MCQ
$\int_0^{\pi /2} {{{\sin }^4}x{{\cos }^6}x\,dx} $ equals
- A$\frac{{5\pi }}{{512}}$
- ✓$\frac{{3\pi }}{{512}}$
- C$\frac{\pi }{{512}}$
- DNone of these
$ \Rightarrow I = \frac{{\Gamma \,(5/2)\,\Gamma \,(7/2)}}{{2\Gamma (6)}}$, (Applying gamma function)
==> $I = \frac{{3/2.1/2.\sqrt \pi .5/2.3/2.1/2.\sqrt \pi }}{{2.5.4.3.2.1}}$
$ = \frac{{3\pi }}{{512}}$.
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$P_6=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$ and $X=\sum_{k=1}^6 P_k\left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^{\top}$
where $P _{ K }^{ T }$ denotes the transpose of the matrix $P _{ K }$. Then which of the following options is/are correct?
$(1)$ $X -30 I$ is an invertible matrix
$(2)$ The sum of diagonal entries of $X$ is 18
$(3)$ If $X \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha=30$
$(4)$ $X$ is a symmetric matrix