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Potential funtion $\varphi$ is givne as $\varphi =x^2-y^2$. What will be the stream funciton $\left(\Psi \right)$ with the condition $\Psi =0$ at $x=y=0$?

The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is .

1. 1

Consider the differential equaion $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-4y=0$ with the boundary condition of $y\left(0\right)=0$ and $y\left(1\right)=1$. The complete solution of the differential equation is

If $f\left(z\right)=(x^2+a{y}^2)+ibxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then

The solution of the system of equations $x+y+z=4,x-y+z=0,2x+y+z=5$ is

1. 4.17

If $f\left(z\right)=c_0+c_1{Z}^{-1}$, the ${\oint }_{unitcircle}\frac{1+f\left(z\right)}{z}dz$ is given by

To evaluate the double integral ${\int }_0^8\left({\int }_{y/2}^{\left(y/2\right)+1}\left(\frac{2x-y}{2}\right)dx\right)dy,$ we make the substitution $u=\left(\frac{2x-y}{2}\right)$ and $v=\frac{y}{2}.$ The integral will reduce to

The mean square of a zero mean random process is kT/C, where k is Boltzman's constant. T is the absolute temperature, and C is capacitance. The standard deviation of the random.

If a vector field $\overrightarrow{V}$ is related to another field $\overline{A}$ through $\overrightarrow{V}$ = $\nabla \times \overline{A}$, which of the following is true?



Note: C and $S_c$ refer to any closed contour and any surface whose boundary is C.

The life of a bulb (in hours) is a random variable with an exponential distribution $f\left(t\right)=\alpha e^{-\alpha t},0\le t\le \infty .$ The probability that its value lies between 100 and 200 hours is

The complete integral of

$(z-px-qy{)}^3=pq+2(p^2+q{)}^2$ is

The value of the integral ${\oint }_C\frac{z+1}{z^2-4}dz$ in counter clockwise direction around a circle $C$ of radius $1$ with center at the point $z=-2$

The matrix $M=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& 6\\ -1& -2& 0\end{array}\right]$ has eigen values 3, -3, 5. An eigen vector corresponding to the eigen value 5 is $[12-1{]}^T$. One of the eigen vector of the matrix $M^3$ is

Consider two functions: x = $\Psi ln\varphi$ and y = $\varphi$ ln$\Psi$ Which one of the following is the corrent expression for $\frac{\partial \Psi }{\partial x}$ ?

An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the second draw is

1. 0.142

The residue of $f\left(z\right)=\frac{z^3}{(z-1{)}^4(z-2\left)\right(z-3)}$ at $z=3$ is

A box contains 5 black balls and 3 red balls. A total of three balls are picked from the box one after another, without replacing them back. The probability of getting two black balls and one red ball is

1. 85.33

1. 1.2

Let $A=\left[\begin{array}{cc}2& -0.1\\ 0& 3\end{array}\right]$ and $A^{-1}=\left[\begin{array}{cc}\frac{1}{2}& a\\ 0& b\end{array}\right]$. Then $(a+b)=$

The value of the contour integral ${\oint }_{|z-j|=2}\frac{1}{z^2+4}dz$ in positive sense is

Consider the following linear system.

$x+2y-3z=a$

$2x+3y+3z=b$

$5x+9y-6z=c$

The system is consistent if a,b and c safisfy the equation .

What is the area common to the circle $r=a$ and $r=2acos\theta ?$

The value of ${\oint }_C\frac{sinz}{z}dz,$ where the contour of the integration is a simple closed curve around the origin is

1. 0.67

A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is

A scalar field is given by $f=x^{2/3}+y^{2/3},$ where $x$ and $y$ are the Cartesian coordinates. The derivative of ${}^{\prime }{f}^{\prime }$ along the line $y=x$ directed away from the origin at the point (8,8) is

A sphere of unit radius is centred at the origin. The unit normal at a pint $(x,y,z)$ on the surface of the sphere is the vector.

1. 0.185

For $f\left(z\right)=\frac{sin\left(z\right)}{z^2}$, the residue of the pole $z=0$ is

1. 1

If $Z=x+jy$ where $x,y$ are real when the value of $|e^{iz}|$ is

The derivative of $f(x,y)$ at point (1,2) in the direction of vector $i+j$ is $2\sqrt{2}$ and in the direction of the vector $-2j$ is $-3$. Then the derivative of $f(x,y)$ in direction $-i-2j$ is

The input X to the Binary Symmetric Channel (BSC) shown in the figure '1' with probability 0.8.

The crossover probability is 1/7. If the received bit Y = 0,. the conditional probability that '1' was transmitted is .

1. 0.4

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is .

1. 0.25

1. 226.19

Two random variables X and Y are distributed according to



$f_{x,y}(x,y)=(\begin{array}{cc}(x,y),& 0\le x\le 1,0\le y\le 1\\ 0,& \mathrm{otherwise}.\end{array}$



The probability $P(X+Y\le 1)\text{is}$ .

1. 0.33

Consider two events $E_{1}and{E}_2$ such that $P\left(E_1\right)=\frac{1}{2},P\left(E_2\right)=\frac{1}{3}andP(E_1\cap E_2)=\frac{1}{5}.$ Which of the following statements is true?

Consider an unbiased cubic dice with opposite faces coloured identically an each face coloured red, blue or green such that each colour appears only two times on the dice. If the dice is thrown thrice, the probability of obtaining red colour on top face of the dice at least twice is

1. 0.2593

The value of the contour integral in th complex-plane $\oint \frac{z^3-2z+3}{z-2}dz$ along the contour $|z|=3$, taken counter-clockwise is:

A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads?

An urn contains 5 red and 7 green balls. A ball is drawn at random and its colour is noted. The ball is placed back into the urn along with another ball of the same colour. The probability of getting a red ball in the next draw is

A fair coin is tossed N times. The probability that head does not turn up in any of the tosses is