Greatest value of term independent of x in expansion of {left( {xsin θ + frac{{cos θ }

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7.Binomial Theorem

normal

The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta + \frac{{\cos \theta }}{x}} \right)^{10}}$ is

$^{10}C_5$

$2^5$

$2^5 · ^{10}C_5$

$\frac{{^{10}{C_5}}}{{{2^5}}}$

Solution

${\left( {x\sin \theta + \frac{{\cos \theta }}{x}} \right)^{10}}$ ;$ T_{r + 1} = ^{10}C_r(x sin \theta )^{10 – r} ·{\left( {\frac{{\cos x}}{x}} \right)^r} = ^{10}C_r (sin \theta )^{10 – r} · x^{10 – 2r} (cos\theta )^r$

hence for independent of $x, r = 5$

$T_6 = ^{10}C_5 (sin \theta )^5 · (cos\theta )^5$.

$=\frac{{^{10}{C_5}{{(\sin 2\theta )}^5}}}{{{2^5}}}$

Standard 11

Mathematics

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(JEE MAIN-2020)

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Solution

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