GATE EC Engineering Mathematics questions

If the characteristic equation of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}+2\alpha \frac{\mathrm{d}y}{\mathrm{d} x}+y= 0$$ has two equal roots, then the value of $\alpha$ are

The capacity of a band-limited additive white Gaussian noise (AWGN) channel is given by $C = W \log_{2}\left ( 1+\frac{p} {\sigma ^{2}w} \right )$ bits per second (bps), where $W$ is the channel bandwidth, $P$ is the average power received and $\sigma ^{2}$ is the one-sided power spectral density of the AWGN. For a fixed …

The system of linear equations $\begin{pmatrix} 2 & 1 & 3\\ 3&0 &1 \\ 1& 2 &5 \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix} = \begin{pmatrix} 5\\ -4\\ 14 \end{pmatrix}$ has

The real part of an analytic function $f(z)$ where $z = x + jy$ is given by $e^{-y} \cos(x)$. The imaginary part of $f(z)$ is

The maximum value of the determinant among all $2 \times 2$ real symmetric matrices with trace $14$ is __________.

If $\overrightarrow {r}= x\hat{a_{x}}+y\hat{a_{y}}+z\hat{a_{z}}$ and $\mid \overrightarrow{r} \mid= r$ , then $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .

The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.

The input to a $1$ – bit quantizer is a random variable $X$ with pdf $f_{X}( x )= 2e^{-2x}$ for $x\geq 0$ and $f_{X} (x )= 0$ for $x< 0$. For outputs to be of equal probability, the quantizer threshold should be ______.

The maximum value of the function $f(x) = \text{ln } (1+x) – x$ (where $x >-1$) occurs at $x=$_______.

Which $ONE$ of the following is a linear non-homogeneous differential equation, where $x$ and $y$ are the independent and dependent variables respectively?

Match the application to appropriate numerical method. …

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

If $z= xy \text{ ln} (xy)$, then

The maximum value of $f(x)$= $2x^{3}$ – $9x^{2}$ + $12x – 3$ in the interval $0\leq x\leq 3$ is _______.

Which one of the following statements is NOT true for a square matrix $A$?

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is _______ .

Let $X_{1},X_{2},$ and $X_{3}$ be independent and identically distributed random variables with the uniform distribution on $[0,1]$. The probability $P\left \{ X_{1}+X_{2}\leq X_{3}\right \}$ is _________.

The state equation of a second-order linear system is given by $$\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0}$$ For $x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix},$ $x(t)= \begin{bmatrix} e^{-t}\\ -e^{-t} \end{bmatrix},$ and for $x_{0}= \begin{bmatrix} 0\\ 1 \end{bmatrix},$ $x(t) \begin{bmatrix} e^{-t}-e^{-2t}\\ -e^{-t}+2e^{-2t} \end{bmatrix}$. When …

A binary random variable $X$ takes the value of $1$ with probability $1/3$. $X$ is input to a cascade of $2$ independent identical binary symmetric channels (BSCs) each with crossover probability $1/2$. The output of BSCs are the random variables $Y_{1}$ and $Y_{2}$ as shown in the figure. The value of $H( Y_{1} )+H( Y_{2} )$ in bits is …

Given the vector $\textbf{A}= ( \cos x ) ( \sin y )\hat{a_{x}}+( \sin x )( \cos y )\hat{a_{y}},$ where $\hat{a_{x}},$ $\hat{a_{y}}$ denote unit vectors along $x$, $y$ directions, respectively. The magnitude of curl of $\textbf{A}$ is __________