Question

If $p$ and $q$ are positive real numbers such that $p^2+q^2=1$, then the maximum value of $(p+q)$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\sqrt{2}$
• D. $2$

Answer 1

Answer: (C)

$\sqrt{2}$

given that $p^2+q^2=1$

Let $p=\cos \theta$, $q=\sin \theta$

now, $p+q=\cos \theta +\sin \theta$

$=\sqrt{2}[\frac{1}{\sqrt{2}}\cos \theta +\frac{1}{\sqrt{2}}\sin \theta ]$

$=\sqrt{2}\sin (\theta +\frac{\pi }{4})$

we know that maximum of $\sin$ avlue is $1$

$\therefore$ maximum value of $(p+q)=\sqrt{2}$
Ans: C