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Question
The term without x in the expansion of \[\left( 2x - \frac{1}{2 x^2} \right)^{12}\] is
Options
495
−495
−7920
7920
Solution
7920
\[\text{ Suppose the } (r + 1)\text{ th term in the given expansion is independent of } x . \]
\[\text{ Then, we have: } \]
\[ T_{r + 1} = ^{12}{}{C}_r (2x )^{12 - r} \left( \frac{- 1}{2 x^2} \right)^r \]
`= ( - 1 )^r "^12 C _r 2^{12 - 2r} x^{12 - r - 2r}`
\[\text{ For this term to be independent of x, we must have: } \]
\[12 - 3r = 0\]
\[ \Rightarrow r = 4\]
\[ \therefore \text{ Required term: } \]
`( - 1 )^4 "^12C_4 2^{12 - 8}`
\[ = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2} \times 16\]
\[ = 7920\]
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