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Which of the following defines mean deviation?

If $x^3-x^2+5x-1=0$ has roots $\alpha ,\beta ,\gamma$ and $x^3+a{x}^2+bx+c=0$ has roots $\alpha \beta ,\beta \gamma ,\gamma \alpha$, then the value of $(a+b+c)$ is

Let $z=x+iy$ be a complex number where $x$ and $y$ are intergers. The area of the rectangle whose vertices are the roots of the equation $\overline{z}{z}^3+z{\overline{z}}^3=350$ is

For all the values of $A,B,C$ and $P,Q,R$, the value of $\left|\begin{array}{ccc}\cos (A-P)& \cos (A-Q)& \cos (A-R)\\ \cos (B-P)& \cos (B-Q)& \cos (B-R)\\ \cos (C-P)& \cos (C-Q)& \cos (C-R)\end{array}\right|$ is:

In any $\Delta ABC,$ prove that

$(b^2-c^2)cotA+(c^2-a^2)cotB+(A^2-B^2)cotC=0$

A uniform square of side $2l$ is divided into four squares. If one them is cut off (represented by shaded region in figure) then the position of centre of mass of the remaining portion from the origin is

If $2^{x+y}=6^y$, then the value of $\frac{x}{y}$ is

The value of $\cos [{\sin }^{-1}\left(\frac{-4}{5}\right)-{\cos }^{-1}\left(\frac{4}{5}\right)]=$

$\text{If}\left|\begin{array}{ccc}ka& kb& kc\\ kd& ke& kf\\ kg& kh& ki\end{array}\right|=A\text{and}$

$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=B$

Then the value of $A/B$ is

Integrate the function.
$\int \sqrt{1-4x^2}dx.$

Orthogonal trajectories of family of the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}},$ where $a$ is any arbitrary constant, is:

(where $c$ is integration constant)

If y = $\sqrt{\left(\frac{1+sinx}{1-sinx}\right),}\frac{\pi }{2}<x<\pi$, then $\frac{dy}{dx}$ equals

If $p\equiv$ "It rains today", $q\equiv$ "I go to school", $r\equiv$ "I shall meet my friends"

What can be concluded from "I will go to school and meet my friends if it doesn't rain today"?

The solution set of the equation $2{\sin }^{2}x+\sqrt{3}\cos x+1=0$ is $\{2n\pi \pm \frac{a\pi }{b},n\in Z;\frac{a}{b}\in [0,1\left]\right\}$ where $a,b$ are co-prime. Then the value of $a+b$ is

1. -1.25

Prove that $\sqrt{\frac{1+\text{sin}x}{1-\text{sin}x}}=\text{tan}(\frac{\pi }{4}+\frac{x}{2})$

The differential equation by eliminating the arbitary constants from the equation $y=$ $a\cos x+b\sin x+x\sin x,$ where $y_1=\frac{dy}{dx},y_2=\frac{d^{2}y}{d{x}^2}$ is

An $n\times n$ array v is defined as follows:



$v[i,j]=i-jforalli,j$

$1\le i\le n,1\le j\le n$



The sum of the elements of the array v is

Let,
show that,
where I is
the identity matrix of order 2 and n
∈
N

Convert the given complex number in polar form: 1 – i

If variance of the first $n$ natural numbers is $10,$ then the value of $n$ is

The distance between the two lines represented by the equation $9x^2-24xy+16y^2-12x+16y-12=0$ is __ units

The value of ${\sin }^4\frac{\pi }{8}+{\sin }^4\frac{3\pi }{8}+{\sin }^4\frac{5\pi }{8}+{\sin }^4\frac{7\pi }{8}$ is

In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees that each section of each class will plant, will be same as the class, in which they are studying, e.g., a section of class $\text{I}$ will plant $1$ tree, a section of class $\text{II}$ will plant $2$ trees and so on till class $\text{XII}$, If there are three sections of each class, then the number of trees planted is

Consider two points whose ordinates differ by $3$, on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$. The tangents at these variable points, intersect at $P$. find the locus of $P$.

The least value of $2x^2+y^2+2xy+2x-3y+8$ for real numbers $x$ and $y$ is

Laplace transform for the function $f\left(x\right)=\cos h\left(ax\right)$ is

The value of x in the expression $[x+x^{lo{g}_{10}\left(x\right)}{]}^5$, if the third term in the expansion is 10,00,000

We know that for a linear differential equation of first order, I.F = $e^{\int Pdx}$. Then value of p for $\frac{xdy}{dx}+x^{2}y=xlogx$ is

Determine whether the argument used to check the validity of the following statement is correct :

p : " If $x^2$ is irrational, then x is rational "

The statement is true because the number $x^2={\pi }^2$ is irrational, therefore $x={\pi }^2$ is irrational.

for $x>1$, If $(2x{)}^{2y}=4e^{2x-2y}$, then

$(1+{\log }_{e}2x{)}^2\frac{dy}{dx}$ is equal to :

In a triangle $ABC$, if $|\overrightarrow{BC}|=3,|\overrightarrow{CA}|=5$ and $|\overrightarrow{BA}|=7,$ then the projection of the vector $\overrightarrow{BA}$ on $\overrightarrow{BC}$ is equal to:

In a certain code, $REQUEST$ is written as $S2R52TU.$ How is $ACID$ written in that code?