Consider the quadratic equation $(\mathrm{n}^{2}-2 \mathrm{n}+2) \mathrm{x}^{2}-3 \mathrm{x}+(\mathrm{n}^{2}-2 \mathrm{n}+2)^{2}=0, \mathrm{n} \in \mathrm{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is :

JEE Main · 2026 · 06 Apr 2026 · Shift 2 · Quadratic Equations and Inequalities

Consider the quadratic equation

(n^{2} - 2 n + 2) x^{2} - 3 x + (n^{2} - 2 n + 2)^{2} = 0, n \in R

. Let

α

be the minimum value of the product of its roots and

β

be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is

α

and the common ratio is

\frac{α}{β}

, is :

Answer: (C)

Let

m = n^{2} - 2 n + 2 = (n - 1)^{2} + 1 \geq 1

. The product of the roots is

\frac{m ^{2}}{m} = m

, least when

m = 1

, so

α = 1

; the sum of the roots is

\frac{3}{m}

, greatest when

m = 1

, so

β = 3

. The G.P. has first term

1

and common ratio

\frac{α}{β} = \frac{1}{3}

, so

S_{6} = \frac{1 - ( 1/3 ) ^{6}}{1 - 1/3} = \frac{3}{2} (1 - \frac{1}{729}) = \frac{3}{2} \cdot \frac{728}{729} = \frac{364}{243}

JEE Main · 2026 · 06 Apr 2026 · Shift 2 · Single correct option