CAT 2023 Slot 3 QA Q5 Equations

Question 5 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

A quadratic equation \(x^2+b x+c=0\) has two real roots. If the difference between the reciprocals of the roots is \(\frac{1}{3}\), and the sum of the reciprocals of the squares of the roots is \(\frac{5}{9}\), then the largest possible value of \((b+c)\) is

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

Answer: 9 Equations TITA

Solution

Let \(u, v\) be the reciprocals of the roots. \((u-v)^2 = \frac19\) and \(u^2+v^2=\frac59\).
So \(2uv = \frac59-\frac19 = \frac49\), \(uv=\frac29\), and \((u+v)^2 = \frac59+\frac49 = 1\), so \(u+v=\pm1\).
Product of roots \(c = \frac{1}{uv} = \frac92\). Sum of roots \(= \frac{u+v}{uv} = \pm\frac92\), so \(b = \mp\frac92\).
Largest \(b+c = \frac92+\frac92 = 9\) (discriminant \(\frac{81}{4}-18>0\), so the roots are real).