Question

If a, b, c are respectively the $p^{th},q^{th},r^{th}$ terms of an A.P., the $\left|\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right|=$

• A. 1
• B. -1
• C. pqr

Answer 1

Let first term = A and common difference = D
$\therefore a=A+(p-1)D,b=A+(q-1)D,c=A+(r-1)D\left|\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right|=\left|\begin{array}{ccc}A+(p-1)D& p& 1\\ A+(q-1)D& q& 1\\ A+(r-1)D& r& 1\end{array}\right|operate{C}_1\to C_1-D{C}_2+D{C}_3=\left|\begin{array}{ccc}A& p& 1\\ A& q& 1\\ A& r& 1\end{array}\right|=A\left|\begin{array}{ccc}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|=0.$