Question

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

• A. $3$
• B. $2$
• C. $4$
• D. $8$

Answer 1

The correct option is B $2$

$(81{)}^{{\sin }^{2}x}+(81{)}^{1-{\sin }^{2}x}=30(81{)}^{{\sin }^{2}x}+\frac{81}{(81{)}^{{\sin }^{2}x}}=30\text{Let}(81{)}^{{\sin }^{2}x}=tt+\frac{81}{t}=30\Rightarrow t^2+81=30t\Rightarrow t^2-30t+81=0\Rightarrow t^2-27t-3t+81=0\Rightarrow (t-3)(t-27)=0\Rightarrow t=3,27\Rightarrow (81{)}^{{\sin }^{2}x}=3,3^3\Rightarrow 3^{4{\sin }^{2}x}=3^1,3^3\Rightarrow 4{\sin }^{2}x=1,3\Rightarrow {\sin }^{2}x=\frac{1}{4},\frac{3}{4}\text{in}[0,\pi ],\sin x\ge 0\sin x=\frac{1}{2},\frac{\sqrt{3}}{2}x=\frac{\pi }{6},\frac{5\pi }{6},\frac{\pi }{3},\frac{2\pi }{3}$
Number of solutions $=4$