Moderate -0.3 This is a straightforward application of the binomial expansion requiring students to rewrite the quotient as (1-x)^{1/2}(1+x)^{-1/2}, expand each factor to x^2, and multiply the series. While it involves fractional indices and careful algebraic manipulation, it's a standard textbook exercise with no novel insight required—slightly easier than average due to being purely procedural.
3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) 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 \((1-x)^{\frac{1}{2}}\) or \((1+x)^{-\frac{1}{2}}\)
B1
State correct unsimplified expansion of \((1-x)^2\) up to the term in \(x^2\)
B1
State correct unsimplified expansion of \((1+x)^{-\frac{1}{2}}\) up to the term in \(x^2\)
B1
Obtain sufficient terms of the product of the expansions of \((1-x)^2\) and \((1+x)^{-\frac{1}{2}}\)
M1
Obtain final answer \(1 - x + \frac{1}{2}x^2\)
A1
OR1:
Answer
Marks
State that the given expression equals \((1-x)(1-x^2)^{-\frac{1}{2}}\) and state that the first term of the expansion of \((1-x^2)^{-\frac{1}{2}}\) is 1
B1
State correct unsimplified term in \(x^2\) of \((1-x^2)^{-\frac{1}{2}}\)
B1
State correct unsimplified expansion of \((1-x^2)^{-\frac{1}{2}}\) up to the term in \(x^2\)
B1
Obtain sufficient terms of the product of \((1-x)\) and the expansion
M1
Obtain final answer \(1 - x + \frac{1}{2}x^2\)
A1
OR2:
Answer
Marks
Guidance
State correct unsimplified expansion of \((1+x)^{\frac{1}{2}}\) up to the term in \(x^2\)
B1
Multiply expansion by \((1-x)\) and obtain \(1 - 2x + 2x^2\)
B1
Carry out correct method to obtain one non-constant term of the expansion of \(\left[1 - 2x + 2x^2\right]^{\frac{1}{2}}\)
M1
Obtain a correct unsimplified expansion with sufficient terms
A1
Obtain final answer \(1 - x + \frac{1}{2}x^2\)
A1
[5]
Note: [Treat \((1+x)^{-1}(1-x^2)^{-\frac{1}{2}}\) by the EITHER scheme.]
[Symbolic coefficients, e.g. \(\left(\frac{1}{2}\right)\), are not sufficient for the B marks.]
**Main scheme:**
| State a correct unsimplified term in $x$ or $x^2$ of $(1-x)^{\frac{1}{2}}$ or $(1+x)^{-\frac{1}{2}}$ | B1 | |
| State correct unsimplified expansion of $(1-x)^2$ up to the term in $x^2$ | B1 | |
| State correct unsimplified expansion of $(1+x)^{-\frac{1}{2}}$ up to the term in $x^2$ | B1 | |
| Obtain sufficient terms of the product of the expansions of $(1-x)^2$ and $(1+x)^{-\frac{1}{2}}$ | M1 | |
| Obtain final answer $1 - x + \frac{1}{2}x^2$ | A1 | |
**OR1:**
| State that the given expression equals $(1-x)(1-x^2)^{-\frac{1}{2}}$ and state that the first term of the expansion of $(1-x^2)^{-\frac{1}{2}}$ is 1 | B1 | |
| State correct unsimplified term in $x^2$ of $(1-x^2)^{-\frac{1}{2}}$ | B1 | |
| State correct unsimplified expansion of $(1-x^2)^{-\frac{1}{2}}$ up to the term in $x^2$ | B1 | |
| Obtain sufficient terms of the product of $(1-x)$ and the expansion | M1 | |
| Obtain final answer $1 - x + \frac{1}{2}x^2$ | A1 | |
**OR2:**
| State correct unsimplified expansion of $(1+x)^{\frac{1}{2}}$ up to the term in $x^2$ | B1 | |
| Multiply expansion by $(1-x)$ and obtain $1 - 2x + 2x^2$ | B1 | |
| Carry out correct method to obtain one non-constant term of the expansion of $\left[1 - 2x + 2x^2\right]^{\frac{1}{2}}$ | M1 | |
| Obtain a correct unsimplified expansion with sufficient terms | A1 | |
| Obtain final answer $1 - x + \frac{1}{2}x^2$ | A1 | [5] |
**Note:** [Treat $(1+x)^{-1}(1-x^2)^{-\frac{1}{2}}$ by the EITHER scheme.]
[Symbolic coefficients, e.g. $\left(\frac{1}{2}\right)$, are not sufficient for the B marks.]
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3 Expand $\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2012 Q3 [5]}}