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The product $\left[P\right][Q{]}^T$ of the following two matrices [P] and [Q] is



$\left[P\right]=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$ ; $\left[Q\right]=\left[\begin{array}{cc}4& 8\\ 9& 2\end{array}\right]$

Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$. centred at origin.

$I={\oint }_c\frac{1}{5z-4}dz=A\pi i$

the value of $A$ is

There are 25 calculators in a box. Two of them are defective. suppose 5 calculators are randomly picked for inspection (i.e., each has the same chance of being selected), what is the probability that only one of the defective calculators will be include in the inspection?

A point $z$ has been potted in the complex plane, as shown in figure below







$\frac{1}{Z}$ lies in the curve

Passengers try repeatedly to get a seat reservaton in any train running between two stations until they are successful. If there is 40% chance of getting reservation in any attempt by a passenger then the average number of attempts that passengers need to make to get a seat reserved is

1. 3

Starting from $x_0=1$, one step of Newton-Raphson method in solving the equation $2x^3+5x-8=0$ gives the next value $\left(x_1\right)$ as

A box contains 25 parts of which 10 are defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is

The solution to the differential equation $\frac{d^{2}u}{d{x}^2}-k\frac{du}{dx}=0$ where k is constant, subjected to the boundary conditions $u\left(0\right)=0$ and $u\left(L\right)=U$, is

A complex variable $z=x+y\left(0.1\right)$ has its real part $x$ varying in the range $-\infty$ to $\infty$. Which one of the following is the locus (show in thick lines) of $\frac{1}{z}$ in the complex palne?

1. 3

A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is _____ .

1. 0.265

Let X be a random variable with probability density function

$f\left(x\right)=(\begin{array}{cc}0.2,& \mathrm{for}\left|x\right|\le 1\\ 0.1,& \mathrm{for}1<\left|x\right|\le 4\\ 0,& \mathrm{otherwise}\end{array}$



The probability P(0.5 < X < 5) is .

1. 0.45

An orthogonal set of vectors having a span that contains P, Q, R is , where P = (-10,-1,3)', Q = (-2,-5,9)' & R = (2,-7,12)' are three vectors.

The solution for the following differential equation with boundary conditions $y\left(0\right)=2$ and $y^{\prime }\left(1\right)=-3$ is, $\frac{d^{2}y}{d{x}^2}=3x-2$

If a random variable X has a Poisson distribution with mean 5, then the expectation



$E\left[\right(X+2{)}^2]\text{equals}$.

1. 54

The divergence of the vector field $\overrightarrow{A}=x{\hat{a}}_x+y{\hat{a}}_y+z{\hat{a}}_z$ is

In the following integral, th contour $C$ encloses the points $2\pi j$ and $-2\pi j$.

$-\frac{1}{2\pi }{\oint }_C\frac{sinz}{(z-2\pi j{)}^3}dz$

The value of the integral is

1. 133.87

With reference to the conventional cartesian$(x,y)$ coordinate system, the vertices of a triangle have the following coordinates;

$(x_1,y_1)=(3,2);(x_2,y_2)=(4,4)and(x_3,y_3)=(6,5)$.

The area of the triangle is equal to

The mean of a numerical data set is $\overline{x}$ and the standard deviation is S. if a number P is added to each term in the data set then the mean and standard deviation become.

The value of the integral ${\int }_0^{\pi }xco{s}^{2}xdx$ is

If E denotes expectation,the variance of a random variable X is given by

An integra $I$ over a counterclockwise circle $C$ is given by $I={\oint }_C\frac{z^2-1}{z^2+1}{e}^{z}dz$

If $C$ is defined as $|z|=3$, then the value of $I$ is

If $y$ is the soultion of the differential equation $y^3\frac{dy}{dx}+x^3=0,y\left(0\right)=1,$ the value of $y(-1)$ is

The value of the directional derivative of the $\varphi$ function $(x,y,z)=x{y}^2+y{z}^2+z{x}^2$ at the point $(2,-1,1)$ in the direction of the vector $p=i+2j+2k$ is

For the matrix $\left[\begin{array}{cc}4& 2\\ 2& 4\end{array}\right]$, the eigen value corresponding to the eigen vector $\left[\begin{array}{c}101\\ 101\end{array}\right]$ is

The value of ${\int }_0^{\infty }{\int }_0^{\infty }{e}^{-x^2}.e^{-y^2}dxdy$ is

The countour $C$ in the adjoining figure is described by $x^2+y^2=16$. Then the value of ${\int }_c\frac{z^2+8}{\left(0.5\right)z-\left(1.5\right)j}dz$

Vehicles arriving at an intersection from one of the approach roads follow the Poisson distribution. The mean rate of arrival is 900 vehicles per hour. If a gap is defined as the time difference between two successive vehicle arrivals (with vehicles assumed to be points), the probability (up to four decimal places) that the gap is greater than 8 seconds is

1. 0.1353

Given $X\left(z\right)=\frac{z}{(z-a{)}^2}$ with $|Z|>a$, the residue fo $X\left(z\right)z^{n-1}$ at $z=a$ for $n\ge 0$ will be

The solution of the differential equation is

$\frac{d^{2}y}{d{x}^2}=\frac{dy}{dx}$

Where $C_{1}and{C}_2$ are arbitrary constants.

The probability density function of evaporation E on any day during a year in watershed is given by $f\left(E\right)=(\begin{array}{cc}\frac{1}{5}& 0\le E\le 5\mathrm{mm}/\mathrm{day}\\ 0& \mathrm{otherwise}\end{array}$

The probability that E lies in between 2 and 4 mm/day in a day in watershed is (in decimal)

1. 0.4

It is given that $y^{\prime \prime }+2y^{\prime }+y=0,y\left(0\right)=0,y\left(1\right)=0$. What is $y\left(0.5\right)$?

1. 10

The direction of vector $A$ is radially outward from the origin, with $|A|=K{r}^n$ where $r^2=x^2+y^2+z^2$ and $K$ is constant. The value of $n$ for which $▽.A=0$ is

Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

Two dice are thrown simultaneously. The probability that the sum of numbers on both exceeds 8 is

If A and B are square matrices of size n$\times$n, then which of the following statements is not true.

A bag contains 10 blue marbles, 20 black marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag. This process is repeated 3 times. The probability that no two of the marbles drawn have the same colour is

The probability that a part manufactured by a company will be defective is 0.05. If 15 such parts are selected randomly and inspected, then the probability that at least two parts will be defective is (round off to two decimal places).

1. 0.17

1. 0.26

1. 2

A curve y = $\frac{1}{2\sqrt{x}}$ is allowed to revolve around x axis. The volume of solid of revolution for $3\le x\le 4$ is

A box contains 5 black and 5 red balls. Two balls are randomly picked one after another from the box, without replacement. The probability for both balls being red is

The area enclosed between the straight line $y=x$ and the parabola $y=x^2$ in the $x-y$ plane is

Suppose A and B are two independent events with probabilities $P\left(A\right)\ne 0\text{and}P\left(B\right)\ne 0.$ $\text{Let}\overline{A}\text{and}\overline{B}$ be their complements. Which one of the following statements is FALSE?

Which one of the following differential equations has a solution given by the function y = 5 sin $(3x+\frac{\pi }{3})$

Integrtion of the complex funciton $f\left(z\right)=\frac{z^2}{z^2-1}$, in the counterclockwise direction, around $|z-1|=1$, is

Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation, E[x], is

1. 50