Moderate -0.3 This is a straightforward application of the binomial expansion for (1-2x)^(1/2) followed by multiplication with (1+x). It requires routine algebraic manipulation and knowledge of the generalised binomial theorem, but involves no problem-solving insight—just methodical execution of a standard technique with simple coefficients.
2 Expand \(( 1 + x ) \sqrt { } ( 1 - 2 x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
EITHER: State correct unsimplified first two terms of the expansion of \(\sqrt{(1-2x)}\), e.g. \(1 + \frac{1}{2}(-2x)\)
B1
State correct unsimplified term in \(x^2\), e.g. \(\frac{1}{2}(\frac{1}{2}-1)(-2x)^2/2!\)
B1
Obtain sufficient terms of the product of \((1 + x)\) and the expansion up to the term in \(x^2\) of \(\sqrt{(1-2x)}\)
M1
Obtain final answer \(1 - \frac{3}{2}x^2\)
A1
[The B marks are not earned by versions with symbolic binomial coefficients such as \(\binom{1/2}{1}\).]
[SR: An attempt to rewrite \((1+x)\sqrt{(1-2x)}\) as \(\sqrt{(1-3x^2)}\) earns M1 A1 and the subsequent expansion \(1 - \frac{3}{2}x^2\) gets M1 A1.]
OR: Differentiate expression and evaluate \(f(0)\) and \(f'(0)\), having used the product rule
M1
Obtain \(f(0) = 1\) and \(f'(0) = 0\) correctly
A1
Obtain \(f''(0) = -3\) correctly
A1
Obtain final answer \(1 - \frac{3}{2}x^2\), with no errors seen
A1
[4]
**EITHER:** State correct unsimplified first two terms of the expansion of $\sqrt{(1-2x)}$, e.g. $1 + \frac{1}{2}(-2x)$ | B1 |
State correct unsimplified term in $x^2$, e.g. $\frac{1}{2}(\frac{1}{2}-1)(-2x)^2/2!$ | B1 |
Obtain sufficient terms of the product of $(1 + x)$ and the expansion up to the term in $x^2$ of $\sqrt{(1-2x)}$ | M1 |
Obtain final answer $1 - \frac{3}{2}x^2$ | A1 |
[The B marks are not earned by versions with symbolic binomial coefficients such as $\binom{1/2}{1}$.] | |
[SR: An attempt to rewrite $(1+x)\sqrt{(1-2x)}$ as $\sqrt{(1-3x^2)}$ earns M1 A1 and the subsequent expansion $1 - \frac{3}{2}x^2$ gets M1 A1.] | |
**OR:** Differentiate expression and evaluate $f(0)$ and $f'(0)$, having used the product rule | M1 |
Obtain $f(0) = 1$ and $f'(0) = 0$ correctly | A1 |
Obtain $f''(0) = -3$ correctly | A1 |
Obtain final answer $1 - \frac{3}{2}x^2$, with no errors seen | A1 | [4]
2 Expand $( 1 + x ) \sqrt { } ( 1 - 2 x )$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2008 Q2 [4]}}