Question

The coefficient of \( x^2 \) in the expansion of \( (1 - 3x)^{\frac{1}{3}} (1 + 2x)^{\frac{1}{2}} \) is:

• A. \( \frac{-3}{2} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{-1}{2} \)

Hint:

When expanding binomial expressions, focus on the terms that contribute to the desired power, and multiply corresponding terms from each expansion.

Answer 1

The Correct Option is B

We are asked to find the coefficient of \( x^2 \) in the expansion of \( (1 - 3x)^{\frac{1}{3}} (1 + 2x)^{\frac{1}{2}} \). Step 1: Use the binomial expansion for both terms. The binomial series expansion for \( (1 - 3x)^{\frac{1}{3}} \) is: \[ (1 - 3x)^{\frac{1}{3}} = 1 - \frac{1}{3}(3x) + \frac{1}{9}(3x)^2 + \dots \] Step 2: The binomial expansion for \( (1 + 2x)^{\frac{1}{2}} \) is: \[ (1 + 2x)^{\frac{1}{2}} = 1 + \frac{1}{2}(2x) - \frac{1}{8}(2x)^2 + \dots \] Step 3: Multiply the two expansions, and focus on the terms that contribute to \( x^2 \). After expanding and simplifying, we find that the coefficient of \( x^2 \) is \( \frac{3}{2} \). Thus, the correct answer is \( \frac{3}{2} \).