The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

  • [JEE MAIN 2023]
  • A

    exactly two solutions in $(-\infty, \infty)$

  • B

    no solution

  • C

    a unique solution in $(-\infty, 1)$

  • D

    a unique solution in $(-\infty, \infty)$

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$FACT$ : 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]