| Date | May 2009 | Marks available | 8 | Reference code | 09M.2.hl.TZ2.7 |
| Level | HL only | Paper | 2 | Time zone | TZ2 |
| Command term | Find and Show that | Question number | 7 | Adapted from | N/A |
Question
The cubic curve \(y = 8{x^3} + b{x^2} + cx + d\) has two distinct points P and Q, where the gradient is zero.
(a) Show that \({b^2} > 24c\) .
(b) Given that the coordinates of P and Q are \(\left( {\frac{1}{2},{\text{ }} - 12} \right)\) and \(\left( { - \frac{3}{2},{\text{ }}20} \right)\) respectively, find the values of \(b\) , \(c\) and \(d\) .
Markscheme
(a) \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 24{x^2} + 2bx + c\) (A1)
\(24{x^2} + 2bx + c = 0\) (M1)
\(\Delta = {\left( {2b} \right)^2} - 96\left( c \right)\) (A1)
\(4{b^2} - 96c > 0\) A1
\({b^2} > 24c\) AG
(b) \(1 + \frac{1}{4}b + \frac{1}{2}c + d = - 12\)
\(6 + b + c = 0\)
\( - 27 + \frac{9}{4}b - \frac{3}{2}c + d = 20\)
\(54 - 3b + c = 0\) A1A1A1
Note: Award A1 for each correct equation, up to \(3\), not necessarily simplified.
\(b = 12\), \(c = - 18\), \(d = - 7\) A1
[8 marks]
Examiners report
Many candidates throughout almost the whole mark range were able to score well on this question. It was pleasing that most candidates were aware of the discriminant condition for distinct real roots of a quadratic. Some who dropped marks on part (b) either didn't write down a sufficient number of linear equations to determine the three unknowns or made arithmetic errors in their manual solution – few GDC solutions were seen.
