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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2019 · Mathematics · Binomial Theorem

In the expansion of (1+a x)^ n , the first three terms are...

In the expansion of (1+ax)n(1+a x)^{\mathrm{n}}, the first three terms are respectively 1,12x1,12 x and 64x264 x^2. What is n equal to?

Options

  1. A6
  2. B9
  3. C10
  4. D12

Correct answer

B. 9

Step-by-step solution

The first three terms in expansion of (1+ax)n(1+\mathrm{ax})^{\mathrm{n}} are nC0,nC1ax,nC2a2x2{ }^n C_0,{ }^n C_1 a x,{ }^n C_2 a^2 x^2 Given, nC0=1;nC1ax=12x;nC2a2x2=64x2{ }^n C_0=1 ;{ }^n C_1 a x=12 x ;{ }^n C_2 a^2 x^2=64 x^2⇒nax=12x;n(n−1)2a2=64⇒na=12⇒a=12n∴n(n−1)2a2=64⇒n(n−1)2×144n2=64⇒n−1n=64×2144=89∴n=9\begin{aligned} & \Rightarrow \mathrm{nax}=12 \mathrm{x} ; \frac{n(n-1)}{2} a^2=64 \\ & \Rightarrow \mathrm{na}=12 \Rightarrow a=\frac{12}{n} \\ & \therefore \frac{n(n-1)}{2} a^2=64 \Rightarrow \frac{n(n-1)}{2} \times \frac{144}{n^2}=64 \\ & \Rightarrow \frac{n-1}{n}=\frac{64 \times 2}{144}=\frac{8}{9} \\ & \therefore \mathrm{n}=9\end{aligned}$$
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