Multiple choice

If $p,q$ be non-zero real numbers and $f\left(x\right)\neq 0$ in $\left[0,2\right]$ and ${\int}{0}^{1}f\left(x\right).\left(x^{2}+px+q\right)dx={\int}{0}^{2}f\left(x\right).\left(x^{2}+px+q\right)dx=0$ then equation $x^{2}+px+q=0$ has

  1. Two imaginary roots
  2. No root in $\left(0,2\right)$
  3. One root in $\left(0,1\right)$ and other in $\left(1,2\right)$
  4. One root in $\left(-\infty,0\right)$ and other in $\left( 2,\infty\right)$
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