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1. 0.08

1. 0.8119

If $x=\sqrt{-1}$, then the value of $x^x$ is

1. 3

Probability density function of a random variable x is given below:



$f\left(x\right)=(\begin{array}{cc}0.25\mathrm{if}& 1\le x\le 5\\ 0& \mathrm{otherwise}\end{array}$



$P(x\le 4)$ is

1. 0.5

Consider the hemi-spherical tank of radius $13m$ as shown in the figure (not drawn to scale). What is the volume of water $\left(in{m}^3\right)$ when the depth of water at the centre of the tank is $6m?$

In a given day in the rainy season, it may rain 70% of the time. If the rains, chance that a village fair make a loss on that day is 80%. However, if it does not rain, chance that the fair will make a loss on that day is only 10%. If the fair has not made a loss on a given day in the rainy season, what is the probability that it has not rained on that day?

In the given matrix $\left[\begin{array}{ccc}1& -1& 2\\ 0& 1& 0\\ 1& 2& 1\end{array}\right]$, one of the eigen value is 1. The eigen vectors corresponding to the eigen value 1 are

If $x$ is uniformly distributed over (0, 15), the probability that $5<x<9$ is _____

The divergence of the vector field $3xz\hat{i}+2xy\hat{j}-y{z}^2\hat{k}$ at a point $(1,1,1)$ is equal to

1. 1

The value of ${\int }_c\frac{z^2}{z^4-1}dz$, using Cauchy's integral formula, around the circle $|z+1|=1$ where $z=x+iy$ is

From a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be Kings, if first card is NOT replaced?

The double integral ${\int }_0^a{\int }_0^{y}f(x,y)dxdy$ is equivalent to

1. 3

A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till '3' is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is .

1. 6

A two-faced fair coin has its faces designated as head (H) and tail (T). This coin is tossed three times in succession to record the following outcomes: H.H.H. If the coin is tossed one more time, the probability (up to one decimal place) of obtaining H again given the previous realizations of H, H and H. would be

1. 0.5

1. 2

The probability of a resistor being defective is 0.02. There are 50 such aresistors in a circuit. The probability of two or more defective resistors in the circuit (round off to two decimal places) is .

1. 0.26

$\text{Let}{X}_1,X_2,X_3\text{and}{X}_4$ be independent normal random variables with zero mean and unit variance. The probability that $X_4$ is the smallest among the four is .

1. 0.25

1. 5.732

Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is

1. 0.75

Let X be a zero mean unit variance Gaussian random variable. E[|X|] is equal to .

1. 0.7978

1. 4

The contour integral ${\int }_c{e}^{1/z}dz$ with $C$ as the counter clock wise unit circle in the z-plnae is equal to

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

Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is $\mu .$ The standard deviation for this distribution is given by

If a fair coin is tossed 4 times, what is the probability that two heads and two tails will result?

Let X be a real-valued random variable with E[X] and $E\left[X^2\right]$ denoting the mean values of X and $X^2,$ respectively. The relation which always holds true is

Which one of the following is not true for complex number $z_1$ and $z_2$?

Let X be the Poisson random variable with parameter $\lambda =1$, then the probability $P(0\le x\le 2)$ equals

The eigen values of the matrix

$\left[\begin{array}{cc}5& 3\\ 3& -3\end{array}\right]$ are:

For a scalar function $f(x,y,z)=x^2+3y^2+2z^2,$ the gradient at the point $P(1,2,-1)$ is

With K as a constant the possible for the first order differential equation $\frac{dy}{dx}=e^{-3x}$ is

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

If a triangle PQR has vertex points P(2, 0), Q(0, 2) and R(0,0), then the value of integral $\iint 5ydxdy$ evaluated over the triangle is

A fair coins is tossed independently four times. the probability of the event "the number of times heads show up is more than the number of times tails show up" is

The solution at $x=1,t=1$ of the partial differential equation $\frac{{\partial }^{2}u}{\partial x^2}=25\frac{{\partial }^{2}u}{\partial t^2}$ subject to initial conditions of $u\left(0\right)=3x$ and $\frac{\partial u}{\partial t}\left(0\right)=3$ is

if z = xy ln(xy), then

1. 3

1. 0.333

The probability that a screw manufactured by a company is defective is 0.1. The company sells screws in packets containing 5 screws and gives a guarantee of replacement if one or more screws in the packet are found to be defective. The probability that a packet would have to be replaced is .

1. 0.41

If the vector function $\overrightarrow{F}=\widehat{a_x}(3y-k_{1}z)+\widehat{a_y}(k_{2}x-2z)-\widehat{a_z}(k_{3}y+z)$ is irrotational, then the values of the constants $k_1,k_2$ and $k_3$, respectively, are

The curl of the gradient of the scalar field defined by $V=2x^{2}y+3y^{2}z+4z^{2}x$ is

1. 4.71

The value $x$ and $y$ where $\frac{(\cos \theta +i\sin \theta {)}^7}{(\sin \theta +i\cos \theta {)}^4}=x+iy$ is

Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number on the faces showing up is (up to third decimal place)

1. 0.0278

If $z$ is a complex variable, the value of ${\int }_5^{3i}\frac{dz}{z}$ is