Expand and simplify \({\left( {\frac{x}{y} - \frac{y}{x}} \right)^4}\).

\({\left( {\frac{x}{y} - \frac{y}{x}} \right)^4} = {\left( {\frac{x}{y}} \right)^4} + 4{\left( {\frac{x}{y}} \right)^3}\left( { - \frac{y}{x}} \right) + 6{\left( {\frac{x}{y}} \right)^2}{\left( { - \frac{y}{x}} \right)^2} + 4\left( {\frac{x}{y}} \right){\left( { - \frac{y}{x}} \right)^3} + {\left( { - \frac{y}{x}} \right)^4}\)     (M1)(A1)

Note: Award M1 for attempt to expand and A1 for correct unsimplified expansion.

\( = \frac{{{x^4}}}{{{y^4}}} - 4\frac{{{x^2}}}{{{y^2}}} + 6 - 4\frac{{{y^2}}}{{{x^2}}} + \frac{{{y^4}}}{{{x^4}}}\,\,\,\,\,\left( { = \frac{{{x^8} - 4{x^6}{y^2} + 6{x^4}{y^4} - 4{x^2}{y^6} + {y^8}}}{{{x^4}{y^4}}}} \right)\)     A1A1

Note: Award A1 for powers, A1 for coefficients and signs.

Note: Final two A marks are independent of first A mark.

[4 marks]

This was generally very well answered. Those who failed to gain full marks often made minor sign slips. A surprising number obtained the correct simplified expression, but continued to rearrange their expressions, often doing so incorrectly. Fortunately, there were no penalties for doing so.