Standard +0.3 This is a straightforward application of the binomial expansion requiring students to rewrite the quotient as a product (1+2x)^{1/2}(1-2x)^{-1/2}, expand each factor to x^2, and multiply the series. While it involves fractional indices and careful coefficient arithmetic, it's a standard textbook exercise with no novel insight required—slightly easier than average due to its mechanical nature.
2 Expand \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
State a correct unsimplified term in \(x\) or \(x^2\) of the expansion of either \((1+2x)^{\frac{1}{2}}\) or \((1-2x)^{-\frac{1}{2}}\)
B1
State correct unsimplified expansion of \((1+2x)^{\frac{1}{2}}\) up to the term in \(x^2\)
B1
State correct unsimplified expansion of \((1-2x)^{-\frac{1}{2}}\) up to the term in \(x^2\)
B1
Obtain sufficient terms of the product of the expansions
M1
Obtain final answer \(1+2x+2x^2\)
A1
Alternative method:
State that the expression equals \((1+2x)(1-4x^2)^{-\frac{1}{2}}\) and state a term of the expansion
B1
State correct unsimplified expansion of \((1-4x^2)^{-\frac{1}{2}}\) up to the term in \(x^2\)
B1+B1
Obtain sufficient terms of the product of \((1+2x)\) and the expansion
M1
Obtain final answer \(1+2x+2x^2\)
A1
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| State a correct unsimplified term in $x$ or $x^2$ of the expansion of either $(1+2x)^{\frac{1}{2}}$ or $(1-2x)^{-\frac{1}{2}}$ | B1 | |
| State correct unsimplified expansion of $(1+2x)^{\frac{1}{2}}$ up to the term in $x^2$ | B1 | |
| State correct unsimplified expansion of $(1-2x)^{-\frac{1}{2}}$ up to the term in $x^2$ | B1 | |
| Obtain sufficient terms of the product of the expansions | M1 | |
| Obtain final answer $1+2x+2x^2$ | A1 | |
| **Alternative method:** | | |
| State that the expression equals $(1+2x)(1-4x^2)^{-\frac{1}{2}}$ and state a term of the expansion | B1 | |
| State correct unsimplified expansion of $(1-4x^2)^{-\frac{1}{2}}$ up to the term in $x^2$ | B1+B1 | |
| Obtain sufficient terms of the product of $(1+2x)$ and the expansion | M1 | |
| Obtain final answer $1+2x+2x^2$ | A1 | |
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2 Expand $\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.\\
\hfill \mbox{\textit{CAIE P3 2022 Q2 [5]}}