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A and B friends. They decide to meet between 1 PM and 2 PM on a given day. There is a condition that whoever arrives first will not wait for the other for more than 15 minutes. The probability that they will meet on that day is

The probability that there are 53 sundays in a randomly chosen leap year is

Divergence of the three dimensional radial vector field $\overrightarrow{r}$ is

What is curl of the vector field $2x^{2}y\hat{i}+5z^2\hat{j}-4yz\hat{k}?$

The value of the integral ${\int }_{\infty }^{-\infty }\frac{sinx}{x^2+2x+2}dx$ evaluated using contour integration and the residue theorem is

The differential equation $\frac{dy}{dx}+4y=5$ is valied in the domain $0\le x\le 1$ with $y\left(0\right)=2.25$. The solution of the differential equation is

The value of the contour integral

$\frac{1}{2\pi j}\oint {(z+\frac{1}{z})}^{2}dz$

evaluated over the unit circle $|z|=1$ is

1. 0

Given a vector field F, the divergence theorem states that

The three characteristic roots of the following matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 0& 2\end{array}\right]$ are

The value of ${\oint }_C\frac{dz}{(1+z^2)}$ where $C$ is the contour $|z-\frac{1}{2}|=1$ is

Seven car accidents occurred in a week. What is the probability that they all occurred on the same day?

1. 0.99

The eigen values of $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ are

1. 25

Consider the following simultaneous equation (wirh $c_1$ and $c_2$ beings constants):



$3x_1+2x_2=c_1$



$4x_1+x_2=c_2$



The characteristic equation for these simultaneous equations is

For a random variable $x(-\infty <x<\infty )$ following normal distribution, the mean is $\mu =100.$ If the probability is $P=\alpha \text{for}x\ge 110.$ Then the probability of x lying between 90 and 110 i.e. $P(90\le x\le 110)$ and equal to

The probability density function on the interval [a, 1] is given by $1/x^2$ and outside this interval the value of the function is zero. The value of a is

1. 0.5

The solution of the ordinary differential equation $\frac{dy}{dx}+3y=0$ for the boundary condition, $y=4atx=1$ is

The magnitude of the directional derivative of the function $f(x,y)=x^2+3y^2$ in a direction normal to the circle $x^2+y^2=2,$ at the point $(1,1),$ is

1. 5

Which one of the following does NOT equal

$\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|$ ?

Let $z^3=\overline{z}$, wherer $z$ is a complex number not equal to zero. Then $z$ is a solution of

A random variable X has the density function $f\left(x\right)=K\frac{1}{1+x^2},\text{where}-\infty <x<\infty .$ Then the value of K is

The spot speed (expressed in km/hr) observed at a road section are 66, 62, 45, 79, 32, 51, 56, 60, 53 and 49. The median speed expressed in km/hr is .

[Note Answer with one decimal accuracy]

1. 54.5

The solution to the ordinary differential equation $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}-6y=0$ is

Let Z be an exponential random variable with mean 1. That is, the cumulative distribution function of Z is given by

$F_z\left(x\right)=(\begin{array}{cc}1-e^{-x}& \mathrm{if}x\ge 0\\ 0& \mathrm{if}x<0\end{array}$

Then Pr(Z > 2 | Z > 1), rounded off to two decimal places, is equal to .

1. 0.367

The eigen values of the matrix $M$ given below are $15,3$ and $0.M=\left[\begin{array}{ccc}8& -6& 2\\ -6& 7& -4\\ 2& -4& 3\end{array}\right]$

The value of the determinant of the matrix is

1. 2

By a change of variable $x(u,v)=uv,y(u,v)=v/u$ in double integral, the integrand $f(x,y)$ changes to $f(uv,v/u)\varphi (u,v)$. Then, $\varphi (u,v)$ is

The modulus of the complex number is $\left(\frac{6+8i}{1-2i}\right)$

For an analytic function, $f(x+iy)=u(x,y)+iv(x,y),u$ is given by $u=3x^2-3y^2$. Th expression for $v$, considering $K$ to be a constant is

The arrival of customers over fixed time intervals in a bank follow a poisson distribution with an average of 30 customers/hour. The probability that the time between successive customer arrival is between 1 and 3 m minutes is (correct to two decimal places).

1. 0.38

Which of the following integrals is unbounded?

Find the value of $\lambda$ such that the function f(x) is a valid probability density function.

$f\left(x\right)=\lambda (x-1)(2-x)\text{for}1\le x\le 2$

=0 otherwise

1. 6

Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

The Eigen vectors of the matrix

$\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ is/are

1. 4

If Z = f(x,y), dZ is equal to

The value of ${lim}_{x\to 8}\frac{x^{1/3}-2}{(x-8)}$

A random variable X has probability density function f(x) as give below:

$f\left(x\right)=(\begin{array}{c}a+\mathrm{bx}\mathrm{for}0<x<1\\ 0\mathrm{otherwise}\end{array}$



If the expected value E[X] = 2/3, then Pr[X < 0.5] is .

1. 0.25

The value of the integral ${\int }_{-\infty }^{\infty }\frac{dx}{1+x^2}$ is

Suppose X is a normal random variable with mean 0 and variance 4. Then the mean of the absolute value of X is

The standard deviation of a uniformly distributed random variable between 0 and 1 is

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

The solution of differential equation $\frac{d^2\Psi }{d{t}^2}+\frac{6d\Psi }{dt}+13\Psi =0is,given\Psi \left(0\right)=0$ is,

Two dice are thrown. What is the probability that the sum of the numbers on the two dice is eight?

Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is .

1. 1.5

The solution of the differential equation $\frac{dy}{dx}=\left(sin2x\right)y^{1/3}$ satisfying y(0)= 0 is

The value of the integral given below is ${\int }_0^{\pi }{x}^{2}cosxdx$