| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| 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 leading coefficient, requiring only algebraic manipulation. Part (ii) involves straightforward differentiation of a polynomial and interpreting the sign of the derivative—both are standard textbook procedures with no problem-solving insight required. This is easier than average for A-level. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (i) \([3]\,[(x-1)^2]\,[-1]\) | B1B1B1 | |
| [3] | ||
| (ii) \(f'(x) = 3x^2 - 6x + 7\) | B1 | Ft *their* (i) + 5 |
| \(= 3(x-1)^2 + 4\) | B1\(\checkmark\) | |
| \(> 0\) hence increasing | DB1 | Dep B1\(\checkmark\) unless other valid reason |
| [3] |
## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(i)** $[3]\,[(x-1)^2]\,[-1]$ | B1B1B1 | |
| **[3]** | | |
| **(ii)** $f'(x) = 3x^2 - 6x + 7$ | B1 | Ft *their* (i) + 5 |
| $= 3(x-1)^2 + 4$ | B1$\checkmark$ | |
| $> 0$ hence increasing | DB1 | Dep B1$\checkmark$ unless other valid reason |
| **[3]** | | |
---3 (i) Express $3 x ^ { 2 } - 6 x + 2$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are constants.\\
(ii) The function f , where $\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 7 x - 8$, is defined for $x \in \mathbb { R }$. Find $\mathrm { f } ^ { \prime } ( x )$ and state, with a reason, whether f is an increasing function, a decreasing function or neither.
\hfill \mbox{\textit{CAIE P1 2015 Q3 [6]}}