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Chapter 7: Problem 550

The number of rational terms in expansion of \((1+\sqrt{2}+3 \sqrt{5})^{6}\) are (a) 22 (b) 12 (c) 11 (d) 7

Short Answer

Expert verified
The number of rational terms in the expansion of \((1+\sqrt{2}+3 \sqrt{5})^{6}\) is 12.

Step by step solution

01

Recall the Binomial Expansion

Recall that the Binomial Expansion for any expression \((a+b)^n\) is given by: \[(a+b)^n = \binom{n}{0} a^{n} b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n} a^0 b^n\] Where \(\binom{n}{k}\) denotes the binomial coefficient, which can be computed by \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
02

Apply the Binomial Expansion to our expression

Using the Binomial Expansion for \((1+\sqrt{2}+3 \sqrt{5})^{6}\), we are going to have \(3^6=729\) terms. For our purpose, we don't need the coefficients, we only need to check the powers.
03

Analyze the powers in the expansion

Basically, we are to check how many of the 729 terms have the form of a rational number. To get a rational term, the powers of both the irrational components \(\sqrt{2}\) and \(3\sqrt{5}\) must be even. Examine the three tokens (without coefficients): 1. \((1^2)^m (\sqrt{2})^n (3\sqrt{5})^p\) 2. \((1^2)^m (1)^n (3\sqrt{5})^p\) 3. \((1^2)^m (\sqrt{2})^n (1)^p\) For rational terms: 1. Both n and p have to be even. 2. p has to be even. 3. n has to be even.
04

Count the valid terms

In the expansion, the total power must be 6. Let's count the number of valid cases: 1. With \(m+n+p=6\), if both n and p are even, then m must be even as well. There are 4 possibilities here: \((m,n,p) = (0,2,4), (2,2,2), (4,2,0), (6,0,0)\). That is 4 rational terms. 2. In this case, \(m + n = 6 - p\). Here, p must be even. The possibilities are: \((m,n) = (0,6), (2,4), (4,2), (6,0)\). That is 4 rational terms. 3. In this case, \(m + p = 6 - n\). Here, n must be even. The possibilities are: \((m,p) = (0,6), (2,4), (4,2), (6,0)\). That is 4 rational terms.
05

Find the total number of rational terms

Finally, we sum the cases from Step 4: there are \(4 + 4 + 4 = 12\) rational terms in the expansion of \((1+\sqrt{2}+3 \sqrt{5})^{6}\). So, the correct answer is (b) 12.

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Most popular questions from this chapter

\(\left[\left(a^{1 / 3} / b^{1 / 6}\right)+\left(b^{1 / 2} / a^{1 / 6}\right)\right]^{21}\) has same power of a and \(b\) for \((\mathrm{r}+1)\) th term then \(\mathrm{r}=\) (a) 8 (b) 9 (c) 10 (d) 11

In the expansion of \((\mathrm{x}+\mathrm{y})^{13}\) the co-efficient of \(3 \mathrm{rd}\) term and th terms are equal. (a) 12 (b) 11 (c) 8 (d) 13

\(\left|\begin{array}{l}\mathrm{n} \\\ 0\end{array}\right|+3\left|\begin{array}{l}\mathrm{n} \\\ 1\end{array}\right|+5\left|\begin{array}{l}\mathrm{n} \\\ 2\end{array}\right|+\ldots+(2 \mathrm{n}+1)\left|\begin{array}{l}\mathrm{n} \\\ \mathrm{n}\end{array}\right|=\ldots ; \mathrm{n} \in \mathrm{N}\) (a) \((\mathrm{n}+2) 2^{\mathrm{n}}\) (b) \((n+1) 2^{n}\) (c) \(\mathrm{n} 2^{\mathrm{n}}\) (d) \((\mathrm{n}+1) 2^{\mathrm{n}+1}\)

If middle term is \(\mathrm{kx}^{\mathrm{m}}\) in expansion of \([\mathrm{x}+(1 / \mathrm{x})]^{12}\) then \(\mathrm{m}=\) (a) \(-2\) (b) \(-1\) (c) 0 (d) 1

It coefficients of middle terms in expansion of \((1+\lambda \mathrm{x})^{8}\) and \((1-\lambda \mathrm{x})^{6}\) are equal then \(\lambda=\) (a) \((2 / 7)\) (b) \([(-2) / 7]\) (c) \([(-3) / 7]\) (d) None of these
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