CAT 2022 Slot 3 QA Q14 Equations

Question 14 of the Quantitative Ability (QA) section, CAT 2022 Slot 3, with CAT's official answer and a worked solution.

If \((3+2 \sqrt{2})\) is a root of the equation \(a x^2+b x+c=0\), and \((4+2 \sqrt{3})\) is a root of the equation \(a y^2+m y+n=0\), where \(a, b, c, m\) and \(n\) are integers, then the value of \(\left(\frac{b}{m}+\frac{c-2 b}{n}\right)\) is

A
3
B
1
C
4
D
0

Answer: (C) 4 Equations MCQ

Solution

With integer coefficients the conjugate \(3-2\sqrt2\) is also a root: sum 6, product 1, so \(b=-6a,\ c=a\).
Similarly \(4\pm2\sqrt3\): sum 8, product \(16-12=4\), so \(m=-8a,\ n=4a\).
\(\frac bm=\frac{-6a}{-8a}=\frac34\) and \(\frac{c-2b}n=\frac{a+12a}{4a}=\frac{13}4\).
Sum \(=\frac{16}4=4\).