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The value of $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^n{\sin }^{2k}\left(\frac{r\pi }{2n}\right),$ where $k$ is a non-negative integer, is equal to

Each of $2010$ boxes in a line contains one red marble, and for $1\le k\le 2010,$ the box in the $k^{\text{th}}$ position also contains $k$ white marbles. Ram begins at the first box and successively draws a single marble at random from each box, in order. He stops when he first draws a red marble. Let $P\left(n\right)$ be the probability that he stops after drawing exactly $n$ marbles. Then the possible value(s) of $n$ for which $P\left(n\right)<\frac{1}{2010},$ is

The number of arrangements of the letters of the word $BANANA$ in which two $N^{\prime }s$ do not appear adjacently is :

Which of the following should be the FOURTH sentence after rearrangement?

What is mean by (-infinity, infinity)?

The cost of two red balls and five blue balls is ₹160. Then find the correct equation for this situation among the given options.

$(r:\text{price of one red ball})$

$(b:\text{price of one blue ball})$

The integral value of $x$, that satisfies $1<{\log }_2(x-2)\le 2$, is

A normal chord of the parabola $y^2=4x$ subtending a right angle at the vertex makes an acute angle $\theta$ with the $x-$ axis the angle $\theta$ is equal to

If $z=\frac{1+2i}{1-(1-i{)}^2}$, then arg (z) equals

Solve
system of linear equations, using matrix method.

2x
+ 3y + 3z = 5

x −
2y + z = −4

3x
− y − 2z = 3

Complete set of values of $x$, satisfying the in equality $x^2+\frac{x^2}{(x+1{)}^2}<\frac{5}{4},$ is

If a straight line parallel to the line $y=\sqrt{3}x$ passes through $Q(2,3)$ and cuts the line $2x+4y-27=0$ at $P$, then the length of $PQ$ is (units)

If the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3y-\alpha z+\beta =0$. Then $(\alpha ,\beta )$ equals

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

Find the shortest
distance between the lines

and

The angle between the line $\frac{x-1}{1}=\frac{y+2}{1}=\frac{z-4}{0}$ and the plane $y+z+2=0$ is

In the expansion of ${(3\sqrt{\frac{a}{b}}+3\sqrt{\frac{b}{\sqrt{a}}})}^{21}$ the terms cantaining same powers of a and b is

Find the equation of the line of intersection of planes 4x + 4y – 5z = 12 and 8x + 12y – 13z = 32 in the symmetric form.

A loop transfer function is given by

$G\left(s\right)H\left(s\right)=\frac{K(s+2)}{s^2(s+10)}$

The point of intersection of the asymptotes of G(s)H(s) on the real axis in the s-plane is at

1. -4

The angle between the vectors $\left(\hat{i}+\hat{j}\right)$ and $\left(\hat{j}+\hat{k}\right)$ is

Water is filled into a right inverted conical tank at a constant rate of $3{\text{m}}^3\text{/sec},$ whose semi vertical angle is ${\cos }^{-1}\frac{4}{5}.$ The rate $\left(\text{in}\text{m}\text{/sec}\right),$ at which the level of water is rising at the instant when the depth of water in the tank is $4\text{m},$ is

Prove that the line through A(0,-1,-1) and B(4,5,1) intersects the line through C(3,9,4) and D(-4,4,4).

Find the derivative of y = x^x

f(x) = 11-7sinx.

Find the range

If in a $△ABC,$ the incircle passing through the point of intersection of perpendicular bisector of sides $BC,AB.$ Then $4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$ is equal to

$\underset{x\to \infty }{lim}\frac{sinx}{x}=$

Solve
the equation for x,
y, z
and t if

An interference is observed due to two coherent sources ‘A’ and ‘B’ having phase constant zero separated by a distance $4\lambda$ along the y-axis where $\lambda$ is the wavelength of the source. A detector D is moved on the positive x-axis. The number of points on the x-axis excluding the points $x=0$ and $x=\infty$ at which maxima will be observed is