Determinants Matrices Question 148
Question: If $ a>0,b>0,c>0 $ are respectively the pth, qth,rth terms of GP, then the value of the determinant $ \begin{vmatrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \\ \end{vmatrix} $ is
Options:
A) $ 0 $
B) $ 1 $
C) $ -1 $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Let A be the 1st term and R the common ration of G.P., then; $ a=T _{p}=A{R^{p-1}}\therefore \log a=\log A+(p-1)logR $ Similarly, $ \log b=\log A+(q-1)logR $ and $ \log c=logA+(r-1)logR $
$ \therefore \Delta = \begin{vmatrix} & \log A+(p-1)logRp1 \\ & logA+(q-1)logRq1 \\ & \log A+(r-1)logRr1 \\ \end{vmatrix} $ Split into two determinants and in the first take log A common and in the second take log R common $ \Delta =\log A \begin{vmatrix} 1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1 \\ \end{vmatrix}+\log R \begin{vmatrix} p-1 & p & 1 \\ q-1 & q & 1 \\ r-1 & r & 1 \\ \end{vmatrix} $ Apply $ C_1\to C_1-C_2+C_3 $ in the second $ \Delta =0+\log R \begin{vmatrix} 0 & p & 1 \\ 0 & q & 1 \\ 0 & r & 1 \\ \end{vmatrix}=0 $
BETA
