Question

The term independent of \( x \) in the expansion of \( \left( 2x^4 - \frac{1}{x^2} \right)^{12} \) is

• A. 6920
• B. 7920
• C. 7900
• D. 3960

Hint:

Set power of x = 0 to find independent term.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A term independent of \( x \) is the term where the power of \( x \) is zero. We use the general term formula and set the exponent of \( x \) to zero.
: Key Formula or Approach:
General term \( T_{r+1} = {}^{12}C_r (2x^4)^{12-r} (-x^{-2})^r \).
Step 2: Detailed Explanation:
Combine the powers of \( x \): \[ T_{r+1} = {}^{12}C_r \cdot 2^{12-r} \cdot x^{4(12-r)} \cdot (-1)^r \cdot x^{-2r} \] \[ T_{r+1} = {}^{12}C_r \cdot 2^{12-r} \cdot (-1)^r \cdot x^{48 - 4r - 2r} \] For the term to be independent of \( x \): \[ 48 - 6r = 0 \implies 6r = 48 \implies r = 8 \] Now calculate the coefficient for \( r = 8 \): \[ T_9 = {}^{12}C_8 \cdot 2^{12-8} \cdot (-1)^8 \] \[ T_9 = {}^{12}C_4 \cdot 2^4 \cdot 1 \] \[ {}^{12}C_4 = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \] \[ T_9 = 495 \times 16 = 7920 \].
Step 3: Final Answer:
The term independent of x is 7920.