MCQ
$\int_0^a {{x^2}{{({a^2} - {x^2})}^{3/2}}dx = } $
- ✓$\frac{{\pi {a^6}}}{{32}}$
- B$\frac{{2{a^5}}}{{15}}$
- C$\frac{{{a^6}}}{{32}}$
- DNone of these
Put $x = a\sin \theta \Rightarrow dx = a\cos \theta \,d\theta $
$I = \int_0^{\pi /2} {{a^2}{{\sin }^2}\theta .{a^3}{{\cos }^3}\theta .a\cos \theta \,d\theta } $
$ = {a^6}\int_0^{\pi /2} {{{\sin }^2}\theta {{\cos }^4}\theta \,d\theta= {a^6}\frac{{\Gamma \frac{3}{2}.\,\Gamma \frac{5}{2}}}{{2.\Gamma \frac{8}{2}}}} $
$= {a^6}\frac{{\frac{1}{2}.\sqrt \pi .\frac{3}{2}.\frac{1}{2}.\sqrt \pi }}{{2.3.2.1}} $
$= \frac{{\pi {a^6}}}{{32}}$.
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$\left|\begin{array}{lll}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c\end{array}\right|$ is
$g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$
where
$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$.
Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?
$(A)$ $p \left(\frac{3+\sqrt{2}}{4}\right)<0$
$(B)$ $p \left(\frac{1+3 \sqrt{2}}{4}\right)>0$
$(C)$ $p \left(\frac{5 \sqrt{2}-1}{4}\right)>0$
$(D)$ $p \left(\frac{5-\sqrt{2}}{4}\right)<0$