JEE PYQ: Quadratic Equation Question 26
Question 26 - 2020 (07 Jan Shift 2)
Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - x - 1 = 0$. If $p_k = (\alpha)^k + (\beta)^k$, $k \geq 1$, then which one of the following statements is not true?
(1) $p_3 = p_5 - p_4$
(2) $P_5 = 11$
(3) $(p_1 + p_2 + p_3 + p_4 + p_5) = 26$
(4) $p_5 = p_2 \cdot p_3$
Show Answer
Answer: (4)
Solution
$\alpha^5 = 5\alpha + 3$
$\beta^5 = 5\beta + 3$
$p_5 = 5(\alpha + \beta) + 6 = 5(1) + 6 = 11$
$p_5 = 11$ and $p_3 = \alpha^3 + \beta^3 = \alpha + 1 + \beta + 1 = 4$
$p_1 = 1$ and $p_2 = \alpha^2 + \beta^2 = 2\alpha + 1 + 2\beta + 1 = 4$ (wait: actually $p_2 = (\alpha+\beta)^2 - 2\alpha\beta = 1 + 2 = 3$)
$p_2 \times p_3 = 12$ and $p_5 = 11 \Rightarrow p_5 \neq p_2 \times p_3$
BETA
