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?

Q39 / 75 · JEE Main 2020 (07 Jan Shift 2) · Mathematics

Let $α$ and $β$ be the roots of the equation $x^{2} - x - 1 = 0$ . If $p_{k} = {(α)}^{k} + {(β)}^{k}, k \geq 1,$ then which one of the following statements is not true?

A
$p_{3} = p_{5} - p_{4}$
B
$p_{5} = 11$
C
$(p_{1} + p_{2} + p_{3} + p_{4} + p_{5}) = 26$
Correct answer
D
$p_{5} = p_{2} \cdot p_{3}$

Solution

$α^{5} = 5 α + 3$
$β^{5} = 5 β + 3$

$P_{5} = 5 (α + β) + 6$
$= 5 (1) + 6$
$P_{5} = 11$ and

$P_{2} = α^{2} + β^{2} = α + 1 + β + 1 = 3$
and $P_{3} = α^{3} + β^{3} = 2 α + 1 + 2 β + 1 = 2 (1) + 2 = 4$
$P_{2} \times P_{3} = 12$ and $P_{5} = 11 \Rightarrow P_{5} \neq P_{2} \times P_{3}$

Hence, the correct answer is option: 'D'.

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Let α and β be the roots of the equation x2-x-1=0 . If pk=αk+βk,k≥1, then which one of the following statements is not true?

Solution

Relevant Questions

Let $α$ and $β$ be the roots of the equation $x^{2} - x - 1 = 0$ . If $p_{k} = {(α)}^{k} + {(β)}^{k}, k \geq 1,$ then which one of the following statements is not true?