Maths Q2 – JEE-MAIN 2026

Home Maths Complex & QuadraticQuestion #80

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:

Clue 1 of 3: Quadratic Equation Properties

Identify the coefficients of the quadratic equation and express them in terms of a single variable to simplify the problem.

🧠 Full Solution Path

Step 1: Analyze the quadratic equation and define key terms✦ Active

The given quadratic equation is $(n^{2} - 2 n + 2) x^{2} - 3 x + (n^{2} - 2 n + 2)^{2} = 0$ . Let $k = n^{2} - 2 n + 2$ . This can be rewritten as $k = (n - 1)^{2} + 1$ . Since $(n - 1)^{2} \geq 0$ , the minimum value of $k$ is $1$ (when $n = 1$ ). Thus, $k \geq 1$ . The equation becomes $k x^{2} - 3 x + k^{2} = 0$ . The product of roots is $P = \frac{k^{2}}{k} = k$ . The sum of roots is $S = \frac{- (- 3)}{k} = \frac{3}{k}$ .

Step 2: Determine

α

and

β

○ Expand

$α$ is the minimum value of the product of roots, $P$ . Since $P = k$ and $k \geq 1$ , the minimum value of $P$ is $1$ . So, $α = 1$ . $β$ is the maximum value of the sum of roots, $S$ . Since $S = \frac{3}{k}$ and $k \geq 1$ , to maximize $S$ , $k$ must be minimized. The minimum value of $k$ is $1$ . So, the maximum value of $S$ is $\frac{3}{1} = 3$ . Thus, $β = 3$ .

Step 3: Calculate the sum of the first six terms of the G.P.○ Expand

The first term of the G.P. is $a = α = 1$ . The common ratio is $r = \frac{α}{β} = \frac{1}{3}$ . The sum of the first six terms of a G.P. is $S_{6} = a \frac{1 - r^{6}}{1 - r}$ . Substituting the values:

S_{6} = 1 \cdot \frac{1 - {(\frac{1}{3})}^{6}}{1 - \frac{1}{3}}

S_{6} = \frac{1 - \frac{1}{729}}{\frac{2}{3}}

S_{6} = \frac{\frac{729 - 1}{729}}{\frac{2}{3}}

S_{6} = \frac{\frac{728}{729}}{\frac{2}{3}}

S_{6} = \frac{728}{729} \times \frac{3}{2}

S_{6} = \frac{364}{243}

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Maths Question 2 – JEE-MAIN 2026

Clue 1 of 3: Quadratic Equation Properties

🧠 Full Solution Path