| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward stationary points question requiring standard differentiation, solving quadratic equations, and using the second derivative test. Part (iii) involves a discriminant condition which is slightly beyond pure routine but still a common A-level technique. The multi-part structure and 10 marks suggest slightly above average difficulty, but all methods are standard bookwork with no novel insight required. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks |
|---|---|
| (i) | y = x3 + px2 |
| Answer | Marks |
|---|---|
| 3 27 | B1 |
| Answer | Marks |
|---|---|
| [4] | cao |
| Answer | Marks |
|---|---|
| (iii) | d2y |
| Answer | Marks |
|---|---|
| → 0 <p< 3 aef | M1 |
| Answer | Marks |
|---|---|
| [3] | Other methods include; clear |
| Answer | Marks | Guidance |
|---|---|---|
| Page 9 | Mark Scheme | Syllabus |
| Cambridge International AS/A Level – May/June 2015 | 9709 | 11 |
Question 9:
--- 9
(i) ---
9
(i) | y = x3 + px2
dy
= 3x² + 2px
dx
2p
Sets to 0 → x = 0 or −
3
2p 4p3
→ (0, 0) or − ,
3 27 | B1
M1
A1 A1
[4] | cao
Sets differential to 0
cao cao, first A1 for any correct
turning point or any correct pair of
x values. 2nd A1 for 2 complete
TPs
(ii)
(iii) | d2y
= 6x + 2p
dx2
At (0, 0) → 2p +ve Minimum
2p 4p3
At − , → −2p –ve Maximum
3 27
y = x3 + px2 + px → 3x² + 2px + p (= 0)
Uses b2 −4ac
→ 4p² − 12p< 0
→ 0 <p< 3 aef | M1
A1
A1
[3]
B1
M1
A1
[3] | Other methods include; clear
demonstration of sign change of
gradient, clear reference to the
shape of the curve
www
Any correct use of discriminant
cao (condone ø)
Page 9 | Mark Scheme | Syllabus | Paper
Cambridge International AS/A Level – May/June 2015 | 9709 | 11The equation of a curve is $y = x^3 + px^2$, where $p$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of $p$. [4]
\item Find the nature of each of the stationary points. [3]
\end{enumerate}
Another curve has equation $y = x^3 + px^2 + px$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the set of values of $p$ for which this curve has no stationary points. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2015 Q9 [10]}}