| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 Part (i) is a routine completing the square exercise with a coefficient on x², requiring factoring out 9 first—standard bookwork. Part (ii) requires differentiating a cubic and analyzing the sign of the derivative (which relates to part (i)), but this is a straightforward application of calculus techniques with no conceptual difficulty. Both parts are mechanical procedures well below average A-level difficulty. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \((3x - 2)^2 + 1\) | B1B1B1 | For either of 1st 2 marks bracket must be in the form \((ax + b)^2\) except for SCB2 for \(9\left(x - \frac{2}{3}\right)^2 + 1\) [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = 9x^2 - 12x + 5 = \text{their } (3x - 2)^2 + 1 > 0\) (or \(\geq 1\)) hence an increasing function | B1, M1, A1 | Ft from (i). Some reference/recognition required. Allow \(> 1\). Allow their 1 provided positive. Allow a complete all method (2/2 or 0/2) [3] |
### (i)
$(3x - 2)^2 + 1$ | **B1B1B1** | For either of 1st 2 marks bracket must be in the form $(ax + b)^2$ except for SCB2 for $9\left(x - \frac{2}{3}\right)^2 + 1$ [3]
### (ii)
$f'(x) = 9x^2 - 12x + 5 = \text{their } (3x - 2)^2 + 1 > 0$ (or $\geq 1$) hence an increasing function | **B1, M1, A1** | Ft from (i). Some reference/recognition required. Allow $> 1$. Allow their 1 provided positive. Allow a complete all method (2/2 or 0/2) [3]
---\begin{enumerate}[label=(\roman*)]
\item Express $9x^2 - 12x + 5$ in the form $(ax + b)^2 + c$. [3]
\item Determine whether $3x^3 - 6x^2 + 5x - 12$ is an increasing function, a decreasing function or neither. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q3 [6]}}