Question

The coefficient of x² in the binomial expansion of (2x² + 1/x)¹⁰ is:

• A. 3260
• B. 3360
• C. 1760
• D. 1890

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We use the general term formula for a binomial expansion to find the specific term where the power of $x$ is 2.
Step 2: Key Formula or Approach:
General term \( T_{r+1} = \binom{n}{r} a^{n-r} b^r \).
Step 3: Detailed Explanation:
1. Write the general term for \( (2x^2 + x^{-1})^{10} \): \[ T_{r+1} = \binom{10}{r} (2x^2)^{10-r} (x^{-1})^r \] \[ T_{r+1} = \binom{10}{r} 2^{10-r} x^{20-2r} x^{-r} = \binom{10}{r} 2^{10-r} x^{20-3r} \] 2. Set the power of $x$ to 2: \[ 20 - 3r = 2 \implies 3r = 18 \implies r = 6 \] 3. Calculate the coefficient for $r = 6$: \[ \text{Coeff} = \binom{10}{6} 2^{10-6} = \binom{10}{4} 2^4 \] \[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 10 \times 3 \times 7 = 210 \] \[ \text{Coeff} = 210 \times 16 = 3360 \]
Step 4: Final Answer:
The coefficient of \( x^2 \) is 3360.