Question

The number of roots of the equation, ${\left(81\right)}^{{\sin }^2\left(x\right)}+{\left(81\right)}^{{\cos }^2\left(x\right)}=30$ in the interval $\left[0,\mathrm{\pi }\right]$ is equal to

• A. $3$
• B. $2$
• C. $4$
• D. $8$

Answer 1

The correct option is C

$4$

Explanation for correct option

Given expression is,

$$\begin{gathered} {\left(81\right)}^{{\sin }^2\left(x\right)}+{\left(81\right)}^{{\cos }^2\left(x\right)}=30 \\ \Rightarrow {\left(81\right)}^{{\sin }^2\left(x\right)}+\frac{81}{{\left(81\right)}^{{\sin }^2\left(x\right)}}=30 \end{gathered}$$

Let ${\left(81\right)}^{{\sin }^2\left(x\right)}=t$

$$\begin{gathered} \Rightarrow t+\frac{81}{t}=30 \\ \Rightarrow t^2-30t+81=0 \\ \Rightarrow t^2-27t-3t+81=0 \\ \Rightarrow t\left(t-27\right)-3\left(t-27\right)=0 \\ \Rightarrow \left(t-27\right)\left(t-3\right)=0 \end{gathered}$$

$\therefore t=27$or $t=3$

If $t=27$

$$\begin{gathered} {\left(81\right)}^{{\sin }^2\left(x\right)}=t \\ \Rightarrow {\left(81\right)}^{{\sin }^2\left(x\right)}=27 \\ \Rightarrow 3^{4{\sin }^2\left(x\right)}=3^3 \\ \Rightarrow 4{\sin }^2\left(x\right)=3 \\ \Rightarrow {\sin }^2\left(x\right)=\frac{3}{4} \\ \Rightarrow \sin \left(x\right)=\frac{\sqrt{3}}{2} \\ \therefore x=\frac{\pi }{3},\frac{2\mathrm{\pi }}{3}\left[\because x\in \left[0,\mathrm{\pi }\right]\right] \end{gathered}$$

If $t=3$

$$\begin{gathered} {\left(81\right)}^{{\sin }^2\left(x\right)}=t \\ \Rightarrow {\left(81\right)}^{{\sin }^2\left(x\right)}=3 \\ \Rightarrow 3^{4{\sin }^2\left(x\right)}=3^1 \\ \Rightarrow 4{\sin }^2\left(x\right)=1 \\ \Rightarrow {\sin }^2\left(x\right)=\frac{1}{4} \\ \Rightarrow \sin \left(x\right)=\frac{1}{2} \\ \therefore x=\frac{\pi }{6},\frac{5\mathrm{\pi }}{6}\left[\because x\in \left[0,\mathrm{\pi }\right]\right] \end{gathered}$$

Therefore, total possible solutions is $4$.

Hence, the correct option is $\left(\mathrm{C}\right)$.