Maths · General term and middle term and simple applications
The coefficient of in the expansion of ^{11}, \) where \)
The coefficient of \( x^{9} \) in the expansion of \( \left(x^{3}+\frac{1}{2^{t}}\right)^{11}, \) where \( t=\log _{\sqrt{2}}\left(x^{\frac{3}{2}}\right) \)
- A. -5
- B. 330
- C. 520 \)
- D. \( 5+\log _{\sqrt{2}}(3) \)
Step-by-step solution
Simplify t using logarithm properties: t = log_{√2}(x^{3/2}) = 3 log_2 x, so 2^t = x^3. The expression becomes (x^3 + x^{-3})^11. General term: C(11,r) x^{33-6r}. Set exponent 9 → r=4. Coefficient = C(11,4)=330.
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