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Find $x$, if $x\in (0,1)$

$3{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)-4{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)+2{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)=\frac{\pi }{3}$

Let $f_k\left(x\right)=\frac{1}{k}({\sin }^{k}x+{\cos }^{k}x)$ for $k=1,2,3,\dots .$ Then for all $x\in R,$ the value of $f_4\left(x\right)-f_6\left(x\right)$ is equal to:

The solution of differential equation $x\frac{dy}{dx}=y+\sqrt{x^2+y^2}$ is:

(where $C$ is integration constant)

Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10.

If the circle $x^2+y^2-6x-10y+k=0$ does not touch or intersect the coordinate axes, and the point $(1,4)$ is inside the circle, then the range of $k$ is

If number of element in set A = p and number of element in set B = q. Find the total number of relation from A to B.

The area bounded by the curve f(x) = x + sin x and its inverse function between x = 0 and $x=2\pi$ is ___ (in sq. units)

A differential equation of first order and first degree is

Find the principal
value of cosec⁻¹
(2)

The value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{x^2\cos x}{1+e^x}dx$ is equal to

If |$Z_1$ + $Z_2$| = |$Z_1$| - |$Z_2$| where $Z_1$, $Z_2$ are two non-zero complex numbers, then arg($Z_1$) - arg($Z_2$) is

If $(21.4{)}^a=(0.00214{)}^b=100$, then the value of $\frac{1}{a}-\frac{1}{b}$ is

cos9y - cos5y =

Find the equation of the circle with centre (–a, –b) and radius

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3$ + 3$x^2$ + 5x - 6 = 0, find the value of $(\alpha -\frac{1}{\beta \gamma })(\beta -\frac{1}{\gamma \alpha })$
$(\gamma -\frac{1}{\alpha \gamma }){(\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma })}^{-1}$

If three students A, B, C independently solve a problem with probabilities $\frac{1}{3},\frac{1}{4}$ and $\frac{1}{5}$ respectively, then the probability that the problem will be solved is

The value of $\underset{x\to \infty }{lim}\frac{e^{1/x^2}-1}{2{\tan }^{-1}\left(x^2\right)-\pi }$ is

The value of $\underset{0}{\overset{\pi }{\int }}|\sin x+\cos x|dx$ is

Number of distinct rational numbers $x$ such that $0<x<1$ and $x=\frac{p}{q},$ where $p,q\in \{1,2,3,4,5,6\}$ is

The differential coefficient of the function f(x) = a^{sin x}, where a is positive constant is:

If 35% of the people residing in a locality are Sikhs then the central angle of the sector representing the Sikh community in the pie chart would be ___.

1. Can't be determined

prove that- tan x+ tan(π/3+x) + tan(2π/3+x)=3tan3x

If normal to $y=f\left(x\right)$ makes an angle $\frac{2\pi }{3}$ with positive $x-$axis at $(2,6)$, then $f^{\prime }\left(2\right)$ is

Consider $f:R_{+}\to [-5,\infty )$ given by $f\left(x\right)=9x^2+6x-5$ show that f is ivnertible with $f^{-1}\left(y\right)=\left(\frac{\left(\sqrt{y+6}\right)-1}{3}\right)$

If $S_n={\cot }^{-1}\left(3\right)+{\cot }^{-1}\left(7\right)+{\cot }^{-1}\left(13\right)+{\cot }^{-1}\left(21\right)+\dots n\text{terms}$, then

The solutions of the equation $\left|\begin{array}{ccc}1+{\sin }^{2}x& {\sin }^{2}x& {\sin }^{2}x\\ {\cos }^{2}x& 1+{\cos }^{2}x& {\cos }^{2}x\\ 4\sin 2x& 4\sin 2x& 1+4\sin 2x\end{array}\right|=0,(0<x<\pi ),$ are :

On the occasion of New Year each student of a class sends greeting cards to the others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is ______.

If $x^2+2x+3<{\cos }^{-1}\left(\cos 4\right)+2{\cot }^{-1}\left(\cot 5\right)\forall x\in Z,$ then number of integral value(s) of $x$ is