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The probability density function of a random variable X is $P_x\left(x\right)=e^{-x}\text{for}x\ge 0\text{and}0\text{otherwise.}$ The expected value of the function $g_x\left(x\right)=e^{3x/4}\text{is}$ .

1. 4

1. 0.9

The value of ${\oint }_r\frac{3z-5}{(z-1)(z-2)}dz$ along a closed path $\Gamma$ is equal ot $4\pi i$, where $z=x+iy$ and $i=\sqrt{-1}$.

The correct $\Gamma$ is

The differential equation $\frac{dy}{dx}=0.75y^2$ is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?

In an experiment, positive and negative value are equally likely to occur. The probability of obtaining at most one negative value in five trials is

The solution of the differential equation $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=0$ is

The value of the integral ${\int }_0^2\frac{(x-1{)}^{2}sin(x-1)}{(x-1{)}^2+cos(x-1)}dx$ is

1. 1

The solution of the differential equation $\sec x\frac{dy}{dx}-y=\sin x$ is given by

If A = $\left[\begin{array}{cc}1& 5\\ 6& 2\end{array}\right]$ and B = $\left[\begin{array}{cc}3& 7\\ 8& 4\end{array}\right]$, then $A{B}^T$ is equal to

1. 0.667

The value of the integral ${\int }_0^{2\pi }\left(\frac{3}{9+si{n}^2\theta }\right)d\theta$ is

Let X be a Gaussian random variable mean 0 and variance ${\sigma }^2.$ Let Y = max(X, 0) where max (a, b) is the maximum of a and b. Th emedian of Y is

1. 0

Four fair coins are tossed simultaneously. The probability that at least one head and at least one tail turn up is

Evaluate $\int \frac{dz}{zsinz},$ which $c$ is $x^2+y^2=1$

If $C$ is a cirlce $|z|=4$ and $f\left(z\right)=\frac{z^2}{(z^2-3z+2{)}^2}$ then $\oint f\left(z\right)dz$ is

Assuming $i=\sqrt{1}$ and $t$ is a real number, ${\int }_0^{\pi /3}{e}^{it}dt$ is.

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

Cancidates were asked to come to an interview with 3 pens each. Black, Blue, greena nd red were the permitted pen colours that the candidate could bring. The probability that a candidate comes with all 3 pens having the same colour is .

1. 0.2

What is the chance that a leap year, selected at random, will contain 53 Saturdays?

If $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 0& 0& 1\end{array}\right]$, then $det\left(A^{-1}\right)$ is ________. (correct to two decimal places)

1. 0.25

The value of the integral ${\oint }_C\frac{-3z+4}{(z^2+4z+5)}dz$ wherer $c$ is the circle $|Z|=1$ is given by

If a complex number $\omega$ satisfies the equation ${\omega }^3=1$ then value of $1+\omega +\frac{1}{\omega }$ is

Consider the random process



X(t) = U + Vt.



where U is a zero-mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are statistically independent. The mean value of the random process at t = 2 is

1. 2

Let $X_1,X_2\text{and}{X}_3$ be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability $P(X_1+X_2\le X_3)$ is the largest} is .

1. 0.16

1. -1

If $g\left(x\right)=1-x$ and $h\left(x\right)=\frac{x}{x-1}$ then $\frac{g\left(h\right(x\left)\right)}{h\left(g\right(x\left)\right)}$ is

For the function $\varphi =a{x}^{2}y-y^3$ to represent the velocity potential of an ideal fluid, ${▽}^2\varphi$ should be equal to zero. In that case, the value of ${}^{\prime }{a}^{\prime }$ has to be

If $\overrightarrow{V}$ is a differentiable vector function and $f$ is a sufficient differentiable scalar function, then curl $\left(f\overrightarrow{V}\right)$ is equal to

If a matrix $A=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$ and matrix $B=\left[\begin{array}{cc}4& 6\\ 5& 9\end{array}\right]$, then the transpose of product of these two matrices I.e., $(AB{)}^T$ is equal to

A one-dimensional domain is discretized into $N$ subdomains of width $\Delta x$ with node numbers $i=0,1,2,3,...,N$. If the time scale is discretized in steps of $\Delta t,$ the forward-time and centered-space finite difference approximation at $i^{th}$ node and $n^{th}$ time step, for the partial differential equation $\frac{\partial v}{\partial t}=\beta \frac{{\partial }^{2}v}{\partial x^2}$ is

If $f(x+iy)=x^3-3x{y}^2+i\varphi (x,y)$ where $i=\sqrt{-1}$ and $f(x+iy)$ is an analytic fucntion then $\varphi (x,y)$ is

Consider a differential function $f\left(x\right)$ on the set of real numbers such that $f(-1)=0$ and $|f^{\prime }\left(x\right)|\le 2$. Given these conditions, which one of the following inequalties is necessarily ture for all $xϵ[-2,2]$?

$\text{Let}{X}_1,X_2\text{and}{X}_3$ be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability $p\left\{x_1\text{is the largest}\right\}\text{is}$

1. 0.33

The real symmetric matric C corresponding to the quadratic form Q = $4x_1{x}_2-5x_2{x}_2$ is

A bag contains 7 red and 4 white balls. Two balls are drawn at random. What is the probability that both the balls are red?

The inverse of the $2\times 2$ matrix $\left[\begin{array}{cc}2& 3\\ 6& 8\end{array}\right]$ is

1. 2

1. 1

Cayley Hamilton theorem states that a square matrix satisfies its own characterstic equation. Consider a matrix, The relation satisfied by the matrix $A$ is

$A=\left[\begin{array}{cc}3& 1\\ -2& 0\end{array}\right]$

Two matrices A and B are given below :



$A=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$ $B=\left[\begin{array}{cc}{p}^2+q^2& pr+qs\\ pr+qs& r^2+s^2\end{array}\right]$



If the rank of matrix A is N, then the rank of matrix B is

The maximum value of 'a' such that the matrix $\left[\begin{array}{ccc}-3& 0& -2\\ 1& -1& 0\\ 0& a& -2\end{array}\right]$ has three linearly independent real eigen vectors is

For a complex number $z$, ${lim}_{z\to i}\frac{z^2+1}{z^3+2z-i(z^2+2)}$ is

Consider the two-dimensional velocity field given by $\overrightarrow{V}=(5+a_{1}x+b_{1}y)\hat{i}+(4+a_{2}x+b_{2}y)\hat{j}$. where $a_1,b_1,a_2$ and $b_2$ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is

A function $n\left(x\right)$ satisfies the differential equaation $\frac{d^{2}n\left(x\right)}{d{x}^2}-\frac{n\left(x\right)}{L^2}=0$ where $L$ is a constant. The boundary conditions are: $n\left(0\right)=K$ and $n\left(\infty \right)=0$. The soluiton to this equation is

The value of the integral ${\int }_0^2{\int }_0^x{e}^{x+y}dydx$ is

The vector $\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$ is an eigen vector of $A=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]$. The corresponding eigen value of $A$ is _____.

Let $j=\sqrt{-1}$. Then one value of $j^j$ is