Question

If $ \frac{a{b} + \frac{b}{a} = 1 $, then $ a^3 + b^3 = $}

• A. $ 1 $
• B. $ -1 $
• C. $ \frac{1}{2} $
• D. $ 0 $

Hint:

When solving problems involving algebraic identities, use known formulas like the sum of cubes and simplify step-by-step. Substituting relationships derived from the given equations can help reduce complexity.

Answer 1

The Correct Option is D

Step 1: Analyze the Given Equation.
We are given: \[ \frac{a}{b} + \frac{b}{a} = 1 \] Rewrite this using a common denominator: \[ \frac{a^2 + b^2}{ab} = 1 \] Multiply both sides by \( ab \) (assuming \( a, b \neq 0 \)): \[ a^2 + b^2 = ab \] Step 2: Use the Identity for \( a^3 + b^3 \).
The identity is: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] From earlier, we know: \[ a^2 + b^2 = ab \] Now substitute into the expression \( a^2 - ab + b^2 \): \[ a^2 - ab + b^2 = (a^2 + b^2) - ab = ab - ab = 0 \] Thus: \[ a^3 + b^3 = (a + b)(0) = 0 \] Step 3: Analyze the Options.
Option (1): \( 1 \) — Incorrect
Option (2): \( -1 \) — Incorrect
Option (3): \( \frac{1}{2} \) — Incorrect
Option (4): \( 0 \) — Correct Step 4: Final Answer.
\[ (4) \quad \mathbf{0} \]