GATE EC Engineering Mathematics questions

The random variable $$Y=\int_{-\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt, \text{ where } \phi \left ( t \right )=\begin{cases} 1; & 5\leq t\leq 7 &\\ 0; & \text{otherwise} \end{cases}$$ and $W(t)$ is a real white Gaussian noise process with two-sided power spectral density $S_{W}\left ( f \right )=3 W/Hz$, for all $f$. …

The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.

Consider the following system of linear equations. $\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$ Which one of the following conditions ensures that a solution exists for the above system?

Which one of the following options contains two solutions of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\left ( y-1 \right )x?$

For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.

$X$ is a random variable with uniform probability density function in the interval $[-2,\:10]$. For $Y=2X-6$, the conditional probability $P\left ( Y\leq 7\mid X\geq 5 \right )$ (rounded off to three decimal places) is __________.

Which one of the following functions is analytic over the entire complex plane?

The families of curves represented by the solution of the equation $$\frac{dy}{dx}=\: – \left(\frac{x}{y} \right)^n$$ for $n=-1$ and $n= +1,$ respectively, are

The value of the contour integral $$\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$$ evaluated over the unit circle $\mid z \mid=1$ is_______.

The number of distinct eigenvalues of the matrix $$A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$$ is equal to ____________.

If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.

The value of the integral $\displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.

Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $$F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.

Consider a differentiable function $f(x)$ on the set of real numbers, such that $f(-1)=0$ and $\mid f’(x) \mid \leq 2.$ Given these conditions, which one of the following inequalities is necessarily true for all $x \in[-2,2]?$

Consider the line integral $$\int_{c} (xdy-ydx)$$ the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ rectangle and a semi-circle of radius $1$. The line integral evaluates to

Consider the homogenous ordinary differential equation $$x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+3y=0, \quad x>0$$ with $y(x)$ as a general solution. Given that $$y(1)=1 \quad \text{and} \quad y(2)=14$$ the value of $y(1.5),$ rounded off to two decimal places, is________.

A random variable $X$ takes values $-1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is independent of $X,$ and is uniformly distributed over the interval $[-2,2].$ The receiver makes a …

Let the input be $u$ and the output be $y$ of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:

Consider $p(s)=s^{3}+ a_{2}s^{2}+a_{1}s+a_{0}$ with all real coefficients. It is known that its derivatives ${p}'(s)$ has no real roots. The number of real roots of $p(s)$ is

Let $\text{M}$ be a real $4\times 4$ matrix. Consider the following statements: $S1: M$ has $4$ linearly independent eigenvectors. $S2: M$ has $4$ distinct eigenvalues. $S3: M$ is non-singular (invertible). Which one among the following is TRUE?