If (3+2\sqrt2) is a root of the equation ax^2+bx+c=0 and (4+2\sqrt3) is a root of the equation ay^2+my+n = 0 where a,b, c, m and n are integers, then the value of (b/m+{c-2b}/n) is

CAT 2022 Slot 3 Paper

Question : 50 of 66

Marks: +1, -0

If $(3+2√2)$ is a root of the equation $ax^2+bx+c=0$ and $(4+2√3)$ is a root of the equation $ay^2+my+n = 0$ where $a, b, c, m$ and $n$ are integers, then the value of $(b/m+{c-2b}/n)$ is

Validate

Solution:

{\a}, {\b}, {\c}, {\m}

and

{\n}

are integers so if one root is

3+2 {√{2}}

then the other root is

3-2 {√{2}}

Sum of roots

=6=-b / a

b=-6 a

Product of roots

=1= {\c} / {\a}

{\c}= {\a}

{\a}, {\b}, {\c}, {\m}

and

{\n}

are integers so if one root is

4+2 {√{3}}

then the other root is

4-2 {√{3}}

Sum of roots

=8=-m \/ a

m=-8 a

product of roots

=4=n \/ a

n=4 a

({b}/{m}+{c-2 b}/{n})

={6 a}/{8 a}+{(a+12 a)}/{4 a}={3}/{4}+{13}/{4}={16}/{4}=4

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