The number of integers $k$ for which the equation $x^3-27 x+k=0$ has at least two distinct integer roots is

  • [KVPY 2016]
  • A

    $1$

  • B

    $2$

  • C

    $3$

  • D

    $4$

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The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is :

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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 $\alpha \neq \beta$. 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]

The complete solution of the inequation ${x^2} - 4x < 12\,{\rm{ is}}$