Question

If $t_{n}$ denotes the $n$th term of an A.P. and $t_{p}=\frac{1}{q}$ and $t_{q}$ $=\frac{1}{p}$, then which of the following is necessarily a root of the equation $(p+2 q-3 r) x^{2}+(q+2 r-3 p) x+(r$ $+2 p-3 q)=0$

• A. $t_{p}$
• B. $t_{q}$
• C. $t_{p q}$
• D. $t_{p+q}$

Answer 1

Answer: (C)

The sum of the coefficients of the equation $=0$
$\therefore x=1$ is a root of the equation.
Let $a$ be the first term and $d$ be the common difference of given A.P.
$t_{p}=a+(p-1) d=\frac{1}{q}\,\,\, (1)$
and, $t_{q}=a+(q-1) d=\frac{1}{p}\,\,\,(2)$
Solving (1) and (2), $a=d=\frac{1}{p q}$
$\therefore t_{p q}=a+(p q-1) d=1$
$\therefore t_{p q}$ is the root of the given equation.