It a, b and c are the p^th, q^th and r^th terms of an HP. then $\left|\begin{array}{ccc}b c & c a & a b \\ p & q & r \\ 1 & 1 & 1\end{array}\right|=$

zero

If A is the first term and D is the common difference of the corresponding A.P. then

$\frac{1}{a}=A+(p-1) D$

$\frac{1}{b}=A+(q-1) D$

$\frac{1}{c}=A+(r-1) D$

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

Operating $R_1 \rightarrow R_1-D\left(R_2\right)-(A-D) R_3 ~\Delta=a b c\left|\begin{array}{lll}0 & 0 & 0 \\ p & q & r \\ 1 & 1 & 1\end{array}\right|=0$

Hence (3) is the correct answer.