| 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 constant using stationary point |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring standard techniques: finding stationary points by setting f'(x)=0, using the second derivative test, then solving simultaneous equations to find constants and integrating. While it has multiple steps, each is routine A-level calculus with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(ax^2 + bx = 0 \rightarrow x(ax+b) = 0 \rightarrow x = \frac{-b}{a}\) | B1 | |
| Find \(f''(x)\) and attempt sub their \(\frac{-b}{a}\) into their \(f''(x)\) | M1 | |
| When \(x = \frac{-b}{a}\), \(f''(x) = 2a\left(\frac{-b}{a}\right) + b = -b\) MAX | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sub \(f'(-2) = 0\) | M1 | |
| Sub \(f'(1) = 9\) | M1 | |
| \(a = 3\), \(b = 6\) | *A1 | Solve simultaneously to give both results |
| \(f'(x) = 3x^2 + 6x \rightarrow f(x) = x^3 + 3x^2\ (+c)\) | *M1 | Sub their \(a\), \(b\) into \(f'(x)\) and integrate 'correctly'. Allow \(\frac{ax^3}{3} + \frac{bx^2}{2}(+c)\) |
| \(-3 = -8 + 12 + c\) | DM1 | Sub \(x=-2\), \(y=-3\). Dependent on \(c\) present. Dependent also on \(a\), \(b\) substituted |
| \(f(x) = x^3 + 3x^2 - 7\) | A1 | |
| Total: 6 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $ax^2 + bx = 0 \rightarrow x(ax+b) = 0 \rightarrow x = \frac{-b}{a}$ | B1 | |
| Find $f''(x)$ and attempt sub their $\frac{-b}{a}$ into their $f''(x)$ | M1 | |
| When $x = \frac{-b}{a}$, $f''(x) = 2a\left(\frac{-b}{a}\right) + b = -b$ MAX | A1 | |
| **Total: 3** | | |
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sub $f'(-2) = 0$ | M1 | |
| Sub $f'(1) = 9$ | M1 | |
| $a = 3$, $b = 6$ | *A1 | Solve simultaneously to give both results |
| $f'(x) = 3x^2 + 6x \rightarrow f(x) = x^3 + 3x^2\ (+c)$ | *M1 | Sub their $a$, $b$ into $f'(x)$ and integrate 'correctly'. Allow $\frac{ax^3}{3} + \frac{bx^2}{2}(+c)$ |
| $-3 = -8 + 12 + c$ | DM1 | Sub $x=-2$, $y=-3$. Dependent on $c$ present. Dependent also on $a$, $b$ substituted |
| $f(x) = x^3 + 3x^2 - 7$ | A1 | |
| **Total: 6** | | |10 A curve has equation $y = \mathrm { f } ( x )$ and it is given that $\mathrm { f } ^ { \prime } ( x ) = a x ^ { 2 } + b x$, where $a$ and $b$ are positive constants.\\
(i) Find, in terms of $a$ and $b$, the non-zero value of $x$ for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.\\
(ii) It is now given that the curve has a stationary point at $( - 2 , - 3 )$ and that the gradient of the curve at $x = 1$ is 9 . Find $\mathrm { f } ( x )$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q10 [9]}}