JEE PYQ: Quadratic Equation Question 15
Question 15 - 2020 (02 Sep Shift 1)
Let $\alpha$ and $\beta$ be the roots of the equation, $5x^2 + 6x - 2 = 0$. If $S_n = \alpha^n + \beta^n$, $n = 1, 2, 3, \ldots$, then:
(1) $6S_6 + 5S_5 = 2S_4$
(2) $6S_6 + 5S_5 + 2S_4 = 0$
(3) $5S_6 + 6S_5 = 2S_4$
(4) $5S_6 + 6S_5 + 2S_4 = 0$
Show Answer
Answer: (3)
Solution
Since, $\alpha$ and $\beta$ are the roots of the equation $5x^2 + 6x - 2 = 0$
Then, $5\alpha^2 + 6\alpha = 2$, $5\beta^2 + 6\beta - 2 = 0$
$5\alpha^2 + 6\alpha = 2$
$5S_6 + 6S_5 = 5(\alpha^6 + \beta^6) + 6(\alpha^5 + \beta^5)$
$= (5\alpha^4 + 6\alpha^5) + (5\beta^6 + 6\beta^5)$
$= \alpha^4(5\alpha^2 + 6\alpha) + \beta^4(5\beta^2 + 6\beta)$
$= 2(\alpha^4 + \beta^4) = 2S_4$
BETA
