Question

Find the value of the determinant $\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$, where a, b and c are respectively the pth, qth and rth terms of a harmonic progression.

• A. 1
• B. pqr
• C. \N
• D. $\frac{1}{abc}$

Answer 1

The correct option is C \N

Since, a, b, c are pth, qth and rth terms of HP.
$\Rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in an AP.
$\Rightarrow \begin{array}{c}\frac{1}{a}=A+(p-1)D\\ \frac{1}{b}=A+(q-1)D\\ \frac{1}{c}=A+(r-1)D\end{array}\}$ . . . (i)
Let $\Delta =\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|=abc\left|\begin{array}{ccc}\frac{1}{a}& \frac{1}{b}& \frac{1}{c}\\ p& q& r\\ 1& 1& 1\end{array}\right|$ [from Eq. (i)]
$=abc\left|\begin{array}{ccc}A+(p-1)D& A+(q-1)D& A+(r-1)D\\ p& q& r\\ 1& 1& 1\end{array}\right|Applying{R}_1\to R_1-(A-D)R_3-D{R}_2=abc\left|\begin{array}{ccc}0& 0& 0\\ p& q& r\\ 1& 1& 1\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|=0$