If 3644k is the term- independent of x- in the binomial expansion of x4 - 12x212- then k is equal to

In expansion of nbsp;(x4 12x^2)^12 nbsp; General term nbsp; T_r+1 = ^12C_r·(x4)^12 r( 12x^2)^r ⇒ nbsp;T_r+1 = ^12C_r·(14)^12 r ( 12)^r · nbsp; x^12 ...

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Byju's Answer

Standard XII

Mathematics

Coefficients of Terms Equidistant from Beginning and End

If 3644k is t...

Question

If $(\frac{3^{6}}{4^{4}}) k$ is the term, independent of $x,$ in the binomial expansion of ${(\frac{x}{4} - \frac{12}{x^{2}})}^{12},$ then $k$ is equal to

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Solution

In expansion of ${(\frac{x}{4} - \frac{12}{x^{2}})}^{12}$
General term
$T_{r + 1} =^{12} C_{r} \cdot {(\frac{x}{4})}^{12 - r} {(- \frac{12}{x^{2}})}^{r} \Rightarrow T_{r + 1} =^{12} C_{r} \cdot {(\frac{1}{4})}^{12 - r} (- 12)^{r} \cdot x^{12 - 3 r}$
Term independent of $x$
$12 - 3 r = 0 \Rightarrow r = 4$
Now,
$(\frac{3^{6}}{4^{4}}) k =^{12} C_{4} {(\frac{1}{4})}^{8} (- 12)^{4} \Rightarrow (\frac{3^{6}}{4^{4}}) k =^{12} C_{4} (\frac{3^{4}}{4^{4}}) \Rightarrow k = \frac{12 \times 11 \times 10 \times 9}{24 \times 3^{2}} ∴ k = 55$

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If (3644)k is the term, independent of x, in the binomial expansion of (x4−12x2)12, then k is equal to

In expansion of (x4−12x2)12 General term Tr+1=12Cr⋅(x4)12−r(−12x2)r⇒Tr+1=12Cr⋅(14)12−r(−12)r⋅x12−3r Term independent of x 12−3r=0⇒r=4 Now, (3644)k=12C4(14)8(−12)4⇒(3644)k=12C4(3444)⇒k=12×11×10×924×32∴k=55

If $(\frac{3^{6}}{4^{4}}) k$ is the term, independent of $x,$ in the binomial expansion of ${(\frac{x}{4} - \frac{12}{x^{2}})}^{12},$ then $k$ is equal to