CAT 2022 Slot 3 QA Question & Solution
AlgebraMedium
Question
If $(3+2\sqrt{2})$ is a root of the equation $ax^{2}+bx+c=0$ and $(4+2\sqrt{3})$ is a root of the equation $ay^{2}+my+n=0$ where a, b, c, m and n are integers, then the value of $(\frac{b}{m}+\frac{c-2b}{n})$ is
Options
0
1
3
4
Solution
a, b, c, m and n are integers so if one root is $3+2\sqrt{2}$ then the other root is $3-2\sqrt{2}$
Sum of roots = 6 = -b/a or b= -6a
Product of roots = 1 = c/a or c=a
a, b, c, m and n are integers so if one root is $4+2\sqrt{3}$ then the other root is $4-2\sqrt{3}$
Sum of roots = 8 = -m/a or m = -8a
product of roots = 4 = n/a or n = 4a
$(\frac{b}{m}+\frac{c-2b}{n})$
= $\frac{6a}{8a}+\frac{\left(a+12a\right)}{4a}=\frac{3}{4}+\frac{13}{4}=\frac{16}{4}=4$