Question

If $a,b,c$ are Pth, Qth, Rth terms of a G.$P$. and $a>0,b>0,c>0$ then $\left|\begin{array}{ccc}loga& P& 1\\ logb& Q& 1\\ logc& R& 1\end{array}\right|$ is equal to

• A. $a^P{b}^Q{c}^R$
• B. $1$
• C. $0$
• D. $PQR$

Answer 1

Answer: (C)

$0$

$a,b,c\to P^{th},Q^{th},R^{th}$ terms in GP

Let first term be x and common difference be d

$a\to x{d}^{P-1}$

$b\to x{d}^{Q-1}$

$c\to x{d}^{R-1}$

$\Rightarrow \frac{x{d}^{Q-1}}{x{d}^{P-1}}=\frac{x{d}^{R-1}}{x{d}^{Q-1}}$

$\Rightarrow \frac{Q\log d}{P\log d}=\frac{R\log d}{Q\log d}$

$\because P,Q$ and R are in G.P.

$Q^2=PR$

$\Rightarrow P,Q,R$ are also in GP

$\Rightarrow P=\log a,Q=\log b,R=\log c$

$\left|\begin{array}{ccc}\log a& P& 1\\ \log b& Q& 1\\ \log c& R& 1\end{array}\right|$

$C_1$ and $C_2$ are same

$\Rightarrow det=0$