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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

\(\mathrm{R}=(3+\sqrt{5})^{2 \mathrm{n}}\) and \(\mathrm{f}=\mathrm{R}-[\mathrm{R}]\), Where [] is an integer part function then \(\mathrm{R}(1-\mathrm{f})=\) (a) \(2^{2 \mathrm{n}}\) (b) \(4^{2 \mathrm{n}}\) (c) \(8^{2 \mathrm{n}}\) (d) \(1^{2 \mathrm{n}}\)

Co-efficient of \(\mathrm{x}^{-3}\) in expansion of \([\mathrm{x}-(\mathrm{a} / \mathrm{x})]^{11}\) is \(\ldots \ldots\) (a) \(-792 \mathrm{a}^{5}\) (b) \(-923 \mathrm{a}^{7}\) (b) \(-792 \mathrm{a}^{6}\) (d) \(-330 \mathrm{a}^{7}\)

Remainder when \(2^{2000}\) is divided by 17 is......... (a) 1 (b) 2 (c) 8 (d) 12

The least positive remainder when \(17^{30}\) is divided by 5 is (a) 2 (b) 4 (c) 3 (d) 1

\(\left[5^{(1 / 2)}+7^{(1 / 8)}\right]^{1024}\) has number of rational terms \(=\) (a) 0 (b) 129 (c) 229 (d) 178
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