The number of roots of the equation,

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:

  1. (1) 3

  2. (2) 4

  3. (3) 8

  4. (4) 2


Correct Option: , 2

Solution:

$(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$

$(81)^{\sin ^{2} x}+\frac{(81)^{1}}{(18)^{\sin ^{2} x}}=30$

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

$\mathrm{t}+\frac{81}{t}=30$

$(t-3)(t-27)=0$

$(81)^{\sin ^{2} x}=3^{1} \quad$ or $\quad(81)^{\sin ^{2} x}=3^{3}$

$3^{4 \sin ^{2} x}=3^{1} \quad$ or $\quad 3^{4 \sin ^{2} x}=3^{3}$

$\sin ^{2} x=\frac{1}{4} \quad$ or $\quad \sin ^{2} x=\frac{3}{4}$

 

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