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Let \( x = \sin\!\left(2 \tan^{-1}\!\alpha\right) \) and \( y = \sin\!\left(\dfrac{1}{2}\tan^{-1}\!\dfrac{4}{3}\right) \). If \( S = \{ \alpha \in \mathbb{R} : y^2 = 1 - x \} \); then \( \displaystyle\sum_{\alpha \in S} 16\alpha^3 \) is equal to __________.

\( y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right) \), then \( (x - y)^2 + 3y^2 \) is equal to ____________.

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

A possible value of \( \tan\!\left(\frac{1}{4}\sin^{-1}\!\frac{\sqrt{63}}{8}\right) \) is:

If \( \alpha>\beta>\gamma>0 \), then the expression \( \displaystyle \cot^{-1} \left\{ \beta + \frac{(1 + \beta^2)}{(\alpha - \beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1 + \gamma^2)}{(\beta - \gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1 + \alpha^2)}{(\gamma - \alpha)} \right\} \) is equal to:

Considering only the principal values of inverse trigonometric functions, the number of positive real values of \( x \) satisfying \(\displaystyle \tan^{-1}(x)+\tan^{-1}(2x)=\frac{\pi}{4} \) is:

If \( \sin^{-1}\frac{\alpha}{17}\ + \cos^{-1}\frac{4}{5}\ - \tan^{-1}\frac{77}{36}\ = 0 ,\; 0<\alpha<13 \), then \( \sin^{-1}(\sin \alpha) + \cos^{-1}(\cos \alpha) \) is equal to:

If \( k = \tan\left( \frac{\pi}{4} + \frac{1}{2}\cos^{-1}\left( \frac{2}{3} \right) \right) + \tan\left( \frac{1}{2}\sin^{-1}\left( \frac{2}{3} \right) \right) \), then the number of solutions of the equation\( \sin^{-1}(kx - 1) = \sin^{-1} x - \cos^{-1} x \) is ____________.

For \( k \in \mathbb{R} \), let the solutions of the equation \( \cos\!\left(\sin^{-1}\!\left(\operatorname{x cot}\!\left(\tan^{-1}\!\left(\cos\!\left(\sin^{-1} x\right)\right)\right)\right)\right) = k, \; 0<|x|<\dfrac{1}{\sqrt{2}} \) be \( \alpha \) and \( \beta \), where the inverse trigonometric functions take only principle values. If the solution of the equation \( x^2 - bx - 5 = 0 \) are \( \dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} \) and \( \dfrac{\alpha}{\beta} \) then \( \dfrac{b}{k^2} \) is equal to _______

If the domain of the function \( f(x) = \log_e\left(4x^2+11x+6\right)+\sin^{-1}(4x+3)+\cos^{-1}\left(\dfrac{10x+6}{3}\right) \) is \( (\alpha,\beta] \), then \( 36|\alpha+\beta| \) is equal to:

The number of solutions of \( \tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6} \), where \( -\frac{1}{2\sqrt{6}} < x < \frac{1}{2\sqrt{6}} \), is equal to:

Let \( x = \dfrac{m}{n} \) (\( m, n \) are co-prime natural numbers) be a solution of the equation \( \cos (2 \sin^{-1} x )= \dfrac{1}{9} \) and let \( \alpha, \beta \) (\( \alpha > \beta \)) be the roots of the equation \( m x^{2} - n x - m + n = 0 \). Then the point \( (\alpha, \beta) \) lies on the line.

Let the maximum value of \( (\sin^{-1}x)^2+(\cos^{-1}x)^2 \) for \( x\in\left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right] \) be \( \frac{m}{n}\pi^2 \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to ____________.

Let \( 0<\alpha<1, \; \beta=\frac{1}{3\alpha} \) and \( \tan^{-1}(1-\alpha)+\tan^{-1}(1-\beta)=\frac{\pi}{4} \). Then \( 6(\alpha+\beta) \) is equal to:

Considering the principal values of the inverse trigonometric functions, \( \sin^{-1}\!\left(\dfrac{\sqrt{3}}{2}\,x + \dfrac{1}{2}\sqrt{1-x^2}\right),\; -\dfrac{1}{2} < x < \dfrac{1}{\sqrt{2}} \) is equal to:

If for some \( \alpha, \beta : \alpha \le \beta \), \( \alpha + \beta = 8 \) and \( \sec^{2}\!\left(\tan^{-1}\! \alpha \right) + cosec^{2}\!\left(\cot^{-1}\! \beta \right) = 36 \), then \( \alpha^{2} + \beta \) is ______.

