Consider the curve C given by \(y = {x^3}\).

The tangent at a point P on \(C\) meets the curve again at Q. The tangent at Q meets the curve again at R. Denote the \(x\)-coordinates of \({\text{P, Q}}\) and R, by \({x_1},{\text{ }}{x_2}\) and \({x_3}\) respectively where \({x_1} \ne 0\). Show that, \({x_1},{\text{ }}{x_2},{\text{ }}{x_3}\) form the first three elements of a divergent geometric sequence.

attempt to find the equation of the tangent at P     M1

\(y - x_1^3 = 3x_1^2(x - {x_1})\)    A1

the tangent meets \(C\) when

\({x^3} - x_1^3 = 3x_1^2(x - {x_1})\)    M1

attempt to solve the cubic     M1

the \(x\)-coordinate of Q satisfies

\({x^2} + x{x_1} - 2x_1^2 = 0\)    A1

hence \({x_2} =  - 2{x_1}\)     A1

hence \({x_3} = 4{x_1}\)     A1

hence \({x_1},{\text{ }}{x_2},{\text{ }}{x_3}\) form the first three terms of a geometric sequence with common ratio \( - 2\) so the sequence is divergent     R1AG

Note:     Final R1 is not dependent on final 3 A1s providing they form a geometric sequence.

Total [8 marks]

This question caused a problem for many candidates and only a small number of fully correct answers were seen. Most candidates were able to find a generalised equation of a tangent, but were then unable to see what could be replaced in order to find a quadratic equation that could be solved.