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CAT 2024 Slot 3 — QA Question 13

Square root of SurdsEasy

(a + b√3)2 = 52 + 30√3, where a and b are natural numbers, then a + b equals

Answer & solution

  • A

    9

  • B

    7

  • 8

  • D

    10

Solution

Easy

Expand (a+b3)2(a+b\sqrt3)^2 and match the rational part with 5252 and the 3\sqrt3-coefficient with 3030. Solve the two equations over the natural numbers.

1

Expand and compare.

(a+b3)2=a2+3b2+2ab3⇒ a2+3b2=52(rational parts)⇒ 2ab=30⇒ab=15(3 coefficients)\begin{aligned} &(a+b\sqrt3)^2=a^2+3b^2+2ab\sqrt3\\ &\Rightarrow\ a^2+3b^2=52\quad\text{(rational parts)}\\ &\Rightarrow\ 2ab=30\Rightarrow ab=15\quad\text{(}\sqrt3\text{ coefficients)} \end{aligned}
2

Solve over naturals. From ab=15ab=15 the natural pairs are (1,15),(3,5),(5,3),(15,1)(1,15),(3,5),(5,3),(15,1). Test in a2+3b2=52a^2+3b^2=52.

a=5, b=3: 25+3(9)=52 ✓⇒ a=5, b=3\begin{aligned} &a=5,\ b=3:\ 25+3(9)=52\ \checkmark\\ &\Rightarrow\ a=5,\ b=3 \end{aligned}
3

Required sum.

a+b=5+3=8\begin{aligned} &a+b=5+3=8 \end{aligned}
88
CAT 2024 Slot 3 QA Q13: (a + b√3) 2 = 52 + 30√3, where a and b are natural numbers, then a + b equals — Solution | TheCATExam