Matrices and Determinants 2 Question 33
37. Find the value of the determinant $\begin{vmatrix}b c & c a & a b \\ p & q & r \\ 1 & 1 & 1\end{vmatrix}$, where $a, b$ and $c$ are respectively the $p$ th, $q$ th and $r$ th terms of a harmonic progression.
(1997C, 2M)
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Answer:
Correct Answer: 37. (0)
Solution:
- Since, $a, b, c$ are $p$ th, $q$ th and $r$ th terms of HP.
$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text { are in an AP. } $
$\frac{1}{a}=A+(p-1) D $
$\frac{1}{b}=A+(q-1) D $
$\frac{1}{c}=A+(r-1) D \quad$ ….(i)
Let $\Delta=\begin{vmatrix}b c & c a & a b \\ p & q & r \\ 1 & 1 & 1\end{vmatrix}=a b c\begin{vmatrix}\frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\ p & q & r \\ 1 & 1 & 1\end{vmatrix}$
$ =a b c\begin{vmatrix} A+(p-1) D & A+(q-1) D & A+(r-1) D \\ p & q & r \\ 1 & 1 & 1 \end{vmatrix} $
Applying $R_{1} \rightarrow R_{1}-(A-D) R_{3}-D R_{2}$
$ =a b c\begin{vmatrix} 0 & 0 & 0 \\ p & q & r \\ 1 & 1 & 1 \end{vmatrix} \Rightarrow 0 \Rightarrow\begin{vmatrix} b c & c a & a b \\ p & q & r \\ 1 & 1 & 1 \end{vmatrix}=0 $
