Question

If $a=\frac{1}{5\cos x+12\sin x}$ , then for all real $x$

• A. the least positive value of $a$ is $\frac{1}{13}$
• B. the greatest negative value of $a$ is $-\frac{1}{13}$
• C. $a\le \frac{1}{13}$
• D. $-\frac{1}{13}\le a\le \frac{1}{13}$

Answer 1

Answer: (A), (B)

The correct options are
A the greatest negative value of $a$ is $-\frac{1}{13}$
B the least positive value of $a$ is $\frac{1}{13}$

$a=\frac{1}{5\cos x+12\sin x}$

$\Rightarrow a=\frac{1}{13(\frac{5}{13}\cos x+\frac{12}{13}\sin x)}$

$\Rightarrow a=\frac{1}{13(\cos \theta \cos x+\sin \theta \sin x)}=\frac{1}{13\cos (\theta -x)}$
Where $\cos \theta =\frac{5}{13}\&\sin \theta =\frac{12}{13}$
$\cos (\theta -x)=\frac{1}{13a}$

$\because -1\le \cos (\theta -x)\le 1$

$\therefore a\ge \frac{1}{13}\&a\le -\frac{1}{13}$
Ans: A,B