cat 2022 Question 14

Question

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

Accepted Answer

$a, b, c, m$ 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 $= 6 = -\dfrac{b}{a}$ or $b = -6a$

Product of roots $= 1 = \dfrac{c}{a}$ 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 $= 8 = -\dfrac{m}{a}$ or $m = -8a$

product of roots $= 4 = \dfrac{n}{a}$ or $n = 4a$

($\frac{b}{m} + \frac{(c-2b)}{n}$)

= $\frac{6a}{8a} + \frac{a+12a}{4a}$ = $\frac34 + \frac{13}{4} = \frac{16}{4} = 4$

Question 14.

Question Explanation