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Let ABCD be a square of a side length l, Let P, Q, R, S be points in the interiors of the sides AD, BC, AB, CD, respectively, such that PQ and RS intersect at right angles. If PQ = $\frac{3\sqrt{3}}{4}$, then RS equals

Let $z_1$ and $z_2$ be roots of the equation $z^2+pz+q=0,$ where p and q may be complex numbers. Let A and B represents $z_1$ and
$z_2$ in the complex plane. Given
$\angle AOB=\alpha \ne 0$ and $OA=OB$ where O is the origin. Using the given information what will be the value of $p^2?$

If $y=f\left(x\right)=\frac{ax-b}{bx-a}$, show that $x=f\left(y\right)$.

$Ifthepoints(a,0),(a{t}_1^2,2a{t}_1^2)and(a{t}_2^2,2a{t}_2)arecollinear,writethevalueof{t}_1{t}_2.$

If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio 2 : 1 , find the value of n.

Find the modulus and conjugate of complex number 4 + 3i.

In the given figure a square OABC is inscribed in a quadrant OPBQ. IF OA = 10 cm, then the perimeter of arc. QPB is

A linear graph is given by the relation "y = 5x + 2". If x = 2, the value of y will be:

At what condition |A U B|= |A| + |B|?

Find the value of the following:

$cot(ta{n}^{-1}a+co{t}^{-1}a)$

Number of solutions of the equation $(2\text{cosec}x-1{)}^{\frac{1}{3}}+(\text{cosec}x-1{)}^{\frac{1}{3}}=1$ in $(-k\pi ,k\pi )$ is $16$, then the possible value of '$k$' is

sin51^o + cos81^o =?

If the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion of $(x+a{)}^n$ are $240,720$ and $1080$ respectively, then the sum of odd numbered terms in the expansion is

$\int \frac{x^5}{x^2+1}dx=$

For the given differential equation find the general solution.

$ydx+(x-y^2)dy=0$

If $si{n}^{-1}(x-\frac{x^2}{2}+\frac{x^3}{4}-\cdots )+co{s}^{-1}(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots )=\frac{\pi }{2}$ for 0 < 1x < $\sqrt{2}$, then x equals

Let $S=\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}$ and
$T=\frac{8xyz}{(1-x^2)(1-y^2)(1-z^2)}$, where $x,y,z\ne \pm 1$. If $x+y+z=xyz$, then the value of $S-T$ is

If $2tan\frac{\alpha }{2}=tan\frac{\beta }{2}$, prove that $cos\alpha =\frac{3+5cos\beta }{5+3cos\beta }$.

The letters of the word ASHISH are permuted and are arranged in an alphabetical order as in an English dictionary. Then, the rank of the word ASHISH is __.

The value of definite integral depends on

Consider a real valued function y = h(x) satisfying the differential equation $\frac{dy}{dx}+g^{\prime }\left(x\right).y=\left(g\right(x)+1)g^{\prime }\left(x\right)$ such that h(0) = 1, then h(1) equals –

If x + $\frac{1}{x}$ = $2cos\theta$, then $x^3$ + $\frac{1}{x^3}$ =

[MP PET 2004]

If a² +b² + c² =-2 and

then f(x) is a polynomial of degree

The value of ${lim}_{n\to \infty }\frac{n!}{(n+1)!-n!}$ is

The radii $r_1,r_2,r_3$ of the escribed circles of the triangle $ABC$ are in H.P. If the area of the triangle is $24{\text{cm}}^2$ and its perimeter is $24\text{cm}$, then the length of its largest side is

Find the angle between the planes
$2x+7y+11z-3=0\text{and}5x+3y+9z+1=0$

The value of the integral:

where [x] denotes the largest integer not exceeding x is :

$b(ccosA-acosC)=c^2-a^2$

The derivative of the function $(x^3+ln6)\left(\sqrt{x}\right)$ is

If $(1-x)(1+x+x^2+x^3+x^4)$ is $\frac{31}{32}$ and $x$ is a rational number. Find the value of $1+x+x^2+x^3+x^4+x^5$.

The distance between the points $(a\sin {30}^{\circ },0)$ and $(0,a\cos {60}^{\circ })$ is

If f(x) be defined on [-2, 2] and is given by $f\left(x\right)=\{\begin{array}{c}-1,-2\le x\le 0\\ x-1,0<x\le 2\end{array}$
and $g\left(x\right)=f\left(|x|\right)+|f\left(x\right)|$. Find $g\left(x\right)$.

The value of p for which the function $f\left(x\right)=\left[\begin{array}{cc}\frac{(4^x-1{)}^3}{sin\frac{x}{p}log[1+\frac{x^2}{3}]},& x\ne 0\\ 12(log4{)}^3,& x=0\end{array}\right],$ $x\ne 0$ may be continuous at x = 0, is

Find the value of $(1.01{)}^{10}+(1-0.01{)}^{10}$ correct to 7 places of decimal.

If $lo{g}_{x}a,a^{\frac{x}{2}}$ and $lo{g}_{b}x$ are in G.P. then write the value of x.

If the 6th, 7th and 8th terms in the expansion $(x+a{)}^n$ are respectivley 112, 7 and $\frac{1}{4}$, find x,a,n.