Question

If $a<\frac{1}{32}$, then the number of solutions of ${\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3=a{\pi }^3$ is

• A. $0$
• B. $1$
• C. $2$
• D. $Infinite$

Answer 1

The correct option is A

$0$

Finding the value:

Given: $a<\frac{1}{32}.....\left(1\right)$

Multiply Equation $1$ with ${\pi }^3$

Now,

$$\begin{gathered} \Rightarrow a<\frac{1}{32} \\ \Rightarrow {\pi }^{3}a<\frac{{\pi }^3}{32} \\ \Rightarrow {\left({\sin }^{-1}x\right)}^3+{\left({\cos }^{-1}x\right)}^3<\frac{1}{32} \end{gathered}$$

Therefore the number of solution is $0$

Hence, correct option is $\mathbf{\left(}\mathit{A}\mathbf{\right)}$.