Question

If $l,m,n$ are respectively the $p^{th},q^{th},r^{th}$ terms of an $H.P$. then$\left|\begin{array}{ccc}mn& ln& lm\\ p& q& r\\ 1& 1& 1\end{array}\right|$ equals to

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

Answer 1

Answer: (C)

Let the first term of an $A.P$. be $A$ and common difference is $d$.
As $l,m,n$ are terms of $H.P$.
$\therefore \frac{1}{l},\frac{1}{m},\frac{1}{n}$ are terms of $A.P$.
$\therefore \frac{1}{l}=A+\left(p-1\right)d$
$\therefore mn\left(q-1\right)=lmn\left[A+\left(p-1\right)d\right]\left(q-r\right)$
(Multiplying & dividing by $I$)
$\frac{1}{m}=A+\left(q-1\right)d$
$\therefore ln\left(r-p\right)=lmn\left[A+\left(q-1\right)d\right]\left(r-p\right)$
$\frac{1}{n}=A+\left(r-1\right)d$
$\therefore lm\left(p-q\right)=lmn\left[A+\left(r-1\right)d\right]\left(p-q\right)$
Adding the above relations, we get
$mn\left(q-r\right)+ln\left(r-p\right)+lm\left(p-q\right)=0$
or $\left|\begin{array}{ccc}mn& ln& lm\\ p& q& r\\ 1& 1& 1\end{array}\right|=0$