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Two players, A and B, alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player A starts the game, the probability that A wins the game is

If $\varphi (x,y)$ and $\Psi (x,y)$ are functions with continuous second derivatives, then $\varphi (x,y)+i\Psi (x,y)$ can be expressed as an analytic function of $x+iy(i=\sqrt{-1})$ when

The probability of a defective piece being produced in a manufacturing proces is 0.01. The probability that out of 5 successive pieces, only one is defective, is

Area bounded by the curve $y=x^2$ and lines $x=4$ and $y=0$ is given by,

The solution ot the differential equation $f^{\prime \prime }\left(x\right)+4f^{\prime }\left(x\right)+4f\left(x\right)=0$ is

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Changing the order of the integration in the double integral $I={\int }_0^8{\int }_{x/4}^{2}f(x,y)dydx$ leads to $I={\int }_r^s{\int }_p^{q}f(x,y)dxdy$. What is $q$?

Four cards are randomly selected from a pack of 52 cards. If the first two cards are kings, what is the probability that third card is a king?

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The trace and determinant of a 2 x 2 matrix are known to be -2 and -35 respectively. Its eigen values are

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Assuming $i=\sqrt{-1}$ and n is a real number, ${\int }_0^{2\pi /3}{e}^{in}dn$ is

Matrix A is expressed as A = B + C where B and C are respectively symmetric & skew symmetric matrix then matrix B is



Given $A=\left[\begin{array}{ccc}1& 3& 2\\ 2& 4& 1\\ 3& 1& 5\end{array}\right]$

Let $x$ be a continous variable defined over the interval $(-\infty ,\infty )$ and $f\left(x\right)=e^{-x-e^{-x}}$. The integral $g\left(x\right)=\int f\left(x\right)dx$ is equal to

The contour $C$ given below is on the complex plane $z=x+iy$, where $j=\sqrt{-1}$.







The value of the integral $\frac{1}{\pi j}{\oint }_C\frac{dz}{z^2-1}$ is

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Two white and two black balls, kept in two bins, are arranged in four ways as shown below. In each arrangement, a bin has to be chosen randomly and only one bal needs to be picked randomly from the chosen bin. Which one of the following arrangements has the highest probability for getting a white ball picked?

Evaluate the integral ${\int }_0^{\infty }\frac{e^{-x}sinbx}{x}dx$

If $C$ denotes the counterclockwise unit circle, the value of the contour integral $\frac{1}{2\pi j}{\oint }_{C}Rezdz$ is

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If $x^2\frac{dy}{dx}+2xy=\frac{2ln\left(x\right)}{x},$ and $y\left(1\right)=0$, then what is $y\left(e\right)$?

The solution of the following partial differential equation $\frac{{\partial }^{2}u}{\partial x^2}=9\frac{{\partial }^{2}u}{\partial y^2}$ is

The partial differential equation that can be formed from $z=ax+by+ab$ has the form $(\text{with}p=\frac{\partial z}{\partial x}\text{and}q=\frac{\partial z}{\partial y})$

Consider the differential equation $\stackrel{..}{y}+2\stackrel{.}{y}+y=0$ with boundary conditions $y\left(0\right)=1$ & $y\left(1\right)=0$. The value of $y\left(2\right)$ is

The area include between the curve $y^2(2a-x)=x^3$ and its asymptote is

An eigen vector of $P=\left[\begin{array}{ccc}1& 1& 0\\ 0& 2& 2\\ 0& 0& 3\end{array}\right]$ is

The solution for ${\int }_0^{\pi /6}co{s}^{4}3\theta si{n}^{3}6\theta d\theta$ is

An observer counts 240 veh/h at a specific highway location. Assume that the vehicles arrival at the location is Poisson distributed, the probability of having one vehicle arriving over a 30-second time interval is _______.

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A parabolic cable is held between two supports at the same level. The horizontal span between the supports is $L.$ The sag at the mid-span is $h.$ The equation of the parabola is $y=4h\frac{x^2}{L^2},$ where $x$ is the horizontal coordinate and $y$ is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is

The standard deviation of linear dimensions P and Q are $3\mu m\text{and}4\mu m$ respectively. When assembled, the standard deviation $\left(in\mu m\right)$ of the resulting linear dimension (P + Q) is .

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The solution of differential equation:



$\frac{d^{2}y}{d{x}^2}+4\frac{dy}{dx}+6y=3^{x}isy\left(x\right)theny\left(x\right)is$

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A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective?

The probability density function of a random variables x is

$\begin{array}{cc}f\left(x\right)=\frac{x}{4}(4-x^2);& \mathrm{for}0\le x\le 2\\ =0;& \mathrm{otherwise}\end{array}$

The mean ${\mu }_x$ of the random variable is

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A parametric curve defined by $x=cos\left(\frac{\pi u}{2}\right),y=sin\left(\frac{\pi u}{2}\right)$ in the range $0\le u\le 1$ is rotated about the $X-$axis by $360$ degrees. Area of the surface generated is

The probability density function $F\left(x\right)=a{e}^{-b|x|},$ where x is a random variable whose allowable value range is from $x=-\infty tox=+\infty .$ The CDF for this function for $x\ge 0$ is

Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

The solution of the equation $\frac{dQ}{dt}+Q=1$ with $Q=0$ at $t=0$ is

The vlaue of ${\oint }_C\frac{1}{z^2}dz$ where the contour is the unit circle traversed clockwise is

A complex function

$f\left(z\right)=u+iv$ is defined as $f\left(z\right)=\frac{z-i}{z+1}$ where $z=x+iy$ then the imaginary part of f(z) is

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A six - face fair dice is rolled a large number of times. The mean value of the outcomes is .

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If $z=f(u,v),u=x^2-2xy-y^2,v=a$, then

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

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Given that $\stackrel{..}{x}+3x=0$, and $x\left(0\right)=1$, $\stackrel{.}{x}\left(0\right)=0,$ whate is $x\left(1\right)$?

Given the shaded triangular region $P$ shown in the figure. What is $\int \int xydxdy$?