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Let $f\left(x\right)=\cos \left(2{\tan }^{-1}\sin \left({\cot }^{-1}\sqrt{\frac{1-x}{x}}\right)\right),0<x<1.$ Then

A person writes a
letter to four of his friends. He asks each one of them to copy the
letter and mail to four different persons with instruction that they
move the chain similarly. Assuming that the chain is not broken and
that it costs 50 paise to mail one letter. Find the amount spent on
the postage when 8^th set of letter is mailed.

If $P\left(A\right)=\frac{6}{11},P\left(B\right)=\frac{5}{11}andP(A\cup B)=\frac{7}{11},$
find $P\left(\frac{B}{A}\right)$

If $f\left(x\right)=\{\begin{array}{cc}{e}^x,& x<2\\ a+bx,& x\ge 2\end{array}$ is differentiable for all $x\in R,$ then which of the following is/are correct

Find the general solution of the equation $sin7\theta =sin3\theta +sin\theta$

$\text{If}y=e^{si{n}^{-1}x},then(1-x^2)y_2-x{y}_1=$

Find x
and y, if

Find the
principal and general solutions of the equation

If,
then x is equal to

(A) 6 (B) ±6 (C) −6 (D) 0

In the expansion of ${(3^{5x/4}+3^{-x/4})}^n$ the sum of binomial coefficient is $64$. If the term with greatest binomial coefficient exceeds the third term by $(n-1),$ then the number of value(s) of $x$ is

The Taylor series expansion of $\frac{sinx}{x-\pi }$ at $x=\pi$ is given by

If $\tan \left(\pi \sin \theta \right)=\cot \left(\pi \cos \theta \right),$ then $|\cot (\theta -\frac{\pi }{4})|$ is

The line 2x-y+4=0 cuts the parabola $y^2=8x$ in P and Q.The mid-point of PQ is

The general solution of a differential equation of the type $\frac{dx}{dy}+P_{1}x=Q_1$ is

(a) $y{e}^{\int P_{1}dy}=\int \left(Q_1{e}^{\int P_{1}dy}\right)dy+C$

(b) $y{e}^{\int P_{1}dx}=\int \left(Q_1{e}^{\int P_{1}dx}\right)dx+C$

(c) $x{e}^{\int P_{1}dy}=\int \left(Q_1{e}^{\int P_{1}dy}\right)dy+C$

(d) $x{e}^{\int P_{1}dx}=\int \left(Q_1{e}^{\int P_{1}dx}\right)dx+C$

In the circuit current through source will be [Given $\left(co{s}^{-1}\right(0.6)={53}^{\circ })$]

$V=10+10\sqrt{2}sin(100\pi t+{45}^{\circ })$

Let $P$ lie on the line $x-y=1;|PA-PB|$ is maximum where $A\equiv (2,-3)$ $\&B\equiv (5,6)$. Then the coordinates of $P$ are

The value of${\sum }_{r=1}^{n+1}\left({\sum }_{k=1}^n{}^k{C}_{r-1}\right)$ where r, k, n $ϵ$ N is equal to

The quadratic equation $\tan \theta x^2+2(\sec \theta +\cos \theta )x+(\tan \theta +3\sqrt{2}\cot \theta )$ always has

If the normals at $A\left(t_1\right)$ and $B\left(t_2\right)$ meet again at $C\left(t_3\right)$ on the parabola $y^2=4ax,$ then the locus of the mid point of $AB$ is

$co{s}^{-1}\left(\frac{1}{2}\right)+2si{n}^{-1}\left(\frac{1}{2}\right)$ is equal to

A circle touches the y-axis at the point (0, 4) and cuts the x-axis in a chord of length 6 units. The radius of the circle is

The OLTF of a feedback system is

$G\left(s\right)H\left(s\right)=\frac{K(s+1)(s+3)}{s^2+4s+8}$

The root locus for the same is

If $A={}^{2n}{C}_0\cdot {}^{2n}{C}_1+{}^{2n}{C}_1{}^{2n-1}{C}_1+{}^{2n}{C}_2{}^{2n-2}{C}_1+\dots {+}^{2n}{C}_{2n-1}\cdot {}^1{C}_1$

then $A$ is

If the two circles which pass through $(0,3)$ and $(0,–3)$ and touch the line $y=mx+5$, cut each other orthogonally, then the value of $m^2$ will be

$\underset{x\to \frac{\pi }{2}}{lim}\frac{(1-tan\left(\frac{x}{2}\right))(1-sinx)}{(1+tan\left(\frac{x}{2}\right))(\pi -2x{)}^3}is$

Let $f\left(x\right)=\{\begin{array}{cc}\left\{x^2\right\},& -1\le x<1\\ |1-2x|,& 1\le x<2\\ (1-x^2)\text{sgn}(x^2-3x-4),& 2\le x\le 4\end{array}$



where $\left\{k\right\}$ and $\text{sgn}\left(k\right)$ denote fractional part function and signum function of $k$ respectively. If $m$ denotes the number of points of discontinuity of $f\left(x\right)$ in $[-1,4]$ and $n$ denotes the number of points of non-differentiability of $f\left(x\right)$ in $(-1,4),$ then $(m+n)$ is equal to

The graph of the function $f\left(x\right)=x^2(x+3)$ is:

What should come in place of both x in the equation $\frac{x}{\sqrt{24}}=\frac{\sqrt{6}}{x}$ ?

For real numbers x and y, we write $xRy\iff x-y+\sqrt{2}$ is an irrational number. Then the relation R is