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NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2014 · Mathematics · Binomial Theorem

Paragraph: In the expansion of (x^3-(1)/(x^2))^n , where n is a...

Paragraph: In the expansion of (x3−1x2)n\left(x^3-\frac{1}{x^2}\right)^n, where nn is a positive integer, the sum of the coefficients of x5andx10x^5 and x^{10} is 0.
Question: What is the value of the independent term?

Options

  1. A5005
  2. B7200
  3. C-5005
  4. D-7200

Correct answer

C. -5005

Step-by-step solution

Value of theindependenttermthe independent term=−nC9(x3)n−9(1x2)9=−15C9(x3)15−9(1x2)9=−15C9x18−18=−15C9x0=−15C9=−15!9!×6!=−15×14×13×12×11×10×9!9!×6×5×4×3×2×1=−91×55=−5005\begin{aligned} & =-{ }^n \mathrm{C}_9\left(x^3\right)^{n-9}\left(\frac{1}{x^2}\right)^9=-{ }^{15} \mathrm{C}_9\left(x^3\right)^{15-9}\left(\frac{1}{x^2}\right)^9 \\ & =-{ }^{15} \mathrm{C}_9 x^{18-18}=-{ }^{15} \mathrm{C}_9 x^0=-{ }^{15} \mathrm{C}_9=-\frac{15!}{9!\times 6!} \\ & =\frac{-15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9!}{9!\times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\ & =-91 \times 55=-5005 \end{aligned}$$
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