Question

Let $a$ be maximum value of $(3 \cos \theta-4 \sin \theta)$ and $\theta \neq \frac{n \pi}{2}$. If $\alpha=a \sin ^{2} \theta \cdot \cos ^{3} \theta$ and $\beta=a \sin ^{3} \theta \cdot \cos ^{2} \theta$, then $\sqrt{\frac{\left(\alpha^{2}+\beta^{2}\right)^{5}}{(\alpha \beta)^{4}}}=$

• A. $5 \sin \frac{\theta}{2} \cos ^{2} \frac{\theta}{2}$
• B. $-3 \sin \theta$
• C. $5$
• D. $16$

Answer 1

Answer: (C)

Maximum value of $3 \cos \theta-4 \sin \theta$
$\therefore a=\sqrt{3^{2}+(-4)^{2}}=5$
$\alpha=5 \sin ^{2} \theta \cos ^{3} \theta $
$\beta=5 \sin ^{3} \theta \cos ^{2} \theta$
Now, $\alpha^{2}+\beta^{2}=5^{2}\left(\sin ^{4} \theta \cos ^{6} \theta+\sin ^{6} \theta \cos ^{4} \theta\right)$
$=25\left(\cos ^{2} \theta+\sin ^{2} \theta\right) \sin ^{4} \theta \cos ^{4} \theta$
$=25 \sin ^{4} \theta \cos ^{4} \theta$
$\therefore \left(\alpha^{2}+\beta^{2}\right)^{5}=\left(25 \sin ^{4} \theta \cos ^{4} \theta\right)^{5}$
$\Rightarrow \left(\alpha^{2}+\beta^{2}\right)^{5 / 2}=\left(5 \sin ^{2} \theta \cos ^{2} \theta\right)^{5}$
$\alpha \beta=25 \sin ^{5} \theta \cos ^{5} \theta$
$\Rightarrow \sqrt{\frac{\left(\alpha^{2}+\beta^{2}\right)^{5}}{(\alpha \beta)^{4}}}=\frac{\left(5 \sin ^{2} \theta \cos ^{2} \theta\right)^{5}}{\left(5^{2} \sin ^{5} \theta \cos ^{5} \theta\right)^{2}}$
$=\frac{5^{5} \sin ^{10} \theta \cos ^{10} \theta}{5^{4} \sin ^{10} \theta \cos ^{10} \theta}=5$