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

The number of roots of the equation, $(81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30$ in the interval $[0, \pi]$, is equal to
  1. 4
  2. 8
  3. 3
  4. 2

Solution

$\begin{aligned} & (81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30 ...(i)\\ & \text { Let } y=81^{\sin ^2 x} \\ \therefore \quad & 81^{\cos ^2 x}=81^{\left(1-\sin ^2 x\right)}=\frac{81}{81^{\sin ^2 x}}=\frac{81}{y}\end{aligned}$ $\therefore \quad$ Equation (i) becomes $\begin{aligned} & y+\frac{81}{y}=30 \\ \therefore \quad & y^2-30 y+81=0 \\ \therefore \quad & (y-27)(y-3)=0 \\ \therefore \quad & y=27 \text { or } 3 \\ \therefore \quad & 81^{\sin ^2 x}=27 \quad \text { or } \quad 81^{\sin ^2 x}=3 \\ \therefore \quad & 3^{4 \sin ^2 x}=3^3 \quad \text { or } \quad 3^{4 \sin ^2 x}=3^1 \end{aligned}$ $\therefore \quad 4 \sin ^2 x=3 \quad$ or $\quad 4 \sin ^2 x=1$ $\therefore \quad \sin x=\frac{\sqrt{3}}{2}, \frac{1}{2}$ $\ldots[\because x \in[0, \pi]]$ $\therefore \quad x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{\pi}{6}, \frac{5 \pi}{6}$ $\ldots[\because x \in[0, \pi]]$ $\therefore \quad$ Required number of roots are 4 .

Asked in: MHT CET 2024 (03 May Shift 1)

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