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Chapter 7: Problem 590

If \(\mathrm{x}\) is so small that terms with \(\mathrm{x}^{3}\) and higher powers of \(\mathrm{x}\) may be neglected then \(\left[\left\\{(1+\mathrm{x})^{(3 / 2)}-\\{1+(\mathrm{x} / 2)\\}^{3}\right\\} /\left\\{(1-\mathrm{x})^{(1 / 2)}\right\\}\right]\) may be approximated as \(\ldots \ldots \ldots\) (a) \([(-3) / 8] \mathrm{x}^{2}\) (b) \((1 / 2) \mathrm{x}-(3 / 8) \mathrm{x}^{2}\) (c) \(1-(3 / 8) x^{2}\) (d) \(3 x+(3 / 8) x^{2}\)

Short Answer

Expert verified
\(-\frac{1}{8}x^2\)

Step by step solution

01

Expansion using binomial theorem

Since the terms with x^3 and higher powers can be neglected, we can use the binomial theorem for the expansion of the given terms up to the second power of x. The binomial theorem states that for any real numbers a, b, and index n: \((a+b)^n = \sum_{k=0}^n {n\choose k} a^{(n-k)}b^k\) Applying this to the given expression, we get: (1) \((1+x)^{(3/2)} \approx 1 + \frac{3}{2}x + \frac{3}{8}x^2\) (2) \((1+(x/2))^3 \approx 1 + \frac{3}{2}x + \frac{3}{4}x^2\) (3) \((1-x)^{(-1/2)} \approx 1 + \frac{1}{2}x + \frac{1}{8}x^2\)
02

Simplification of the expression

Substitute the approximations obtained in step 1 into the given equation, and simplify the expression: \(\frac{[(1+\frac{3}{2}x + \frac{3}{8}x^2)-(1+\frac{3}{2}x + \frac{3}{4}x^2)]}{(1+\frac{1}{2}x + \frac{1}{8}x^2)}\) The numerator simplifies to: \(-\frac{1}{8}x^2\) Now, divide this by the denominator: \(\frac{-\frac{1}{8}x^2}{1+\frac{1}{2}x + \frac{1}{8}x^2} \) As x is very small, we can ignore the terms with x^2 in the denominator, resulting in: \(-\frac{\frac{1}{8}x^2}{1+\frac{1}{2}x}\)
03

Comparing with the given options

Compare the simplified expression from step 2 with the given options: (a) \(-\frac{3}{8}x^2\) (b) \(\frac{1}{2}x - \frac{3}{8}x^2\) (c) \(1-\frac{3}{8}x^2\) (d) \(3x + \frac{3}{8}x^2\) None of the given options matches the simplified expression exactly. However, option (a) is the closest match as it has the same x^2 term but with a slightly different coefficient.

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Most popular questions from this chapter

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

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

Co-efficient of \(\mathrm{x}^{53}\) in \(\sqrt{ }^{100} \sum_{\mathrm{r}=0}{ }^{100} \mathrm{C}_{\mathrm{r}}(\mathrm{x}-5)^{100-\mathrm{r}} 4^{\mathrm{r}}\) is (a) \({ }^{100} \mathrm{C}_{53}\) (b) \({ }^{100} \mathrm{C}_{48}\) (c) \(-{ }^{100} \mathrm{C}_{53}\) (d) \({ }^{100} \mathrm{C}_{51}\)

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) \({ }^{(\mathrm{n}+\mathrm{r}-1)} \mathrm{C}_{\mathrm{r}}\)

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) \((-1)^{\mathrm{n}-1} \mathrm{n}\)
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