GATE EC Engineering Mathematics questions

A simple closed path $C$ in the complex plane is shown in the figure. If $$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$ where $i = \sqrt{-1},$ then the value of $A$ is _________ (rounded off to two decimal places).

The function $f(x) = 8 \log_{e} x – x^{2} + 3$ attains its minimum over the interval $[1, e]$ at $x =$____________. (Here $\log_{e}x$ is the natural logarithm of $x$.)

Let $\alpha, \beta$ be two non-zero real numbers and $v_{1}, v_{2}$ be two non-zero real vectors of size $3 \times 1.$ Suppose that $v_{1}$ and $v_{2}$ satisfy $v_{1}^{T} v_{2} = 0, v_{1}^{T} v_{1} = 1,$ and $v_{2}^{T} v_{2} = 1.$ Let $A$ be the $3 \times 3$ matrix given by: $$A = \alpha v_{1} v_{1}^{T} + \beta v_{2} v_{2}^{T}$$ The …

Consider the following series: $$\sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}}$$ For which of the following combinations of $c, d$ values does this series converge?

A state transition diagram with states $A, B,$ and $C,$ and transition probabilities $p_{1}, p_{2}, \dots, p_{7}$ is shown in the figure (e.g., $\text{p}_{1}$ denotes the probability of transition from state $A$ to $B$). For this state diagram, select the statement(s) which is/are universally true.

The value of the integral $$\iint_{D} 3(x^{2} + y^{2})dxdy,$$ where $D$ is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest integer).

In an electrostatic field, the electric displacement density vector, $\overrightarrow{D},$ is given by $$\overrightarrow{D} (x, y, z) = (x^{3} \overrightarrow{i} + y^{3} \overrightarrow{j} + xy^{2} \overrightarrow{k}) \; \text{C/m}^{2},$$ where $\overrightarrow{i}, \overrightarrow{j}, \overrightarrow{k}$ are the unit vectors along …

The vector function $F\left ( r \right )=-x\hat{i}+y\hat{j}$ is defined over a circular arc $C$ shown in the figure. The line integral of $\int _{C} F\left ( r \right ).dr$ is

Consider the differential equation given below. $$\frac{dy}{dx}+\frac{x}{1-x^{2}}y=x\sqrt{y}$$ The integrating factor of the differential equation is

Two continuous random variables $X$ and $Y$ are related as $$Y=2X+3$$ Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$denote the variances of $X$ and $Y$, respectively. The variances are related as

If the vectors $(1.0,\:-1.0,\:2.0)$, $(7.0,\:3.0,\:x)$ and $(2.0,\:3.0,\:1.0)$ in $\mathbb{R}^{3}$ are linearly dependent, the value of $x$ is __________

Consider the vector field $F\:=\:a_{x}\left ( 4y-c_{1}z \right )+a_y\left ( 4x + 2z\right )+a_{z}\left ( 2y +z\right )$ in a rectangular coordinate system $(x,y,z)$ with unit vectors $a_{x},\:a_{y}$ and $a_{z}$. If the field $F$ is irrotational (conservative), then the constant $c_{1}$ (in integer) is _________________

Consider the integral $$\oint _{c}\frac{sin\left ( x \right )}{x^{2}\left ( x^{2}+4 \right )}dx$$ where $C$ is a counter-clockwise oriented circle defined as $\left | x-i \right |=2$. The value of the integral is

A box contains the following three coins. A fair coin with head on one face and tail on the other face. A coin with heads on both the faces. A coin with tails on both the faces. A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one coin is then picked randomly and tossed. If the first toss results in a head, the …

A real $2\times2$ non-singular matrix $A$ with repeated eigenvalue is given as $$A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}$$ where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________________

For a vector field $D=\rho\cos^{2}\:\varphi \:a_{\rho }+z^{2}\sin^{2}\:\varphi \:a_{\varphi }$ in a cylindrical coordinate system $\left ( \rho ,\varphi ,z \right )$ with unit vectors $a_{\rho },a_{\varphi }$ and $a_{z}$, the net flux of $D$ leaving the closed surface of the cylinder $\left ( \rho =3, 0\leq z\leq 2 \right )$ (rounded off to two …

If $v_{1},v_{2}, \dots ,v_{6}$ are six vectors in $\mathbb{R}^{4}$ , which one of the following statements is $\text{FALSE}$?

For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$?

The partial derivative of the function $$f(x, y, z) = e^{1-x\cos y} + xze^{-1/(1+y^{2})}$$ with respect to $x$ at the point $(1,0,e)$ is

The general solution of $\dfrac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}-6\dfrac{\mathrm{d} y}{\mathrm{d} x}+9y=0$ is