If tn denotes the nth term of an A.P. and tp=(1/q) and tq =(1/p), then which of the following is necessarily a root of the equation (p+2 q-3 r) x2+(q+2 r-3 p) x+(r +2 p-3 q)=0

Question

If tn denotes the nth term of an A.P. and tp=(1/q) and tq =(1/p), then which of the following is necessarily a root of the equation (p+2 q-3 r) x2+(q+2 r-3 p) x+(r +2 p-3 q)=0 (A) tp (B) tq (C) tp q (D) tp+q

Tardigrade Admin · Accepted Answer

The sum of the coefficients of the equation

= 0

∴ 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}{pq}

∴ t_{pq} = a + (pq - 1) d = 1

∴ t_{pq}

is the root of the given equation.

Q. 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 $(p + 2 q - 3 r) x^{2} + (q + 2 r - 3 p) x + (r$ $+ 2 p - 3 q) = 0$

$t_{p}$

$t_{q}$

$t_{pq}$

$t_{p + q}$

Q. If tn​ denotes the nth term of an A.P. and tp​=q1​ and tq​ =p1​, then which of the following is necessarily a root of the equation (p+2q−3r)x2+(q+2r−3p)x+(r +2p−3q)=0

tp​

tq​

tpq​

tp+q​

Q. 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 $(p + 2 q - 3 r) x^{2} + (q + 2 r - 3 p) x + (r$ $+ 2 p - 3 q) = 0$

$t_{p}$

$t_{q}$

$t_{pq}$

$t_{p + q}$