Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative/fractional powers. Students need only substitute n=-1/2 and b=4 into the standard formula and simplify coefficients—a routine procedural task with no problem-solving element, making it easier than average but not trivial since it requires careful arithmetic with fractions.
1 Expand \(( 1 + 4 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
Either: Obtain correct unsimplified version of the \(x\) or \(x^2\) or \(x^3\) term
M1
State correct first two terms \(1 - 2x\)
A1
Obtain next two terms \(6x^2 - 20x^3\)
A1 + A1
[The M mark is not earned by versions with unexpanded binomial coefficients, e.g. \(\binom{-\frac{1}{2}}{2}\)]
4
Or: Differentiate expression and evaluate \(f(0)\) and \(f'(0)\), where \(f'(x) = k(1 + 4x)^{-\frac{2}{3}}\)
M1
State correct first two terms \(1 - 2x\)
A1
Obtain next two terms \(6x^2 - 20x^3\)
A1 + A1
4
**Either:** Obtain correct unsimplified version of the $x$ or $x^2$ or $x^3$ term | M1 |
State correct first two terms $1 - 2x$ | A1 |
Obtain next two terms $6x^2 - 20x^3$ | A1 + A1 |
[The M mark is not earned by versions with unexpanded binomial coefficients, e.g. $\binom{-\frac{1}{2}}{2}$] | | 4
**Or:** Differentiate expression and evaluate $f(0)$ and $f'(0)$, where $f'(x) = k(1 + 4x)^{-\frac{2}{3}}$ | M1 |
State correct first two terms $1 - 2x$ | A1 |
Obtain next two terms $6x^2 - 20x^3$ | A1 + A1 | 4
1 Expand $( 1 + 4 x ) ^ { - \frac { 1 } { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2005 Q1 [4]}}