KVPY Sample Paper KVPY Stream-SX Model Paper-25

The value of \['a'\] for which \[a{{x}^{2}}+\sin {{~}^{-1}}({{x}^{2}}-2x+2)+{{\cos }^{-1}}({{x}^{2}}-2x+2)=0\]has a real solution, is

A) \[\frac{\pi }{2}\]

B) \[-\frac{\pi }{2}\]

C) \[\frac{2}{\pi }\]

D) None of these

Correct Answer: B

Solution :

The given is\[a{{x}^{2}}+{{\sin }^{-1}}\{(x-1)+1\}+{{\cos }^{-1}}\{{{(x-1)}^{2}}+1\}=0\]

\[\because \]\[-1\le {{(x-1)}^{2}}+1\le 1\Rightarrow x=1\]

So, we have \[a+\frac{\pi }{2}=0\Rightarrow a=-\frac{\pi }{2}\]