Question

Statement I : The equation $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3-a{\pi }^3=0$ has a solution for all $a⩾\frac{1}{32}.$
Statement II : For any $xϵR,si{n}^{-1}x+co{s}^{-1}x=\frac{\pi }{2}$ and $0\le (si{n}^{-1}x-\frac{\pi }{4}{)}^2\le \frac{9{\pi }^2}{16}$.

• A. Both statements I and II are true.
• B. Both statements I and II are true but I is not an explanation of II
• C. Statement I is true and statement II is false
• D. Statement I is false and statement II is true.

Answer 1

The correct option is D Statement I is false and statement II is true.

Say $f\left(x\right)={\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3$

$f^{\prime }\left(x\right)=0$ at $x=\frac{\pi }{4}$

$f^{\prime \prime }\left(x\right)\ge 0$ at $x=\frac{\pi }{4}$

so $f\left(x\right)=\frac{{\pi }^3}{32}$ This is least value.

$\therefore f\left(x\right)\ge a{\pi }^3$ has a solution.

$\therefore \frac{1}{32}\ge$ $a$

Statement $I$ is incorrect.

Now $\frac{-\pi }{2}\le {\sin }^{-1}x\le \frac{\pi }{2}$

$0\le {({\sin }^{-1}x-\frac{\pi }{4})}^2\le \frac{9{\pi }^2}{16}$

Min at $x=\frac{\pi }{4}$ and max at $x=\frac{\pi }{2}$. So, Statement $I$ is incorrect and $II$ is correct.