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Chapter 7: Problem 575

If rth term in expansion of \([x+(1 / 2 x)]^{12}\) is constant then \(r\) (a) 5 (b) 6 (c) 7 (d) 8

Short Answer

Expert verified
The r-th term in the expansion of \((x+\frac{1}{2x})^{12}\) is constant when \(r = 7\). So, the correct answer is (c) 7.

Step by step solution

01

General Formula for the r-th Term in a Binomial Expansion

Recall that the general formula for the r-th term of a binomial expansion is given by: \[T_r = ^{n}C_{r-1}(A^{n-r+1})(B^{r-1})\] where n is the exponent, A and B are the terms inside the binomial expression, r is the term index, and ^{n}C_{r-1} represents the combination formula: \[^{n}C_{r-1} = \frac{n!}{(r-1)!(n-r+1)!}\] In our case, A = x, B = \(\frac{1}{2x}\), and n = 12.
02

Substitute the Values and Simplify

Let's substitute these values into the general formula: \[T_r = ^{12}C_{r-1}(x^{12-r+1})(\frac{1}{2x})^{r-1}\] \[T_r = \frac{12!}{(r-1)!(12-r+1)!}(x^{13-r})(\frac{1}{2^{r-1}x^{r-1}})\] Now, let's simplify the expression: \[T_r = \frac{12!}{(r-1)!(13-r)!}(x^{13-r})(\frac{1}{2^{r-1}x^{r-1}})\]
03

Find the Condition for T_r to be a Constant

The given T_r will be constant if the powers of x in the numerator and denominator cancel out. To find that, let's equate the powers of x in both the numerator and the denominator. Power of x in the numerator: \(13 - r\) Power of x in the denominator: \(r-1\) Equate the powers: \(13 - r = r - 1\)
04

Solve for r

Now, let's solve the equation for r: \(13 - r + r = r - 1 + r\) \(13 = 2r - 1\) \(2r = 14\) \(r = \frac{14}{2}\) \(r = 7\) Therefore, the r-th term in the expansion of \((x+\frac{1}{2x})^{12}\) is constant when \(r = 7\). So, the correct answer is (c) 7.

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Most popular questions from this chapter

Number of terms in expansion of \(\left(\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots+\mathrm{x}_{\mathrm{r}}\right)^{\mathrm{n}}\) is (a) \({ }^{(\mathrm{n}+1)} \mathrm{C}_{4}\) (b) \({ }^{(\mathrm{n}+\mathrm{r}-1)} \mathrm{C}_{(\mathrm{r}-1)}\) (c) \(^{(\mathrm{n}-\mathrm{r}+1)} \mathrm{C}_{(\mathrm{r}-1)}\) (d) \({ }^{(\mathrm{n}+\mathrm{r}-1)} \mathrm{C}_{\mathrm{r}}\)

\(3^{\mathrm{rd}}\) term in expansion of \(\left[(1 / \mathrm{x})+\mathrm{x}^{(\log )}{ }_{10}(\mathrm{x})\right]^{5}\) is 1000 then \(\mathrm{x}=\) (a) 10 (b) 100 (c) 1000 (d) None

The Co-efficient of \(\mathrm{x}^{\mathrm{n}}\) in the expansion of \((1+\mathrm{x})(1-\mathrm{x})^{\mathrm{n}}\) is (a) \((-1)^{\mathrm{n}-1}(\mathrm{n}-1)^{2}\) (b) \((-1)^{\mathrm{n}}(1-\mathrm{n})\) (c) \(n-1\) (d) \((-1)^{\mathrm{n}-1} \mathrm{n}\)

Constant term in expansion of \(\left[\left\\{(\mathrm{x}+1) /\left\\{(\mathrm{x})^{(2 / 3)}-(\mathrm{x})^{(1 / 3)}+1\right\\}\right\\}-\left\\{(\mathrm{x}-1) /\left\\{\left(\mathrm{x}-(\mathrm{x})^{(1 / 2)}\right\\}\right\\}\right]^{10}\right.\) is \(\ldots\) (a) 190 (b) 200 (c) 210 (d) 220

The greatest term in expansion of \((3+2 \mathrm{x})^{50}\) is \(;\) where \(\mathrm{x}=(1 / 5)\) (a) \({ }^{50} \mathrm{C}_{7} 343(2 / 5)^{7}\) (b) \({ }^{50} \mathrm{C}_{6} 3^{44}(2 / 5)^{6}\) (c) \({ }^{50} \mathrm{C}_{43} 3^{7}(2 / 5)^{43}\) (d) \({ }^{50} \mathrm{C}_{44} 3^{6}(2 / 5)^{44}\)
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