GATE EC Engineering Mathematics questions

Consider the complex valued function $f(z)=2z^{3}+b\mid z \mid^{3}$ where $z$ is a complex variable. The value of $b$ for which function $f(z)$ is analytic is _________

As $x$ varies from $-1$ to $+3$, which one of the following describes the behaviour of the function $f(x)=x^{3}-3x^{2}+1?$

How many distinct value of $x$ satisfy the equation $\sin(x)=x/2$, where $x$ is in radians?

Consider the time-varying vector $\textbf{I}=\hat{x}15\cos(\omega t)+\hat{y}5\sin(\omega t)$ in Cartesian coordinates, where $\omega> 0$ is a constant. When the vector magnitude $\mid \textbf{I} \mid$ is at its minimum value, the angle $\theta$ that $\textbf{I}$ makes with the $x$ axis (in degrees, such that $0\leq \theta \leq 180)$ ________

The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input $u(t)$ is $y(t)$. The value of $\frac{dy}{dt}$ at $t=0^{+}$ is _________

A discrete memoryless source has an alphabet $\left \{ a_{1},a_{2}, a_{3},a_{4}\right \}$ with corresponding probabilities $\left \{ \frac{1}{2}, \frac{1}{4},\frac{1}{8},\frac{1}{8}\right \}.$ The minimum required average codeword length in bits to represent this source for error-free reconstruction is _________

The ordinary differential equation $$\frac{dx}{dt}=-3x+2, \text{ with }x(0) = 1$$ is to be solved using the forward Euler method. The largest time step that can be used to solve the equation without making the numerical solution unstable is _________

Suppose $C$ is the closed curve defined as the circle $x^{2}+y^{2}=1$ with $C$ oreinted anti-clockwise. The value of $\oint$ ( $xy^{2}$ $dx$ + $x^{2}y$ $dy$ )over the curve $C$ equals _________

Two random variables $X$ and $Y$ are distributed according to $$f_{X,Y}(x,y)=\begin{cases} (x+y),& 0\leq x\leq 1,&0\leq y\leq 1\\ 0, & \text{otherwise.} \end{cases}$$ The probability $P(X+Y\leq 1)$ is ________

The matrix $A=\begin{bmatrix} a & 0 &3 &7 \\ 2& 5&1 &3 \\ 0& 0& 2 &4 \\ 0&0 & 0 &b \end{bmatrix}$ has $\text{det}(A) = 100$ and $\text{trace}(A) = 14$. The value of $\mid a-b \mid$ is ________

A positive charge $q$ is placed at $x=0$ between two infinte metal plates placed at $x=-d$ and at $x=+d$ respectively. The metal plates lie in the $yz$ plane. The charge is at rest at $t=0$, when a voltage $+V$ is applied to the plate at $-d$ and voltage $-V$ is applied to the plate at $x=+d$. Assume that the quantity of the charge $q$ …

Consider a $2\times2$ sqaure matrix $$\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix},$$ where $x$ is unknown. If the eigen values of the matrix $\textbf{A}$ are $(\sigma + j\omega)$ and $(\sigma - j\omega)$, then $x$ is equal to

For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______

The probability of getting a “head” in a single toss of a biased coin is $0.3$. The coin is tossed repeatedly till a head is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _______

The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______

Consider the first order initial value problem $$y’= y+2x-x^2 ,\ y(0)=1,\ (0 \leq x < \infty)$$ with exact solution $y(x) = x^2 + e^x$. For $x = 0.1$, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runga-Kutta method with step-size $h=0.1$ is _______

The particular solution of the initial value problem given below is $$\frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\hspace{0.3cm} \text{ with } \hspace{0.3cm}y(0)=3\hspace{0.3cm} \text{ and }\hspace{0.3cm} \frac{dy}{dx} \bigg| _{x=0} =-36$$

If the vectors $e_1=(1,0,2)$, $e_2=(0,1,0)$ and $e_3=(-2,0,1)$ form an orthogonal basis of the three-dimensional real space $\mathbb{R}^3$, then the vector $\textbf{u}=(4,3,-3)\in \mathbb{R}^3$ can be expressed as

A triangle in the $xy$-plane is bounded by the straight lines $2x=3y, \: y=0$ and $x=3$. The volume above the triangle and under the plane $x+y+z=6$ is _______

The values of the integral $\large\frac{1}{2\pi j}\oint_c\frac{e^z}{(z-2)} \small dz$ along a closed contour $c$ in anti-clockwise direction for the point $z_0=2$ inside the contour $c$, and the point $z_0=2$ outside the contour $c$, respectively,are