Question

The difference between the maximum and minimum values of
${\cos }^2\theta +7{\sin }^2\theta +8\sin \theta \cos \theta$ is

• A. $10$
• B. $6$
• C. $8$
• D. $7$

Answer 1

Answer: (A)

$10$

given expression is
$1+6{\sin }^2\Theta +4\left(2\sin \Theta \right)\left(\cos \Theta \right)$
$=1+3(1-\cos 2\Theta )+4\sin 2\Theta$
$=4+4\sin 2\Theta -3\cos 2\Theta$
now $-\sqrt{a^2+b^2}<a\cos x+b\sin x<\sqrt{a^2+b^2}$
so maximum is $9$ and minimum is $-1$
difference $=10$