GATE EC Engineering Mathematics questions

A surface is given by $z^{2}=2 x^{2}-y^{2}$ and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P}=\hat{\boldsymbol{\imath}}+\sqrt{2} \widehat{\boldsymbol{k}}$. Which of the following vectors can be $\vec{n}$, where $\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{\jmath}}$ and $\widehat{\boldsymbol{k}}$ are the …

Consider the matrix $\mathrm{M}=\left[\begin{array}{ccc}2 & 1 & 1 \\ 1 & 3 & 0 \\ -1 & a & b\end{array}\right]$. Which of the following options is/ are TRUE if $\operatorname{det}(\mathrm{M}) \neq 0 ?$

Consider the two series, $S_{A}$ and $S_{B}$, where $$\begin{array}{l} S_{A}=\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}} \\ S_{B}=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdots \end{array}$$ Which of the following statements is correct for the two given series?

Let $X, N, Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between $-1$ and $1 ; N$ follows Normal distribution with zero mean and unity variance. $Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^{2}+N$. Which of the following pairs represents the values of correlation …

Consider the square region $R$ in the $X-Y$ plane as shown with the dark shading in the Figure. The value of $\iint_{R}\left(x^{2}+y^{2}-1\right) d x d y$ is $\_\_\_\_$ .(rounded off to two decimal places)

Consider the matrix $A$ below: $$A=\left[\begin{array}{llll} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{array}\right]$$ For which of the following combinations of $\alpha, \beta$, and $\gamma$, is the rank of $A$ at least three? Only (i), (iii), and (iv) Only …

Consider the following series: $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ $\sum_{n=1}^{\infty} \frac{1}{n!}$ Choose the correct option.

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$, defined as $$f(x)=2 x^{3}-3 x^{2}-12 x+1$$ Which of the following statements is/are correct? (Here, $\mathbb{R}$ is the set of real numbers.)

The function $y(t)$ satisfies $$t^{2} y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0,$$ where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is $\_\_\_\_\_\_\_$ (rounded off to two decimal …

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable independent of $S$. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$. Suppose the probability of symbol estimation error $P_{e}$ is …

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval. Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality …

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)=\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<1 / 2$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

Two fair dice (with faces labeled $1,2,3,4,5$, and $6$) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is __________ (rounded off to two decimal places).

Consider the vectors $$\boldsymbol{a}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right]$$ For real-valued scalar variable $x$, the value of $$\min _{x}\|\boldsymbol{a} x-\boldsymbol{b}\|_{2}$$ is $\_\_\_\_\_$ (rounded off to two decimal places). $\|\cdot\|_{2}$ denotes the …

The general form of the complementary function of a differential equation is given by $y(t)=(A t+B) e^{-2 t}$, where $A$ and $B$ are real constants determined by the initial condition. The corresponding differential equation is

​​​​​In the context of Bode magnitude plots, $40 \mathrm{~dB} /$ decade is the same as $\_\_\_\_\_\_.$

​​​Let $\hat{i}$ and $\hat{j}$ be the unit vectors along $x$ and $y$ axes, respectively and let $A$ be a positive constant. Which one of the following statements is true for the vector fields $\overrightarrow{F_{1}}=A(\hat{i} y+\hat{j} x)$ and $\overrightarrow{F_{2}}=A(\hat{i} y-\hat{j} x) ?$ .

​​​​​Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho}{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$ Which of the following …