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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:

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:

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:

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 :

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

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: