First slide

Binomial theorem for positive integral Index

Question

If sum of the coefficients of the first, second and third terms of the expansion of ${(x^{2} + \frac{1}{x})}^{m}$ is 46,
then the coefficient of the term that does not contain x is

Moderate

A

84
B

92
C

98
D

106

Solution

Given, $^{m} C_{0} +^{m} C_{1} +^{m} C_{2} = 46$
$\begin{array}{l} \Rightarrow 2 m + m (m - 1) = 90 \\ \Rightarrow m^{2} + m - 90 = 0 \Rightarrow m = 9 as m > 0 \end{array}$
$Now, (r + 1) thtermof {(x^{2} + \frac{1}{x})}^{m} is$ $^{m} C_{r} {(x^{2})}^{m - r} {(\frac{1}{x})}^{r} =^{m} C_{r} x^{2 m - 3 r}$
For this to be independent of x put 2m-3r=0 $\Rightarrow$ r=6
.'. Coefficient of the term independent of x is $^{9} C_{6} = 84$ .

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