Moderate -0.3 This is a straightforward application of the binomial expansion formula with n = -1/2 and b = -2. Students need to recall the formula, substitute values, and simplify coefficients through basic arithmetic. It's slightly easier than average because it's a direct single-part application with no problem-solving required, though the fractional negative power and coefficient simplification require some care.
2 Expand \(\frac { 1 } { \sqrt { ( 1 - 2 x ) } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
State a correct un-simplified 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 \(\frac{3}{4}x^2 + \frac{2}{3}x^3\)
A1 A1
[Symbolic binomial coefficients, e.g. \(\binom{-1/3}{3}\) are not sufficient for the M mark.]
State a correct un-simplified 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 $\frac{3}{4}x^2 + \frac{2}{3}x^3$ | A1 A1 | [Symbolic binomial coefficients, e.g. $\binom{-1/3}{3}$ are not sufficient for the M mark.] | [4]
2 Expand $\frac { 1 } { \sqrt { ( 1 - 2 x ) } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2016 Q2 [4]}}