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

Answer: (B)

$0$

Given: $p,q,r$ are in A.P.
$\therefore 2q=p+r$
$\Rightarrow \left|\begin{array}{ccc}{a}^2+2^{n+1}+2p& b^2+2^{n+2}+3q& c^2+p\\ 2^n+p& 2^{n+1}+q& 2q\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$
$R_1\to R_1-R_3$
$=\left|\begin{array}{ccc}{2}^n+p& 2^{n+1}+q& p+r\\ 2^n+p& 2^{n+1}+q& 2q\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$
$=\left|\begin{array}{ccc}{2}^n+p& 2^{n+1}+q& 2q\\ 2^n+p& 2^{n+1}+q& 2q\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$

$R_1\to R_1-R_2$

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

$=0$