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Using Cauchy's integral formula, the value of the integral (integration beign taken in counter clockwise direction) $\oint \frac{z^3-6}{3z-i}dz$ is within the unit circle

The directional derivative of $f(x,y,z)=2x^2+3y^2+z^2$ at the point $P(2,1,3)$ in the direction of the vector $\overrightarrow{a}=\hat{i}-2\hat{k}$ is

The angle between two unit-magnitude coplanar vectors P(0.866, 0.500, 0) and Q (0.259, 0.966, 0) will be

Let P(E) denote the probability of an event E. Given $P\left(A\right)=1,P\left(B\right)=\frac{1}{2}$, the values of P(A/B) and P(B/A) respectively are

Let $i$ and $j$ be the unit vectors in the $x$ and $y$ directions, respectivley. For the function $F(x,y)=x^3+y^2,$ the gradient of the function $i.e.,▽F$ is given by

Each of the nine words in the sentence, "The Quick brown fox jumps over the lazy dog" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is .

(The answer should be rounded to one decimal place)

1. 3.88

A square matrix $B$ is skew-symmetric if

Let A be an invertible matrix and suppose that the inverse of A is $\left[\begin{array}{cc}-1& 2\\ 4& -7\end{array}\right]$. Then the matrix A is

1. 0.468

If $W=\varphi +i\Psi$ represents the complex potential for an eelectric field.

Given $\Psi =x^2-y^2+\frac{x}{x^2+y^2}$, then the function $\varphi$ is

P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}.$ The probability that P applies for the job given that Q applies for the job is $\frac{1}{2},$ and the probability that Q applies for the job given that P applies for the job is $\frac{1}{3}.$ Then the probability that P does not apply for the job given that Q does not apply for the job is

1. 2

The chance of a student passing an exam is 20%. The chance of a student passing the exam and getting above 90% in it is 5%. Given that a student passes the examination, the probability that the student gets above 90% marks is

The value of ${\int }_0^{\pi }\frac{dx}{c+dcosx}(whenc>0,\left|d\right|<c)$ is

Let $\varphi$ be an arbitrary smooth real valued scalar function and $\overrightarrow{V}$ be an arbitrary smooth vector valued function in a three-dimensional space. Which one of the following is an identify?

A product is an assemble of 5 different components. The product can be sequentially assembled in two possible ways. If the 5 components are placed in a box and these are drawn at random fromt he box, then the probability of getting a correct sequence is

Given $f\left(z\right)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the $z-$ plane such that $|z+1|=1$, the value of $\frac{1}{2\pi j}{\oint }_{C}f\left(z\right)dz$ is

The general solution of the differential equation $(D^2-4D+4)y=0$ is of the form $(givenD=\frac{d}{dx}and{C}_1,C_{2}areconstants)$

1. 0.25

The figure shows the plot of y as a function of x. The function shown is the solution of the differential equation (assuming all initial condition to be zero) is

The figures show diagramatic representations of vector fields, $\overrightarrow{X},\overrightarrow{Y}$ and $\overrightarrow{Z}$, respectively. Which one of the following choices is true?

The value of the integral $\frac{1}{2\pi j}{\oint }_C\frac{z^2+1}{z^2-1}dz$ where $z$ is a complex number and $C$ is a unit circle with center at $1+0j$ in the complex plane is

1. 1

The solution of $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+17y=0$; $y\left(0\right)=1$,$\frac{dy}{dx}\left(\frac{\pi }{4}\right)=0$ in the range $0<x<\frac{\pi }{4}$ is given by

The value of ${\int }_0^3{\int }_0^x(6-x-y)dxdy$ is

For the equation $\frac{dy}{dx}+7x^{2}y=0$, if $y\left(0\right)=3/7$, then the value of $y\left(1\right)$ is

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var(Y). where var(.) denotes the variance, equals:

Probability (up to one decimal place) of consecutively picking 3 red balls without replacement from a box containing 5 red balls and 1 white ball is

1. 0.5

The directional derivative of the following function at $(1,2)$ in the direction $(4\hat{i}+3\hat{j})$ is: $f(x,y)=x^2+y^2$

The inverse Laplace transform of $F\left(s\right)=\frac{s+3}{s^2+2s+1}$ for $t\ge 0$ is

The integral $\frac{1}{2\pi }{\int }_0^{2\pi }sin(t-\tau )cos\tau d\tau$ equals

If $T(x,y,z)=x^2+y^2+2z^2$ defines the temperature at any location $(x,y,z)$ then the magnitude of the temperature gradient at point $P(1,1,1)$ is

If a vector $\overrightarrow{R}\left(t\right)$ has a constant magnitude then

Let $▽.\left(f\overrightarrow{v}\right)=x^{2}y+y^{2}z+z^{2}x,$ where $f$ and $v$ are scalar and vector fields respectively. If $\overrightarrow{v}=y\hat{i}+z\hat{j}+x\hat{k},$ then $\overrightarrow{v}.▽f$ is

The variance of the random variable X with probability density function $f\left(x\right)=\frac{1}{2}|x|e^{-|x|}$ is .

1. 6

Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes is equally likely at both places. He can afford up to 13 minutes of total waiting time if he is to arrive at D on time. What is the probability that Manish will arrive late at D?

If a random variable X satisfies the Poisson's distribution with a mean value of 2, then the probability that X > 2 is

If $C$ is a circle of radius $r$ with centre $z_0$ in the complex $z-$plane and if n is a non-zero integer, then

${\oint }_C\frac{dz}{(z-z_o{)}^{n+1}}$ equals

The value of the integral of the function $g(x,y)=4x^3+10y^4$ along the straight line segment from the point $(0,0)$ to the point $(1,2)$ in the $xy$ plane is

The soluitons of the differential equations $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+2y=0$ are

An ordinary differential equation is given below:

$\left(\frac{dy}{dx}\right)\left(xlnx\right)=y$

The solution for the above equation is

(Note: $K$ denotes a constant in the options)

Consider the following definite integral:

$I={\int }_0^1\frac{(si{n}^{-1}x{)}^2}{\sqrt{1-x^2}}dx$

The value of the integral is

Let U and V be two independent zero mean Gaussian random variables of variances $\frac{1}{4}\text{and}\frac{1}{9}$ respectively. The probability $P(3V\ge 2U)\text{is}$

A box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set contains one red ball and two black balls is

Consider the following complex function:

$f\left(z\right)=\frac{9}{(z-1)(z+1{)}^2}$

Which of the following is one of hte residues of the above function?

The value of the integral $I=\frac{1}{\sqrt{2\pi }}{\int }_0^{\infty }exp(-\frac{x^2}{8})dx$ is

1. 11.9