question_answer

If the area enclosed between \[f(x)=\] Min. \[({{\cos }^{-1}}(\cos x),{{\cot }^{-1}}(\cot x))\] and \[x\]-axis in \[x\in (\pi ,2\pi )\] is \[\frac{{{\pi }^{2}}}{k}\] where \[k\in N\], then k is equal to

A)  2                                

B)  3

C)  4                                

D)  5

Correct Answer: C

Solution :

\[f(x)\,=\left\{ \begin{matrix}    x-\pi ; & \pi <x\le \frac{3\pi }{2}  \\    2\pi -x; & \frac{3\pi }{2}<x<2\pi   \\ \end{matrix} \right.\] Clearly, required area = area of shaded portion of \[\Delta ABC=\frac{1}{2}\times \pi \,=\frac{\pi }{2}=\frac{{{\pi }^{2}}}{4}\,=\frac{{{\pi }^{2}}}{k}\] \[\therefore \] on comparing, we get k = 4