| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Tangent with specified gradient |
| Difficulty | Moderate -0.3 Part (i) requires finding where dy/dx equals -3 (gradient matching), then substituting to find the point and tangent equation—straightforward differentiation and algebra. Part (ii) involves finding where f'(x) ≥ 0 always, requiring completing the square or finding the vertex of a quadratic. Both parts are standard textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 3x^2 - 18x + 24\) | M1A1 | Attempt to differentiate. All correct for A mark |
| \(3x^2 - 18x + 24 = -3\) | M1 | Equate their \(\frac{dy}{dx}\) to \(-3\) |
| \(x = 3\) | A1 | |
| \(y = 6\) | A1 | |
| \(y - 6 = -3(x-3)\) | A1FT | FT on their A. Expect \(y = -3x + 15\) |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((3)(x-2)(x-4)\) SOI or \(x=2,\ 4\). Allow \((3)(x+2)(x+4)\) | M1 | Attempt to factorise or solve. Ignore a RHS, e.g. \(= 0\) or \(> 0\), etc. |
| Smallest value of \(k\) is 4 | A1 | Allow \(k \geqslant 4\). Allow \(k = 4\). Must be in terms of \(k\) |
| 2 |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 3x^2 - 18x + 24$ | M1A1 | Attempt to differentiate. All correct for A mark |
| $3x^2 - 18x + 24 = -3$ | M1 | Equate their $\frac{dy}{dx}$ to $-3$ |
| $x = 3$ | A1 | |
| $y = 6$ | A1 | |
| $y - 6 = -3(x-3)$ | A1FT | FT on their A. Expect $y = -3x + 15$ |
| | **6** | |
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(3)(x-2)(x-4)$ SOI or $x=2,\ 4$. Allow $(3)(x+2)(x+4)$ | M1 | Attempt to factorise or solve. Ignore a RHS, e.g. $= 0$ or $> 0$, etc. |
| Smallest value of $k$ is 4 | A1 | Allow $k \geqslant 4$. Allow $k = 4$. Must be in terms of $k$ |
| | **2** | |8 (i) The tangent to the curve $y = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12$ at a point $A$ is parallel to the line $y = 2 - 3 x$. Find the equation of the tangent at $A$.\\
(ii) The function f is defined by $\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12$ for $x > k$, where $k$ is a constant. Find the smallest value of $k$ for f to be an increasing function.\\
\hfill \mbox{\textit{CAIE P1 2018 Q8 [8]}}