The solution set of the equation $pq{x^2} - {(p + q)^2}x + {(p + q)^2} = 0$ is

  • A

    $\left\{ {\frac{p}{q},\,\frac{q}{p}} \right\}$

  • B

    $\left\{ {pq,\,\frac{p}{q}} \right\}$

  • C

    $\left\{ {\frac{q}{p},\,pq} \right\}$

  • D

    $\left\{ {\frac{{p + q}}{p},\,\frac{{p + q}}{q}} \right\}$

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  • [KVPY 2019]

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 $a=0=b$.

($1$) $a_{12}=$

$[A]$ $a_{11}-a_{10}$  $[B]$ $a_{11}+a_{10}$  $[C]$ $2 a_{11}+a_{10}$   $[D]$ $a_{11}+2 a_{10}$

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

$[A] 21$   $[B] 14$   $[C] 7$    $[D] 12$

 answer the quetion ($1$) and ($2$)

  • [IIT 2017]

Equation $\frac{3}{{x - {a^3}}} + \frac{5}{{x - {a^5}}} + \frac{7}{{x - {a^7}}} = 0,a > 1$ has