Let α and β be the roots of the equation, 5 x ^ 2 + 6x - 2 = 0 . If S _ n = α ^ n + β ^ n , n = 1, 2, 3, . . . . , then

Correct answer: C. 5 S _ 6 + 6 S _ 5 = 2 S _ 4 If α, β are roots of quadratic equation 5 x ^ 2 + 6x - 2 = 0 ⇒ 5 α ^ 2 + 6α - 2 = 0 ⇒ 5 α ^ 2 + 6α = 2 ⇒ 5 β ^ 2 + 6β - 2 = 0 ⇒ 5 β ^ 2 + 6β = 2 5 S _ 6 + 6 S _ 5 = 5( α ^ 6 + β ^ 6 ) + 6( α ^ 5 + β ^ 5 ) = α ^ 4 (5 α ^ 2 + 6α) + β ^ 4 (5 β ^ 2 + 6β) = 2( α ^ 4 + β ^ 4 ) = 2 S _ 4

JEE Main 20202 SeptemberMorning Shift

Exam: JEE Main
Year: 2020
Date: 2 September
Shift: Morning Shift
Subject: Mathematics
Chapter: Quadratic Equation

JEE Main 2020 · 2 September · Morning Shift · Mathematics · Quadratic Equation

Let α and β be the roots of the equation, 5 x ^ 2 + 6x - 2 = 0 . If...

Let

\alpha

and

\beta

be the roots of the equation,

5{x}^{2} + 6x - 2 = 0

. If

{S}_{n} = {\alpha}^{n} + {\beta}^{n}, n = 1, 2, 3, . . . . ,

then

Options

A $6{S}_{6} + 5{S}_{5} = 2{S}_{4}$
B $5{S}_{6} + 6{S}_{5} + 2{S}_{4} = 0$
C $5{S}_{6} + 6{S}_{5} = 2{S}_{4}$
D $6{S}_{6} + 5{S}_{5} + 2{S}_{4} = 0$

Correct answer

C. $5{S}_{6} + 6{S}_{5} = 2{S}_{4}$

Step-by-step solution

If $\alpha, \beta$ are roots of quadratic equation $5{x}^{2} + 6x - 2 = 0$

$⇒ 5{\alpha}^{2} + 6\alpha - 2 = 0 ⇒ 5{\alpha}^{2} + 6\alpha = 2$

$⇒ 5{\beta}^{2} + 6\beta - 2 = 0 ⇒ 5{\beta}^{2} + 6\beta = 2$

$5{S}_{6} + 6{S}_{5} = 5\left({\alpha}^{6} + {\beta}^{6}\right) + 6\left({\alpha}^{5} + {\beta}^{5}\right)$

$= {\alpha}^{4}\left(5{\alpha}^{2} + 6\alpha\right) + {\beta}^{4}\left(5{\beta}^{2} + 6\beta\right)$

$= 2\left({\alpha}^{4} + {\beta}^{4}\right) = 2{S}_{4}$

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