CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

If $$\displaystyle 0 \leq \theta \leq \pi$$ and $$\displaystyle 81^{\sin^2\theta} + 81^{\cos^2\theta} = 30$$, then $$\displaystyle \theta $$


A
300
loader
B
600
loader
C
1200
loader
D
1500
loader

Solution

The correct options are
A $$\displaystyle 60^0$$
B $$\displaystyle 120^0$$
C $$\displaystyle 30^0$$
D $$\displaystyle 150^0$$

$$ 81^{\sin ^2\theta} + 81^{\cos ^2\theta} = 30$$
$$\Rightarrow 81^{\sin ^2\theta} + 81^{ 1 - \sin ^2\theta} = 30$$
Assume $$ 81^{\sin ^2\theta} = x$$
$$\Rightarrow x + \dfrac{81}{x} = 30 $$
$$\Rightarrow x^2 - 30 x + 81 = 0 $$
$$\Rightarrow (x - 3) (x -27) = 0 $$
$$\Rightarrow x = 3, 27 \Rightarrow 81^{\sin ^2\theta} = 3^{4 \sin ^2\theta} = 3^1, 3^3 $$
$$ \Rightarrow \sin ^2\theta = \dfrac{1}{4}, \dfrac{3}{4}$$
given $$ 0 \leq \theta \leq \pi$$,
 therefore, $$ \sin \theta = \dfrac{1}{2}, \dfrac{\sqrt{3}}{2}$$
Hence,
$$ \theta = 30^\circ, 60^\circ, 120^\circ, 150^\circ,  $$
Hence, options 'A', 'B', 'C' and 'D' are correct.


Physics

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image