$\sin \left[ \cos^{-1} \left( \frac{3}{5} \right) + \tan^{-1} 2 \right] = $

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 $f(x) = \cos \left( {{{\tan }^{ - 1}}\left( {\sin \left( {{{\cos }^{ - 1}}x} \right)} \right)} \right) + \sin \left( {{{\cot }^{ - 1}}\left( {\cos \left( {{{\sin }^{ - 1}}x} \right)} \right)} \right)$ has range $[m, M)$,then the number of solutions of the equation $\operatorname{sgn} (|x - 1| - 2) = \ln |x - 2|$ is (where $\operatorname{sgn}$ denotes the signum function).

The sum $\sum\limits_{n = 1}^\infty {{\cot }^{ - 1}} \left( {\frac{{2\left( {\sum\limits_{k = 1}^n k } \right) - 1}}{3}} \right)$ is equal to

The number of solutions of $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$ is

If $\theta = 2 \tan^{-1} \frac{1}{8} + 2 \tan^{-1} \frac{1}{5} + \tan^{-1} \frac{1}{7}$ and $\tan \frac{\theta}{2} = \sqrt{m} + \sqrt{n}$,where $m$ and $n$ are positive integers such that $m < n$,then $(m^n + n^m)^{m+n}$ is equal to