| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.8 This is a straightforward stationary points question requiring only basic differentiation and integration. Students set dy/dx = 0 to solve a simple quadratic, use the second derivative test (differentiating a polynomial), and integrate to find the curve equation. All steps are routine A-level techniques with no problem-solving insight required, making it easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 0\) | M1 | Sets \(\frac{dy}{dx}\) to 0 and attempts to solve leading to two values for \(x\). |
| \(x = 1,\ x = 4\) | A1 | Both values needed |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d^2y}{dx^2} = -2x + 5\) | B1 | |
| Using both of their \(x\) values in their \(\frac{d^2y}{dx^2}\) | M1 | Evidence of any valid method for both points. |
| \(x=1 \rightarrow (3) \rightarrow\) Minimum, \(x=4 \rightarrow (-3) \rightarrow\) Maximum | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = -\frac{x^3}{3} + \frac{5x^2}{2} - 4x\ \) (+c) | B2, 1, 0 | \(+c\) not needed. \(-1\) each error or omission. |
| Uses \(x=6,\ y=2\) in an integrand to find \(c \rightarrow c = 8\) | M1 A1 | Statement of the final equation not required. |
| Total: 4 |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 0$ | M1 | Sets $\frac{dy}{dx}$ to 0 and attempts to solve leading to two values for $x$. |
| $x = 1,\ x = 4$ | A1 | Both values needed |
| **Total: 2** | | |
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d^2y}{dx^2} = -2x + 5$ | B1 | |
| Using both of their $x$ values in their $\frac{d^2y}{dx^2}$ | M1 | Evidence of any valid method for both points. |
| $x=1 \rightarrow (3) \rightarrow$ Minimum, $x=4 \rightarrow (-3) \rightarrow$ Maximum | A1 | |
| **Total: 3** | | |
## Question 8(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = -\frac{x^3}{3} + \frac{5x^2}{2} - 4x\ $ (+c) | B2, 1, 0 | $+c$ not needed. $-1$ each error or omission. |
| Uses $x=6,\ y=2$ in an integrand to find $c \rightarrow c = 8$ | M1 A1 | Statement of the final equation not required. |
| **Total: 4** | | |8 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 2 } + 5 x - 4$.\\
(i) Find the $x$-coordinate of each of the stationary points of the curve.\\
(ii) Obtain an expression for $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and hence or otherwise find the nature of each of the stationary points.\\
(iii) Given that the curve passes through the point $( 6,2 )$, find the equation of the curve.\\
\hfill \mbox{\textit{CAIE P1 2017 Q8 [9]}}