CAT 2023 Slot 1 QA Q5 Inequalities & Modulus

Question 5 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

The number of integer solutions of equation \( 2|x|\left(x^2+1\right)=5 x^2 \) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 3 Inequalities & Modulus TITA

Solution

\(x=0\) satisfies the equation (both sides 0).
For \(x\ne0\), divide by \(|x|\) (using \(x^2=|x|^2\)): \(2(|x|^2+1) = 5|x|\), so \(2t^2-5t+2=0\) with \(t=|x|\).
\(t = 2\) or \(t = \frac12\); only \(t=2\) gives integers, \(x=\pm2\).
Integer solutions: \(-2, 0, 2\), i.e. 3.

Shortcut. Don't forget \(x=0\), which is lost when you divide by \(|x|\).