Maths

Let $\left\{x\right\}$ and $\left[x\right]$ denote the fractional part of $x$ and the greatest integer $\le x$ respectively of $a$ real number $x$ . if ${\int }_0^n \left\{x\right\}dx,{\int }_0^n \left[x\right]dx$ and $10\left(n^2-n\right),(n\in N,n>1)$ are three consecutive terms of a G.P. then $\mathrm{n}$ is equal to(Integration)

If the variance of the following frequency distribution :

$\begin{array}{lllll}\text{Class}& :& 10-20& 20-30& 30-40\\ \text{Frequency}& :& 2& x& 2\end{array}$ is 50 , then $x$ is equal to:

Let $PQ$ be a diameter of the circle $x^2+y^2=9$ . If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ respectively, then the maximum value of $\alpha \beta$ is.

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is

If $\vec{a}=2\hat{i}+\hat{j}+2\hat{k}$ , then the value of $|\hat{i}*(\vec{a}*\hat{i}){|}^2+|\hat{j}*(\vec{a}*\hat{j}){|}^2+|\hat{k}*(\vec{a}*\hat{k}){|}^2$ is equal to

Let $a_1,a_2,\dots a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2$ $+\dots +a_n$ . If $a_1=1,a_1=300$ and $15\le n\le 50$ , then the ordered pair $\left(S_{n-4},a_{n-4}\right)$ is equal to:

If the perpendicular bisector of the line segment joining the points $P\left(\mathrm{1,4}\right)$ and $Q(k,3)$ has $y$ -intercept equal to -4 , then a value of $k$ is:

A water drop of diameter 2 cm is broken into 64 equal droplets. The surface tension of water is 0.075N/m. In this process the gain in surface energy will be

Let ${\bigcup }_{i=1}^{50} X_i={\bigcup }_{i=1}^n Y_i=T$ , where each $X_i$ contains 10 elements and each $Yi$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X_i$ 's and exactly 6 of sets $Y_i$ 's then $n$ is equal to:

Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$ . If $P(1,\beta ),\beta >0$ is a point on this ellipse, then the equation of the normal to it at $P$ is: