Question

Number of values of $\theta \in[0,2 \pi)$ such that $5^{1-\sin \theta-\cos \theta}, \frac{1}{3}, 5^{-2+\sin \theta+\cos \theta}$ (taken in that order) are in harmonic progression, is equal to

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (A)

$ \frac{1}{3}=\frac{2 \cdot 5^{1-\sin \theta-\cos \theta} \cdot 5^{-2+\sin \theta+\cos \theta}}{5^{1-\sin \theta-\cos \theta}+5^{2+\sin \theta+\cos \theta}}$
$ \text { Let } \lambda=5^{\sin \theta+\cos \theta}$
$\therefore \lambda^2-30 \lambda+125=0$
$\therefore \lambda=5,25 $
$\therefore \sin \theta+\cos \theta=1, \sin \theta+\cos \theta=2 $
$\therefore \theta=0, \frac{\pi}{2} $