Question

Let $a_n={\int }_0^{\pi /2}\frac{1-\cos 2nx}{1-cos2x}dx$, then the value of $\left|\begin{array}{ccc}\frac{\pi }{2}& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ equals to,

• A. $0$
• B. $1$
• C. $3$
• D. None of these

Answer 1

The correct option is A $0$

Consider $a_n+a_{n+2}-2a_{n+1}=A\left(Say\right)$
$\therefore A={\int }_0^{\pi /2}(\frac{1-\cos 2nx}{1-cos2x}+\frac{1-\cos 2(n+2)x}{1-cos2x}-2\frac{1-\cos 2(n+1)x}{1-cos2x})dx$
$A={\int }_0^{\pi /2}\left[\frac{2\cos (2n+2)x(1-\cos 2x)}{1-\cos 2x}\right]dx$
$A={\int }_0^{\pi /2}2\cos (2n+2)xdx$
$A=2{\left(\frac{\sin (2n+2)x}{2n+2}\right)}_0^{\pi /2}=0$
$\therefore a_n{a}_n+1a_n-2\epsilon A.P.$
Along with $a_1={\int }_0^{\pi /2}\frac{1-\cos 2x}{1-\cos 2x}dx$
$a_1=\frac{\pi }{2}$so $\Delta =\left|\begin{array}{ccc}\pi /2& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$
Applying $C_1\to C_1+C_3-2C_2$
$=\left|\begin{array}{ccc}0& a_2& a_3\\ 0& a_5& a_6\\ 0& a_8& a_9\end{array}\right|=0$
Therefore, $a_n,a_{n+1},a_{n+2}$ are in A.P
Hence, option 'A' is correct.