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CAT 2022 Slot 3 Paper

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Question : 50 of 66

Marks: +1, -0

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

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=-\frac{b}{a}

or

b=-6a

Product of roots

=1=\frac{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=-\frac{m}{a}

or

m=-8a

product of roots

=4=\frac{n}{a}

or

n=4a

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

=\frac{6a}{8a}+\frac{a+12a}{4a}=\frac{3}{4}+\frac{13}{4}=\frac{16}{4}=4

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