CAT 2022 Slot 2 QA Q9 Equations

Question 9 of the Quantitative Ability (QA) section, CAT 2022 Slot 2, with CAT's official answer and a worked solution.

Let \(r\) and \(c\) be real numbers. If \(r\) and \(-r\) are roots of \(5 x^3+c x^2-10 x+9=0\), then \(c\) equals

A
\(-\frac{9}{2}\)
B
\(\frac{9}{2}\)
C
\(-4\)
D
\(4\)

Answer: (A) -9/2 Equations MCQ

Solution

Let the roots be \(r,-r,s\). Sum of pairwise products: \(-r^2+rs-rs=-r^2=\frac{-10}{5}\Rightarrow r^2=2\).
Product: \(-r^2s=-\frac95\Rightarrow s=\frac9{10}\).
Sum: \(s=-\frac c5\Rightarrow c=-\frac92\).