If $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$ , where $a$ and $b$ are natural numbers, then $a + b$ equals

CAT 2024 Slot 3 QA Question 10

Multiple choice (+3 / −1) · Algebra · Identities · Try it, then check the answer and solution below.

CAT 2024 Slot 3QAAlgebra • IdentitiesModerate

(a + b\sqrt{3})^2 = 52 + 30\sqrt{3} , where

and

are natural numbers, then

a + b$ equals

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Official Convenor Answer Key: A

Answer and solution

Answer: A) 8

📌 Core Concept

The problem involves expanding a binomial expression and equating rational and irrational parts to solve for variables. The governing formula is the expansion of

(x + y)^2 = x^2 + 2xy + y^2

🔢 Step-by-Step Solution

Expand the given expression:

(a + b\sqrt{3})^2 = a^2 + 2ab\sqrt{3} + 3b^2.

Set the expanded form equal to the given value:

a^2 + 3b^2 + 2ab\sqrt{3} = 52 + 30\sqrt{3}.

Equate rational and irrational parts:

Rational part:

a^2 + 3b^2 = 52

Irrational part:

2ab = 30

Solve the irrational part equation:

2ab = 30 \implies ab = 15.

Find natural number pairs (a, b) such that ab = 15:

Possible pairs: (1, 15), (3, 5), (5, 3), (15, 1).

Test each pair in the rational part equation:

(1, 15):

CAT 2024 Slot 3 QA Question 10

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