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NDA Math Binomial Theorem

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For the next three (03) items that follow:

In the expansion of

\left(x^3 - \frac{1}{x^2}\right)^n

where n is a positive integer,

the sum of the coefficients of

x^5

and

x^{10}

is 0

[NDA I 2014]

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Question : 49 of 50

Marks: +1, -0

What is the value of the independent term?

5005
7200
-5005
-7200

Solution:

T_{r+1} = {}^{n}C_{r} (x_3)^{n-r} \left(\frac{-1}{x^2}\right)^{r} = (-1)^{r} \; {}^{n}C_{r} x^{3n-5r}

3n-5r=0

\Rightarrow r = \frac{3n}{5} = 9

T_{9+1} = -15C_9 = -5005

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