Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative/fractional powers. Students need only substitute n=-1/3 and b=3 into the standard formula and simplify coefficients—a routine procedural task with no problem-solving element, making it easier than average but not trivial due to the fractional arithmetic involved.
2 Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
State a correct unsimplified version of the \(x\) or \(x^2\) or \(x^3\) term
M1
State correct first two terms \(1 - x\)
A1
Obtain the next two terms \(2x^2 - \frac{14}{3}x^3\)
A1 + A1
4 marks
[Symbolic binomial coefficients, e.g. \(\binom{-1}{3}\) are not sufficient for the M mark.]
State a correct unsimplified version of the $x$ or $x^2$ or $x^3$ term | M1 |
State correct first two terms $1 - x$ | A1 |
Obtain the next two terms $2x^2 - \frac{14}{3}x^3$ | A1 + A1 | 4 marks
[Symbolic binomial coefficients, e.g. $\binom{-1}{3}$ are not sufficient for the M mark.] | |
2 Expand $( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2014 Q2 [4]}}