Question

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

• A. $\le 1$
• B. $=1$
• C. $\ge 1$
• D. $\le \frac{1}{2}$

Answer 1

Answer: (A)

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