In the expansion of x2-3 - 1-x 30- a term containing the power x13 -

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Coefficients of Terms Equidistant from Beginning and End

In the expans...

Question

In the expansion of ${(x^{2 / 3} - \frac{1}{\sqrt{x}})}^{30}$ , a term containing the power $x^{13} =$

Does not exists

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Exists and co-efficient is divisible by

29

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Exists and the co-efficient is divisible by

63

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Exists and the co-efficient is divisible by

65

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Solution

The correct options are
B Exists and co-efficient is divisible by $29$
C Exists and the co-efficient is divisible by $63$
D Exists and the co-efficient is divisible by $65$
Writing the general term, we get
$T_{r + 1} =^{30} C_{r} . x^{\frac{120 - 7 r}{6}}$
Hence for $x^{13}$
$\frac{120 - 7 r}{6} = 13$
$120 - 7 r = 78$
$7 r = 42$
$r = 6$
Hence coefficient will be
$^{30} C_{6}$
$= \frac{30.29.28.27.26.25}{6!}$
Hence it is divisible by 29,63,65.

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In the expansion of (x2/3−1√x)30, a term containing the power x13=

In the expansion of ${(x^{2 / 3} - \frac{1}{\sqrt{x}})}^{30}$ , a term containing the power $x^{13} =$