# CAT 2022 Set-3 | Quantitative Aptitude | Question: 3

430 views

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

1. $0$
2. $3$
3. $4$
4. $1$

The roots of ax$^{2}$+bx+c=0 are (-b+$\sqrt{b^{2} -4ac}$)/2a and (-b-$\sqrt{b^{2} -4ac}$)/2a. And one of the root is (3+2$\sqrt{2}$). The value (3+2$\sqrt{2}$) is equal to (-b+$\sqrt{b^{2} -4ac}$)/2a because sign of irrational part of both are same. ($\sqrt{2}$ is irrational)

Compare rational part and irrational part separately,

(-b+$\sqrt{b^{2} -4ac}$)/2a = (3+2$\sqrt{2}$)

=> (-b+$\sqrt{b^{2} -4ac}$) = (6a+$\sqrt{32a^{2}}$)  => b = -6a  &  c = a (b$^{2}$-4ac = 32a$^{2}$  => 36a$^{2}$ – 4ac = 32a$^{2}$  => c = a).

Similarly we have to do for 2nd euqation :-

The roots of ax$^{2}$+mx+n=0 are (-m+$\sqrt{m^{2} -4an}$)/2a and (-m-$\sqrt{m^{2} -4an}$)/2a. And one of the root is (4+2$\sqrt{3}$). The value (4+2$\sqrt{3}$) is equal to (-m+$\sqrt{m^{2} -4an}$)/2a because sign of irrational part of both are same. ($\sqrt{3}$ is irrational)

Compare rational part and irrational part separately,

(-m+$\sqrt{m^{2} -4an}$)/2a = (4+2$\sqrt{3}$)

=> (-m+$\sqrt{m^{2} -4an}$) = (8a+$\sqrt{48a^{2}}$)  => m = -8a  &  n = 4a (m$^{2}$-4an = 48a$^{2}$  => 64a$^{2}$ – 4an = 48a$^{2}$  => n = 4a).

Therefore the value of (b/m)+(c-2b)/n = (-6a/-8a)+(a+12a)/4a = 4.

## Related questions

A donation box can receive only cheques of ₹$100$, ₹$250$, and ₹$500$. On one good day, the donation box was found to contain exactly $100$ cheques amounting to a t...
If $c=\dfrac{16 x}{y}+\dfrac{49 y}{x}$ for some non-zero real numbers $x$ and $y,$ then $c$ cannot take the value$-60$ $-50$ $60$ $-70$
Suppose the medians $\mathrm{BD}$ and $\mathrm{CE}$ of a triangle $\mathrm{ABC}$ intersect at a point $\mathrm{O}$. If area of triangle $\mathrm{ABC}$ is $108$ $\mathrm{s... 1 votes 0 answers 4 431 views Bob can finish a job in$40$days, if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in the same job. Suppose Alex and Bob work together on the f... 1 votes 0 answers 5 486 views A glass contains$500 \mathrm{cc}$of milk and a cup contains$500 \mathrm{cc}$of water. From the glass,$150 \mathrm{cc}\$ of milk is transferred to the cup and mixed th...