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Let $a$ be maximum value of $(3 \cos \theta-4 \sin \theta)$ and $\theta \neq \frac{n \pi}{2}$. If $\alpha=a \sin ^2 \theta$. $\cos ^3 \theta$ and $\beta=a \sin ^3 \theta \cdot \cos ^2 \theta$, then $\sqrt{\frac{\left(\alpha^2+\beta^2\right)^5}{(\alpha \beta)^4}}=$
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5
Maximum value of $3 \cos \theta-4 \sin \theta$
$a=\sqrt{3^2+(-4)^2}=5$
$\therefore \quad \alpha=5 \sin ^2 \theta \cos ^3 \theta$
$\beta=5 \sin ^3 \theta \cos ^2 \theta$
Now, $\alpha^2+\beta^2=5^2\left(\sin ^4 \theta \cos ^6 \theta+\sin ^6 \theta \cos ^4 \theta\right)$
$=25\left(\cos ^2 \theta+\sin ^2 \theta\right) \sin ^4 \theta \cos ^4 \theta$
$=25 \sin ^4 \theta \cos ^4 \theta$
$\therefore \quad\left(\alpha^2+\beta^2\right)^5=\left(25 \sin ^4 \theta \cos ^4 \theta\right)^5$
$\Rightarrow\left(\alpha^2+\beta^2\right)^{5 / 2}=\left(5 \sin ^2 \theta \cos ^2 \theta\right)^5$
$\alpha \beta=25 \sin ^5 \theta \cos ^5 \theta$
$\Rightarrow \quad(\alpha \beta)^2=\left(5^2 \sin ^5 \theta \cos ^5 \theta\right)^2$
$\therefore \sqrt{\frac{\left(\alpha^2+\beta^2\right)^5}{(\alpha \beta)^4}}=\frac{\left(5 \sin ^2 \theta \cos ^2 \theta\right)^5}{\left(5^2 \sin ^5 \theta \cos ^5 \theta\right)^2}$
$=\frac{5^5 \sin ^{10} \theta \cos ^{10} \theta}{5^4 \sin ^{10} \theta \cos ^{10} \theta}=5$
$a=\sqrt{3^2+(-4)^2}=5$
$\therefore \quad \alpha=5 \sin ^2 \theta \cos ^3 \theta$
$\beta=5 \sin ^3 \theta \cos ^2 \theta$
Now, $\alpha^2+\beta^2=5^2\left(\sin ^4 \theta \cos ^6 \theta+\sin ^6 \theta \cos ^4 \theta\right)$
$=25\left(\cos ^2 \theta+\sin ^2 \theta\right) \sin ^4 \theta \cos ^4 \theta$
$=25 \sin ^4 \theta \cos ^4 \theta$
$\therefore \quad\left(\alpha^2+\beta^2\right)^5=\left(25 \sin ^4 \theta \cos ^4 \theta\right)^5$
$\Rightarrow\left(\alpha^2+\beta^2\right)^{5 / 2}=\left(5 \sin ^2 \theta \cos ^2 \theta\right)^5$
$\alpha \beta=25 \sin ^5 \theta \cos ^5 \theta$
$\Rightarrow \quad(\alpha \beta)^2=\left(5^2 \sin ^5 \theta \cos ^5 \theta\right)^2$
$\therefore \sqrt{\frac{\left(\alpha^2+\beta^2\right)^5}{(\alpha \beta)^4}}=\frac{\left(5 \sin ^2 \theta \cos ^2 \theta\right)^5}{\left(5^2 \sin ^5 \theta \cos ^5 \theta\right)^2}$
$=\frac{5^5 \sin ^{10} \theta \cos ^{10} \theta}{5^4 \sin ^{10} \theta \cos ^{10} \theta}=5$
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