Question

If m & M denotes the minimum & maximum value of $|z|$ where $z=e^{i2\Phi }sin\Phi +cos\Phi$ where $\Phi ϵR$, then

• A. $m^2+M^2=4$
• B. $m^2+M^2=2$
• C. $M^2-m^2=1$
• D. $m^2-M^2=2$

Answer 1

Answer: (B), (C)

$M^2-m^2=1$
$m^2+M^2=2$

$z=e^{i2\varphi }\sin \varphi +\cos \varphi$

$m$ and $M$ denotes the max and min value of $\left|z\right|$

$\Rightarrow \left|z\right|=\sqrt{{\left(e^{i2\varphi }\sin \varphi \right)}^2+{\left(\cos \varphi \right)}^2}$

$=\sqrt{e^{i4\varphi }\sin \varphi +{\cos }^2\varphi }$

$=\sqrt{(\cos 4\varphi +i\sin 4\varphi ){\sin }^2\varphi +{\cos }^2\varphi }$

$=\sqrt{\cos 4\varphi {\sin }^2\varphi +{\cos }^2\varphi \cos 4\varphi +i\sin 4\varphi {\sin }^2\varphi +i\sin 4\varphi {\cos }^2\varphi }$

When $\varphi =0,$

The maximum value $=1$

when $\varphi =\frac{\pi }{2}$

The minimum value $=\sqrt{-i}=i$

When $\varphi =\pi .$

The minimum value $=1$

$\therefore m^2+M^2=1^2+1^2=2$

$\Rightarrow M^2-m^2=1^2+i^2-1^2=1$

Hence, the answer is $M^2-m^2=1.$