Question

If $a<\dfrac{1}{32}$ , then the number of solutions of ${{\left( {{\sin }^{-1}}x \right)}^{-3}}+{{\left( {{\cos }^{-1}}x \right)}^{3}}=a{{\pi }^{3}}$ is
1. $0$
2. $1$
3. $2$
4. Infinite.

Answer 1

In this problem we need to calculate the number of solutions for the given equation according to the value of $a$. First we will consider the expression ${{\left( {{\sin }^{-1}}x \right)}^{3}}+{{\left( {{\cos }^{-1}}x \right)}^{3}}$ and calculate the maximum and minimum value of the function based on the different functions in the given expression. Based on the maximum and minimum value we will calculate the range of the variable $a$ and compare that value with the given value to know whether the given equation is defined for the given value of $a$.

Complete step by step answer:
Given equation is ${{\left( {{\sin }^{-1}}x \right)}^{-3}}+{{\left( {{\cos }^{-1}}x \right)}^{3}}=a{{\pi }^{3}}$ and the range of the variable $a$ is $a<\dfrac{1}{32}$.
Consider the expression ${{\left( {{\sin }^{-1}}x \right)}^{3}}+{{\left( {{\cos }^{-1}}x \right)}^{3}}$ and assume it as a function, then we will have
$f\left( x \right)={{\left( {{\sin }^{-1}}x \right)}^{3}}+{{\left( {{\cos }^{-1}}x \right)}^{3}}$
By observing the above function, the minimum value of the above function will occur at $x=\dfrac{1}{\sqrt{2}}$ and the minimum value of the function is given by
$\begin{aligned} & f\left( \dfrac{1}{\sqrt{2}} \right)={{\left( {{\sin }^{-1}}\left( \dfrac{1}{\sqrt{2}} \right) \right)}^{3}}+{{\left( {{\cos }^{-1}}\left( \dfrac{1}{\sqrt{2}} \right) \right)}^{3}} \\\ & \Rightarrow f\left( \dfrac{1}{\sqrt{2}} \right)={{\left( \dfrac{\pi }{4} \right)}^{3}}+{{\left( \dfrac{\pi }{4} \right)}^{3}} \\\ & \Rightarrow f\left( \dfrac{1}{\sqrt{2}} \right)=\dfrac{2{{\pi }^{3}}}{64} \\\ & \Rightarrow f\left( \dfrac{1}{\sqrt{2}} \right)=\dfrac{{{\pi }^{3}}}{32} \\\ \end{aligned}$
Now the maximum value of the function $f\left( x \right)$ will occur at $x=-1$ and the maximum value will be
$\begin{aligned} & f\left( -1 \right)={{\left( {{\sin }^{-1}}\left( -1 \right) \right)}^{3}}+{{\left( {{\cos }^{-1}}\left( -1 \right) \right)}^{3}} \\\ & \Rightarrow f\left( -1 \right)=-\dfrac{{{\pi }^{3}}}{8}+{{\pi }^{3}} \\\ & \Rightarrow f\left( -1 \right)=\dfrac{7{\pi}^ {3}}{8} \\\ \end{aligned}$
From the maximum and minimum values of the function $f\left( x \right)$ , we can write the range of the variable $a$ as
$a\in \left[ \dfrac{1}{32},\dfrac{7}{8} \right]$
From the above range of the variable $a$ , we can say that there is no solution for the given equation when $a<\dfrac{1}{32}$.

So, the correct answer is “Option 1”.

Note: For this problem we have asked to calculate the number of solutions for the given equation according to the range of the value $a$. Sometimes they may ask to calculate the range of the variable $a$ in which the given equation has solutions. Then also we can follow the above procedure.