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CAT 2021 Slot 2 — QA Question 17

Forming a Quadratic Equation and Relation between roots and coefficientsEasy

Suppose one of the roots of the equation ax2 – bx + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c3 then |a| equals

Answer & solution

  • 2

  • B

    4

  • C

    3

  • D

    1

Solution

Easy

With rational coefficients, an irrational root 2+32+\sqrt3 forces its conjugate 2−32-\sqrt3 to also be a root. Use sum and product of roots of ax2−bx+c=0ax^{2}-bx+c=0 to express bb and cc in terms of aa, then apply b=c3b=c^{3}.

1

Sum of roots. For ax2−bx+c=0ax^{2}-bx+c=0, sum =ba=\dfrac{b}{a}.

ba=(2+3)+(2−3)=4⇒ b=4a\begin{aligned} &\frac{b}{a}=(2+\sqrt3)+(2-\sqrt3)=4\\ &\Rightarrow\ b=4a \end{aligned}
2

Product of roots. Product =ca=\dfrac{c}{a}.

ca=(2+3)(2−3)=4−3=1⇒ c=a\begin{aligned} &\frac{c}{a}=(2+\sqrt3)(2-\sqrt3)=4-3=1\\ &\Rightarrow\ c=a \end{aligned}
3

Apply b=c3b=c^{3}. Substitute step 1 and step 2.

4a=a3⇒ a2=4(divide by a≠0)⇒ a=±2 ⇒ ∣a∣=2\begin{aligned} &4a=a^{3}\\ &\Rightarrow\ a^{2}=4 \quad\text{(divide by } a\neq 0)\\ &\Rightarrow\ a=\pm 2\ \Rightarrow\ |a|=2 \end{aligned}
∣a∣=2|a|=2
CAT 2021 Slot 2 QA Q17: Suppose one of the roots of the equation ax 2 – bx + c = 0 is 2 + √3, where a, b and c are rationa — Solution | TheCATExam