Question

If $p,q,r$ are in A.P., then the value of determinant $\left|\begin{array}{ccc}{a}^2+2^{n+1}+2p& b^2+2^{n+2}+3q& c^2+p\\ 2^n+p& 2^{n+1}+q& 2q\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$ is

• A. 1
• B. 0
• C. $a^2{b}^2{c}^2-2^n$
• D. $(a^2+b^2+c^2)-2^{n}q$

Answer 1

The correct option is B

0

In the given determinant, using the operation, $R_1\to R_1-R_3$ and $2q=p+r$, we get

$\left|\begin{array}{ccc}{2}^{n+1}-2^n+p& 2^{n+2}-2^{n+1}+q& p+r\\ 2^n+p& 2^{n+1}+q& p+r\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$

$=\left|\begin{array}{ccc}{2}^n(2-1)+p& 2^{n+1}(2-1)+q& p+r\\ 2^n+p& 2^{n+1}+q& p+r\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$

$=\left|\begin{array}{ccc}{2}^n+p& 2^{n+1}+q& p+r\\ 2^n+p& 2^{n+1}+q& p+r\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|=0$ $[\because R_1\equiv R_2]$