\( \tan \left( 2\tan^{-1}\dfrac{1}{5}+\sec^{-1}\dfrac{\sqrt{5}}{2}+2\tan^{-1}\dfrac{1}{8} \right) \) is equal to:

The value of \( \cot^{-1}\!\left(\dfrac{\sqrt{1+\tan^{2}(2)}-1}{\tan(2)}\right) - \cot^{-1}\!\left(\dfrac{\sqrt{1+\tan^{2}\!\left(\dfrac{1}{2}\right)}+1}{\tan\!\left(\dfrac{1}{2}\right)}\right) \) is equal to:

Let $(a,b) \subset (0, 2\pi)$ be the largest interval for which $\sin^{-1}(\sin\theta) - \cos^{-1}(\sin\theta) > 0, \theta \in (0, 2\pi)$, holds. If $\alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0$ and $\alpha - \beta = b - a$, then $\alpha$ is equal to:

Given that the inverse trigonometric function assumes principal values only. Let \( x, y \) be any two real numbers in \( [-1,1] \) such that \( \cos^{-1}x - \sin^{-1}y = \alpha, \dfrac{-\pi}{2} \le \alpha \le \pi \).
Then, the minimum value of \( x^2 + y^2 + 2xy\) \(\sin \alpha \) is:

Let the range of the function \( f(x) = \dfrac{1}{2 + \sin 3x + \cos 3x}, \; x \in \mathbb{R} \) be \( [a, b] \). If \( \alpha \) and \( \beta \) are respectively the A.M. and the G.M. of \( a \) and \( b \), then \( \dfrac{\alpha}{\beta} \) is equal to

The set of all values of \( k \) for which \( (\tan^{-1}x)^{3} + (\cot^{-1}x)^{3} = k\pi^{3},\ x \in \mathbb{R} \), is the interval:

The real values function \( f(x) = \dfrac{\text{cosec}^{-1} x}{\sqrt{x - [x]}} \), where \( [x] \) denotes the greatest integer less than or equal to \( x \), is defined for all \( x \) belonging to:

\( \cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right) \) is equal to:

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation \( \cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x) \) is equal to:

Let $S = \left\{ x \in R : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \right\}$. If $n(S)$ denotes the number of elements in $S$ then:

If the domain of the function \( f(x) = \cos^{-1}\!\left(\dfrac{2 - |x|}{4}\right) + \{\log_{e}(3 - x)\}^{-1} \) is \( [ -\alpha, \beta ) - \{\gamma\} \), then \( \alpha + \beta + \gamma \) is equal to:

\( \cos^{-1}(\cos(-5)) + \sin^{-1}(\sin(6)) - \tan^{-1}(\tan(12)) \) is equal to:
(The inverse trigonometric functions take the principal values)

The angle of elevation of the top \( P \) of a tower from the feet of one person standing due South of the tower is \( 45^\circ \) and from the feet of another person standing due west of the tower is \( 30^\circ \). If the height of the tower is \( 5 \, \text{meters} \), then the distance (in meters) between the two persons is equal to:

\( \max\limits_{0 \le x \le \pi} \left\{ x - 2 \sin x \cos x + \dfrac{1}{3} \sin 3x \right\} = \)

Let \( x^* y = x^2 + y^3 \) and \( (x^ * 1) ^* 1 = x^* (1 ^* 1) \). Then a value of \( 2 \sin^{-1}\!\left(\frac{x^4 + x^2 - 2}{x^4 + x^2 + 2}\right) \) is:

If \( \tan\left( \dfrac{\pi}{9} \right), x, \tan\left( \dfrac{7\pi}{18} \right) \) are in arithmetic progression and \( \tan\left( \dfrac{\pi}{9} \right), y, \tan\left( \dfrac{5\pi}{18} \right) \) are also in arithmetic progression, then \( |x - 2y| \) is equal to:

If \( S=\left\{x\in \mathbb{R}:\sin^{-1}\!\left(\dfrac{x+1}{\sqrt{x^{2}+2x+2}}\right)-\sin^{-1}\!\left(\dfrac{x}{\sqrt{x^{2}+1}}\right)=\dfrac{\pi}{4}\right\} \), then \( \sum\limits_{x\in S}\left(\sin\!\left(\left(x^{2}+x+5\right)\dfrac{\pi}{2}\right)-\cos\!\left(\left(x^{2}+x+5\right)\pi\right)\right) \) is equal to _____.

