Paragraph: Let p, q be integers and let α, β be the roots of the equation, x^ 2 -x-1=0 , where α ≠ β . For n=0,1,2, , let a_ n =p α^ n +q β^ n FACT: If a and b are rational numbers and a+b 5 =0 , then a=0=b . Question

Correct answer: C. a _ 11 + a _ 10 α ^ 2 = α + 1 ⇒ α ^ n = α ^ n - 1 + α ^ n - 2 ⇒ p α ^ n + q β ^ n = p( α ^ n - 1 + α ^ n - 2 ) + q( β ^ n - 1 + β ^ n - 2 ) a _ n = a _ n - 1 + a _ n - 2 ⇒ a _ 12 = a _ 11 + a _ 10

JEE Advanced 2017Paper 2

Exam: JEE Advanced
Year: 2017
Paper: Paper 2
Subject: Mathematics
Chapter: Quadratic Equation

JEE Advanced 2017 · Paper 2 · Mathematics · Quadratic Equation

Paragraph: Let p, q be integers and let α, β be the roots of the...

Paragraph: Let

p, q

be integers and let

\alpha, \beta

be the roots of the equation,

x^{2}-x-1=0

, where

\alpha \neq \beta

. For

n=0,1,2, \ldots

, let

a_{n}=p \alpha^{n}+q \beta^{n}

FACT: If

a

and

b

are rational numbers and

a+b \sqrt{5}=0

, then

a=0=b

.
Question:

a_{12}=

Options

A $2{a}_{11} + {a}_{10}$
B ${a}_{11} - {a}_{10}$
C ${a}_{11} + {a}_{10}$
D ${a}_{11} + 2{a}_{10}$

Correct answer

C. ${a}_{11} + {a}_{10}$

Step-by-step solution

${\alpha}^{2} = \alpha + 1 ⇒ {\alpha}^{n} = {\alpha}^{n - 1} + {\alpha}^{n - 2}$

$⇒ p{\alpha}^{n} + q{\beta}^{n} = p\left({\alpha}^{n - 1} + {\alpha}^{n - 2}\right) + q\left({\beta}^{n - 1} + {\beta}^{n - 2}\right)$

${a}_{n} = {a}_{n - 1} + {a}_{n - 2}$

$⇒ {a}_{12} = {a}_{11} + {a}_{10}$

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