Question

lf $\tan \theta ,2\tan \theta +2,3\tan \theta +3$ are in G.P, then the value of $\frac{7-5cot\theta }{9+4\sqrt{se{c}^2\theta -1}}$ is

• A. $\frac{12}{5}$
• B. $\frac{-33}{28}$
• C. $\frac{33}{100}$
• D. $\frac{12}{13}$

Answer 1

The correct option is C $\frac{33}{100}$

Since, $\tan \theta$, $2\tan \theta +2$ and $3\tan \theta +3$ are in G.P.
$(2\tan \theta +2{)}^2=\tan \theta (3\tan \theta +3)$

$\Rightarrow (\tan \theta +1)(2\tan \theta +2-3\tan \theta )=0$

$\Rightarrow \tan \theta =-1,2$

for $\tan \theta =-1$

$\frac{7-5cot\theta }{9-4\sqrt{se{c}^2\theta -1}}=\frac{7-5cot\theta }{9-4\left|\tan \theta \right|}=\frac{7-5(-1)}{9-4}=\frac{12}{5}$

for $\tan \theta =2$

$\frac{7-5cot\theta }{9-4\sqrt{se{c}^2\theta -1}}=\frac{7-5cot\theta }{9-4\left|\tan \theta \right|}=\frac{7-5\left(1/2\right)}{9-4\left(2\right)}=\frac{9}{2}$