The number of points, where the function $$f:\mathbf{R}\to \mathbf{R}$$, $$f(x)=|x-1|\cos |x-2|\sin |x-1|+(x-3)\left|x^2-5x+4\right|$$, is NOT differentiable, is :

$$\text{ Let the function }f(x)=\{\begin{array}{cl}\frac{{\log }_e(1+5x)-{\log }_e(1+\alpha x)}{x}& ;\text{ if }x\ne 0\\ 10& ;\text{ if }x=0\end{array}\text{ be continuous at }x=0.$$ Then $$\alpha$$ is equal to

The domain of the function {\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right) is :

The value of \cot \left( {\sum\limits_{n = 1}^{50} {{{\tan }^{ - 1}}\left( {{1 \over {1 + n + {n^2}}}} \right)} } \right) is :

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

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

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

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

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

The domain of the function f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}} is :

Let m and M respectively be the minimum and the maximum values of f(x) = {\sin ^{ - 1}}2x + \sin 2x + {\cos ^{ - 1}}2x + \cos 2x,\,x \in \left[ {0,{\pi \over 8}} \right]. Then m + M is equal to :

Let \alpha = \tan \left( {{{5\pi } \over {16}}\sin \left( {2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right)} \right) and \beta = \cos \left( {{{\sin }^{ - 1}}\left( {{4 \over 5}} \right) + {{\sec }^{ - 1}}\left( {{5 \over 3}} \right)} \right) where the inverse trigonometric functions take principal values. Then, the equation whose roots are $$\alpha$$ and $$\beta$$ is :

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

If $$0

The domain of the function $$f(x)={\sin }^{-1}\left[2x^2-3\right]+{\log }_2\left({\log }_{\frac{1}{2}}\left(x^2-5x+5\right)\right)$$, where [t] is the greatest integer function, is :

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

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 :

The sum of the absolute maximum and absolute minimum values of the function $$f(x)={\tan }^{-1}(\sin x-\cos x)$$ in the interval $$[0,\pi ]$$ is :

The domain of the function $$f(x)={\sin }^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)$$ is :

Let for some real numbers $$\alpha$$ and $$\beta$$, $$a=\alpha -i\beta$$. If the system of equations $$4ix+(1+i)y=0$$ and 8\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)x + \overline a y = 0 has more than one solution, then {\alpha \over \beta } is equal to