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08/02/2022 3:02 pm
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If \(\Big(\frac{3^6}{4^4}\Big)\)k is the term, independent of x, in the binomial expansion of \(\Big(\frac{x}{4} - \frac{12}{x^2}\Big)^{12}\), then k is equal to .............
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08/02/2022 3:13 pm
\(\Big(\frac{x}{4} - \frac{12}{x^2}\Big)^{12}\)
Tr+1 = (-1)r. \(^{12}C_r(\frac{x}{4})^{12-r}(\frac{12}{x^2})^r\)
Tr+1 = (-1)r. \(^{12}C_r(\frac{1}{4})^{12-r}(12)^r.(x)^{12-3r}\)
Term independent of x ⇒ 12 – 3r = 0 ⇒ r = 4
T5 = (-1)4.\(^{12}C_4(\frac{1}{4})^8(12)^4\)
= \(\frac{3^6}{4^4}\).k
⇒ k = 55