JEE Advanced 2017 · Paper 2 · Mathematics Q18

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MathematicsSingle correct+3Question 54 of 54

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Let $p$ , $q$ be integers and let $\alpha$ , $\beta$ be the roots of the equation, $x^2 - x - 1 = 0$ , where $\alpha \ne \beta$ . 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 $a = 0 = b$ .

If $a_4 = 28$ , then $p + 2q =$

Answer and worked solution

Official answer: D

Fourth powers. $\alpha,\beta=\dfrac{1\pm\sqrt5}2$ . Then $\alpha^2=\dfrac{3+\sqrt5}2$ and $\alpha^4=\dfrac{7+3\sqrt5}2$ ; $\beta^4$ is the same with $-\sqrt5$ .

Use a₄ = 28.
$a_4=\frac{7(p+q)}2+\frac{3\sqrt5(p-q)}2=28.$

Apply the FACT. $p-q=0$ and $7(p+q)=56$ , so $p=q=4$ .

Answer. $p+2q=12$ .

(D)

Topics:Roots and coefficients Number systems

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