Q54 / 160 · TS EAMCET 2022 (20 Jul Shift 2) · Mathematics

If $\alpha, \beta$ are the irrational roots of the equation $3 p^2 x^3+p x^2$ $+q x+3=0$ when $p=1$ and $q=-7$ then $|\alpha-\beta|=$

  • A
    $\frac{3 \sqrt{13}}{2}$
  • B
    $\frac{\sqrt{3}}{2}$
  • Correct answer
    C
    $\frac{2 \sqrt{13}}{3}$
  • D
    4

Solution

Here we are given that $3 p^2 x^3+p x^2+q x+3=0$ when $\mathrm{p}=1$ and $\mathrm{q}=-7$ $$ \begin{aligned} & \text { Now } 3 x^3+x^2-7 x+3=0 \Rightarrow(x-1)\left(3 x^2+4 x-3\right)=0 \\ & \Rightarrow x=1, x=\frac{2 \pm \sqrt{13}}{3} \\ & \Rightarrow \alpha=\frac{2+\sqrt{13}}{3}, \beta=\frac{2-\sqrt{13}}{3} \\ & \Rightarrow|\alpha-\beta|=\left|\frac{2+\sqrt{13}-2+\sqrt{13}}{3}\right|=\frac{2 \sqrt{13}}{3} \end{aligned} $$

Hence, the correct answer is option: 'C'.

Relevant Questions