwise direction, is

Let $\mathbf{v}_1=\begin{bmatrix}1 \\ 2 \\ 0\end{bmatrix}$ and $\mathbf{v}_2=\begin{bmatrix}2 \\ 1 \\ 3\end{bmatrix}$ be two vectors. The value of the coefficient $\alpha$ in the expression $\mathbf{v}_1=\alpha \mathbf{v}_2+\mathbf{e}$, which minimizes the length of the error vector $\mathbf{e}$, is

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strategy to identify these covid-positive individuals. The strategy is to:
(i) Collect saliva samples from all 1000 individuals and randomly group them into sets of 5.
(ii) Mix the samples within each set and test the mixed sample for covid.
(iii) If the test done in (ii) gives a negative result, then declare all the 5 individuals to be covid negative.
(iv) If the test done in (ii) gives a positive result, then all the 5 individuals are separately tested for covid.
Given this strategy, no more than _____________testing kits will be required to identify all the 100 covid positive individuals irrespective of how they are grouped.

Let $w^4 = 16j$. Which of the following cannot be a value of $w$?

The rate of increase, of a scalar field $f(x,y,z) = xyz$, in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is

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).

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?

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?