Question

If $a,b,c$ be respectively the $p^{th},q^{th}$ and $r^{th}$ terms of a H.P, then $\Delta =\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$ equals

• A. $1$
• B. $0$
• C. $-1$
• D. none of these

Answer 1

Answer: (B)

$0$

As a,b,c are pth, qth, rth terms of H.P.

Then $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are pth, qth, rth terms of A.P.

Thus

$\frac{1}{a}=A+(p-1)D,\frac{1}{b}=A+(q-1)D,\frac{1}{c}=A+(r-1)D$

Now

$\Delta =\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$

$=abc\left|\begin{array}{ccc}\frac{1}{a}& \frac{1}{b}& \frac{1}{c}\\ p& q& r\\ 1& 1& 1\end{array}\right|$

$=abc\left|\begin{array}{ccc}A+(p-1)D& A+(q-1)D& A+(r-1)D\\ p& q& r\\ 1& 1& 1\end{array}\right|$

Applying $R_1\to R_1-(A-D)R_3-D{R}_2$, we get

$\Delta =abc\left|\begin{array}{ccc}0& 0& 0\\ p& q& r\\ 1& 1& 1\end{array}\right|=0$