CAT 2024 Slot 2 QA Question 5

Type-in-the-answer (no negative marking) · Algebra · Indices · Try it, then check the answer and solution below.

(x+6\

\sqrt{2}

)^{\frac{1}{2}} - (x-6\

\sqrt{2}

)^{\frac{1}{2}} = 2\$\

\sqrt{2}$$$ , then x equals

TITA Answer:

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: 11

Answer and solution

Answer: 11

📌 Core Concept

We are given the equation:

\sqrt{x + 6\sqrt{2}} - \sqrt{x - 6\sqrt{2}} = 2\sqrt{2}

To solve for $x , we can use the method of isolating the square roots and squaring both sides to eliminate the radicals.

🔢 Step-by-Step Solution

Let Variables for Square Roots:

Let

a = \sqrt{x + 6\sqrt{2}}

and

b = \sqrt{x - 6\sqrt{2}}

. The equation becomes:

a - b = 2\sqrt{2}

Express Squares of Variables:

Square both expressions:

a^2 = x + 6\sqrt{2} b^2 = x - 6\sqrt{2}

Subtract the Equations:

Subtract the second equation from the first:

a^2 - b^2 = (x + 6\sqrt{2}) - (x - 6\sqrt{2}) a^2 - b^2 = 12\sqrt{2}

Factor the Difference of Squares:

Use the identity

a^2 - b^2 = (a - b)(a + b)

(a - b)(a + b) = 12\sqrt{2}

Substitute Known Value:

From the original equation,

a - b = 2\sqrt{2}

2\sqrt{2} \cdot (a + b) = 12\sqrt{2}

Solve for

a + b

Divide both sides by

2\sqrt{2}

a + b = 6

Set Up System of Equations:

We have:

a - b = 2\sqrt{2} a + b = 6

Add the Equations:

$$ (a - b

Keep going