If \( y(x) = \cot^{-1}\!\left(\sqrt{\frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}}\right),\ x \in \left( \frac{\pi}{2}, \pi \right) \) then \( \frac{dy}{dx} \) at \( x = \frac{5\pi}{6} \) is:

Let \( S = \left\{ \sin^{2} 2\theta : \; \left( \sin^{4}\theta + \cos^{4}\theta \right)x^{2} + \left( \sin 2\theta \right)x + \left( \sin^{6}\theta + \cos^{6}\theta \right) = 0 \text{ has real roots} \right\}. \)
if \( \alpha \) and \( \beta \) be the smallest and largest elements of the set \( S \), respectively, then \( 3\big((\alpha - 2)^{2} + (\beta - 1)^{2}\big) \) equals ____________ .

If the domain of the function \( f(x) = \sin^{-1}\!\left(\dfrac{x - 1}{2x + 3}\right) \) is \( \mathbb{R} - (\alpha, \beta) \), then \( 12\alpha\beta \) is equal to:

If \( \displaystyle \sum_{r=1}^{50} \tan^{-1}\!\left(\frac{1}{2r^{2}}\right) = p \), then the value of \( \tan p \) is:

If \( 0 < x < \dfrac{1}{\sqrt{2}} \text{ and } \dfrac{\sin^{-1} x}{\alpha} = \dfrac{\cos^{-1} x}{\beta}\) then the value of \(\sin\left(\dfrac{2\pi\alpha}{\alpha+\beta}\right)\) is:

Considering only the principal values of the inverse trigonometric functions, the domain of the function \( f(x)=\cos^{-1}\!\left(\dfrac{x^2-4x+2}{x^2+3}\right) \) is:

\( \sin^{-1}\!\left(\sin\frac{2\pi}{3}\right)+\cos^{-1}\!\left(\cos\frac{7\pi}{6}\right)+\tan^{-1}\!\left(\tan\frac{3\pi}{4}\right) \text{ is equal to:} \)

The domain of the function \( f(x) = \sin^{-1}\!\left(\dfrac{3x^{2}+x-1}{(x-1)^{2}}\right) + \cos^{-1}\!\left(\dfrac{x-1}{x+1}\right) \) is:

If \( \frac{\pi}{2} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\!\left( \frac{12}{13}\cos x + \frac{5}{13}\sin x \right) \) is equal to:

Let \( M \) and \( m \) respectively be the maximum and minimum values of the function \( f(x)=\tan^{-1}(\sin x+\cos x) \) in \( \left[0,\frac{\pi}{2}\right] \). Then the value of \( \tan(M-m) \) is equal to:

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2},0\right)$ and directrix $y=-\frac{1}{2}$. Then $S=\left\{x \in R: \tan^{-1}\left(\sqrt{f(x)}\right)+\sin^{-1}\left(\sqrt{f(x)+1}\right)=\frac{\pi}{2}\right\}$:

The value of \( \cot\!\left( \displaystyle \sum_{n=1}^{50} \tan^{-1}\!\left(\frac{1}{1+n+n^{2}}\right)\right) \) is

If \( \frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1}\!\left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha \), then \( \tan \alpha \) is equal to ______.

The sum of possible values of \( x \) for \( \tan^{-1}(x+1) + \cot^{-1}\!\left(\dfrac{1}{x-1}\right) = \tan^{-1}\!\left(\dfrac{8}{31}\right) \) is:

Let \( \alpha=3\sin^{-1}\!\left(\dfrac{6}{11}\right) \) and \( \beta=3\cos^{-1}\!\left(\dfrac{4}{9}\right) \), where inverse trigonometric functions take only the principal values. Given below are two statements:
Statement-I: \( \cos(\alpha+\beta)>0 \).
Statement-II: \( \cos(\alpha)<0 \).
In the light of the above statements, choose the correct answer from the options given below:

For \( n \in \)\(\mathbb{N}\) , if \( \cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \frac{\pi}{4} \), then \( n \) is equal to ________.

Given that the inverse trigonometric functions take principal values only. Then the number of real values of \( x \) which satisfy \( \sin^{-1}\!\left(\dfrac{3x}{5}\right)+\sin^{-1}\!\left(\dfrac{4x}{5}\right)=\sin^{-1}x \) is equal to:

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \( 2\sin^{-1}x + 3\cos^{-1}x = \frac{2\pi}{5} \), is _________.

The number of real roots of the equation \( \tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^{2}+x+1} = \dfrac{\pi}{4} \) is :

\( \tan^{-1}\!\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\!\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right) \) is equal to:

The value of \( \tan\!\left(2\tan^{-1}\!\left(\dfrac{3}{5}\right)+\sin^{-1}\!\left(\dfrac{5}{13}\right)\right) \) is equal to:

If the domain of the function \( f(x)=\sec^{-1}\!\left(\dfrac{2x}{5x+3}\right) \) is \( [\alpha,\beta)\cup(\gamma,\delta] \), then \( \left|3\alpha+10(\beta+\gamma)+21\delta\right| \) is equal to ______.

Let \( S \) be the set of all solutions of the equation \( \cos^{-1}(2x) - 2\cos^{-1}\!\left(\sqrt{1-x^2}\right) = \pi , \; x \in \left[ -\frac{1}{2} , \frac{1}{2} \right] .\) Then \( \displaystyle \sum_{x \in S} 2\sin^{-1}(x^2-1) \) is equal to:

If the sum of all the solutions of $\tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}, -1 < x < 1, x \neq 0,$ is $a - \frac{4}{\sqrt{3}}$, then $a$ is equal to _____.

If \( (\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a; \, 0 < x < 1, \, a \neq 0 \), then the value of \( 2x^2 - 1 \) is:

For \( \alpha, \beta, \gamma \ne 0 \), if \( \sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi \) and \( (\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta \), then \( \gamma \) equals:

If \( a=\sin^{-1}(\sin 5)) \) and \( b=\cos^{-1}(\cos 5)) \), then \( a^{2}+b^{2} \) is equal to:

The value of \( \tan^{-1}\!\left( \frac{\cos\!\left( \frac{15\pi}{4} \right)}{\sin\!\left( \frac{\pi}{4} \right)} - 1 \right) \) is equal to:

If the domain of the function \( f(x) = \dfrac{ \cos^{-1}\sqrt{x^2 - x + 1} }{ \sqrt{ \sin^{-1}\!\left( \dfrac{2x - 1}{2} \right) } } \) is the interval \( (\alpha, \beta) \), then \( \alpha + \beta \) is equal to:

If \( \cot^{-1}(\alpha) = \cot^{-1} 2 + \cot^{-1} 8 + \cot^{-1} 18 + \cot^{-1} 32 + \ldots \) upto \( 100 \) terms, then \( \alpha \) is:

\( 50 \tan\!\left(3\tan^{-1}\!\left(\dfrac{1}{2}\right) + 2\cos^{-1}\!\left(\dfrac{1}{\sqrt{5}}\right)\right) + 4\sqrt{2}\,\tan\!\left(\dfrac{1}{2}\tan^{-1}(2\sqrt{2})\right) \) is equal to __________.

Let \( a_{1}=1,\, a_{2}, a_{3}, a_{4}, \ldots \) be consecutive natural numbers.
Then \( \tan^{-1}\!\left(\dfrac{1}{1+a_{1}a_{2}}\right)+\tan^{-1}\!\left(\dfrac{1}{1+a_{2}a_{3}}\right)+\ldots+\tan^{-1}\!\left(\dfrac{1}{1+a_{2021}a_{2022}}\right) \) is equal to:

Let \( g(x) = f(x) + f(1-x) \) and \( f''(x) > 0, \; x \in (0,1) \). If \( g \) is decreasing in the interval \( (0, \alpha) \) and increasing in the interval \( (\alpha, 1) \), then \( \tan^{-1}(2\alpha) + \tan^{-1}\!\left(\dfrac{1}{\alpha}\right) + \tan^{-1}\!\left(\dfrac{\alpha + 1}{\alpha}\right) \) is equal to:

The sum of the infinite series \( \cot^{-1}\!\left(\dfrac{7}{4}\right)+\cot^{-1}\!\left(\dfrac{19}{4}\right)+\cot^{-1}\!\left(\dfrac{39}{4}\right)+\cot^{-1}\!\left(\dfrac{67}{4}\right)+\ldots \) is:

If the inverse trigonometric functions take principal values, then \( \cos^{-1}\!\left(\frac{3}{10}\cos\!\left(\tan^{-1}\!\left(\frac{4}{3}\right)\right)+\frac{2}{5}\sin\!\left(\tan^{-1}\!\left(\frac{4}{3}\right)\right)\right) \) is equal to:

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \( 16\left( (\sec^{-1} x)^{2} + (cosec^{-1} x)^{2} \right) \) is:

For \( x\in(-1,1] \), the number of solutions of the equation \( \sin^{-1}x=2\tan^{-1}x \) is equal to __________.

Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum\limits_{x \in S} (2x - 1)^2 \) is equal to ________.

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JEE Main 2023 25th Jan Morning Shift

If the sum of all the solutions of $\tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}, -1 < x < 1, x \neq 0,$ is $a - \frac{4}{\sqrt{3}}$, then $a$ is equal to _____.