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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:

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.

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

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:

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:

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 ________.

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 _________.

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:

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JEE Main 2024 31st Jan Morning Shift

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: