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CAT 2022 Slot 3 — QA Question 18

Forming a Quadratic Equation and Relation between roots and coefficientsEasy

If (3 + 2√2) is a root of the equation ax2 + bx + c = 0, and (4 + 2√3) is a root of the equation ay2 + my + n = 0, where a, b, c, m and n are integers, then the value of (bm+c-2bn) is

Answer & solution

  • A

    1

  • B

    0

  • 4

  • D

    3

Solution

Easy

Integer-coefficient quadratics force surd roots to come in conjugate pairs. Use sum and product of roots to express b,c,m,nb,c,m,n in terms of aa — then aa cancels in the target expression.

1

First equation. If 3+223+2\sqrt2 is a root, its conjugate 3−223-2\sqrt2 is the other:

sum=6=−ba⇒b=−6aproduct=(3)2−(22)2=9−8=1=ca⇒c=a\begin{aligned} &\text{sum}=6=-\frac{b}{a}\Rightarrow b=-6a\\ &\text{product}=(3)^2-(2\sqrt2)^2=9-8=1=\frac{c}{a}\Rightarrow c=a \end{aligned}
2

Second equation. Roots 4±234\pm 2\sqrt3:

sum=8=−ma⇒m=−8aproduct=16−12=4=na⇒n=4a\begin{aligned} &\text{sum}=8=-\frac{m}{a}\Rightarrow m=-8a\\ &\text{product}=16-12=4=\frac{n}{a}\Rightarrow n=4a \end{aligned}
3

Plug in; aa cancels.

bm+c−2bn=−6a−8a+a−2(−6a)4a=34+13a4a=34+134=164=4\begin{aligned} \frac{b}{m}+\frac{c-2b}{n}&=\frac{-6a}{-8a}+\frac{a-2(-6a)}{4a}\\ &=\frac{3}{4}+\frac{13a}{4a}=\frac{3}{4}+\frac{13}{4}=\frac{16}{4}=4 \end{aligned}

The value is 4\mathbf{4}.

Surd root ⇒\Rightarrow conjugate root for free with integer coefficients. Sum/product of roots gives every coefficient as a multiple of aa, so any ratio of coefficients is independent of aa.

CAT 2022 Slot 3 QA Q18: 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 — Solution | TheCATExam