If $\sinh (2 \tanh ^{-1} x) = \frac{11}{60}$,then $x =$

If $\sum_{k=1}^n \tan^{-1} \left( \frac{1}{k^2+k+1} \right) = \tan^{-1} ( \theta )$,then $\theta =$

The value of $\cos ^{-1} \left[ \cot \left( \sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}} \right) + \cos ^{-1} \left( \frac{\sqrt{12}}{4} \right) + \sec ^{-1} \sqrt{2} \right]$ is

The value of $\frac{d}{dx} \tan^{-1} \left[ \frac{3a^2x - x^3}{a(a^2 - 3x^2)} \right]$ at $x = 0$ is

Let $\cos ^{-1}(x) + \cos ^{-1} (2x) + \cos ^{-1}(3x) = \pi.$ If $x$ satisfies the cubic equation $ax^3 + bx^2 + cx - 1 = 0,$ then the value of $(a + b + c)$ is -

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