Question

If $a,b,c,$ are respectively the $p^{th},q^{th},r^{th}$ terms of an $AP$ , then $\left[\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right]$ is equal to:

• A. 1
• B. -1
• C. 0
• D. pqr

Answer 1

Answer: (C)

Let the first term be $a_1$ and common difference be $d.$
According to the condition $T_p=a_1+(p-1)d$
$\Rightarrow$ $a=a_1+(p-1)d$ ... (i)
$T_q=a_1+(q-1)d$
$\Rightarrow$ $b=a_1+(q-1)d$ ... (ii)
and $T_r=a_1+(r-1)d$
$\Rightarrow$ $c=a_1+(r-1)d$ ... (iii)
$\therefore$ $\left|\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right|=\left|\begin{array}{ccc}{a}_1& +(p-1)d& p1\\ a_1& +(q-1)d& p1\\ a_1& +(r-1)d& r1\end{array}\right|$
Applying $C_1\to C_1-(C_2-C_1)d$
$=\left|\begin{array}{ccc}{a}_1& p& 1\\ a_1& q& 1\\ a_1& r& 1\end{array}\right|$
$=a_1\left|\begin{array}{ccc}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|$
$=0[\because$two columns are identical]
Note: If any two rows or columns are identical or proportional, then the value of determinant will be zero: