CAT 2023 Slot 1 — QA Question 15

Name the other roots $p,q$. Product of all roots of $x^3+(2r+1)x^2+(4r-1)x+2=0$ is $-\dfrac{2}{1}$:

$$(-2)\cdot p\cdot q=-2\ \Rightarrow\ pq=1\ \Rightarrow\ q=\frac1p$$

So the three roots are $-2,\ p,\ \dfrac1p$.

Sum of roots $=-(2r+1)$:

$$\begin{aligned} &-2+p+\frac1p=-(2r+1)\\ &\Rightarrow\ p+\frac1p=-2r+1 \end{aligned}$$

For real $p$, the quantity $p+\dfrac1p$ never lies strictly between $-2$ and $2$:

$$p+\frac1p\ge 2\quad\text{or}\quad p+\frac1p\le -2$$

Hence $-2r+1\ge 2$ or $-2r+1\le -2$:

$$\begin{aligned} &-2r\ge 1\ \Rightarrow\ r\le -\tfrac12\\ &-2r\le -3\ \Rightarrow\ r\ge \tfrac32 \end{aligned}$$

Least non-negative integer $r$: the allowed band is $r\le -\tfrac12$ or $r\ge \tfrac32$. The smallest non-negative integer satisfying $r\ge\tfrac32$ is $2$.