Question

If the function $f ( x )=\left( m ^2-3 m +2\right) \cos x +( m -1) \sin x +2( n -3),( m \neq 1)$ is an odd function then the value of $\sin ^{ m } \theta+\cos ^{ n } \theta$ is always

• A. $\leq 1$
• B. $=1$
• C. $\geq 1$
• D. $\leq \frac{1}{2}$

Answer 1

Answer: (A)

$\Theta f (- x )=- f ( x ), \forall x \in R $
$\Rightarrow \left( m ^2-3 m +2\right) \cos x -( m -1) \sin x +2( n -3)=-\left( m ^2-3 m +2\right) \cos x -( m -1) \sin x -2( n -3) $
$\therefore m ^2-3 m +2=0 \Rightarrow m =1 \text { or } 2 \text { but } m \neq 1 $
$\therefore m =2 \& n -3=0 \Rightarrow n =3 $
$\therefore \sin ^{ m } \theta+\cos ^{ n } \theta=\sin ^2 \theta+\cos ^3 \theta \leq \sin ^2 \theta+\cos ^2 \theta \leq 1 $