Let α and β be the roots of the equation x ^ 2 - x - 1 = 0 . If p _ k = (α) ^ k + (β) ^ k , k ≥ 1, then which one of the following statements is not true?

Correct answer: D. p _ 5 = p _ 2 ⋅ p _ 3 α ^ 5 = 5α + 3 β ^ 5 = 5β + 3 P _ 5 = 5(α + β) + 6 = 5(1) + 6 P _ 5 = 11 and P _ 2 = α ^ 2 + β ^ 2 = α + 1 + β + 1 = 3 and P _ 3 = α ^ 3 + β ^ 3 = 2α + 1 + 2β + 1 = 2(1) + 2 = 4 P _ 2 × P _ 3 = 12 and P _ 5 = 11 ⇒ P _ 5 ≠ P _ 2 × P _ 3

JEE Main 20207 JanuaryEvening Shift

Exam: JEE Main
Year: 2020
Date: 7 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Quadratic Equation

JEE Main 2020 · 7 January · Evening Shift · Mathematics · Quadratic Equation

Let α and β be the roots of the equation x ^ 2 - x - 1 = 0 . If p _...

Let

\alpha

and

\beta

be the roots of the equation

{x}^{2} - x - 1 = 0

. If

{p}_{k} = {\left(\alpha\right)}^{k} + {\left(\beta\right)}^{k}, k \ge 1,

then which one of the following statements is not true?

Options

A ${p}_{3} = {p}_{5} - {p}_{4}$
B ${p}_{5} = 11$
C $\left({p}_{1} + {p}_{2} + {p}_{3} + {p}_{4} + {p}_{5}\right) = 26$
D ${p}_{5} = {p}_{2} ⋅ {p}_{3}$

Correct answer

D. ${p}_{5} = {p}_{2} ⋅ {p}_{3}$

Step-by-step solution

${\alpha}^{5} = 5\alpha + 3$
${\beta}^{5} = 5\beta + 3$

${P}_{5} = 5\left(\alpha + \beta\right) + 6$
$= 5\left(1\right) + 6$
${P}_{5} = 11$ and

${P}_{2} = {\alpha}^{2} + {\beta}^{2} = \alpha + 1 + \beta + 1 = 3$
and ${P}_{3} = {\alpha}^{3} + {\beta}^{3} = 2\alpha + 1 + 2\beta + 1 = 2\left(1\right) + 2 = 4$
${P}_{2} \times {P}_{3} = 12$ and ${P}_{5} = 11 ⇒ {P}_{5} \ne {P}_{2} \times {P}_{3}$

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