| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Moderate -0.3 This is a straightforward C1 integration question requiring basic antidifferentiation, substitution of boundary conditions to find constants, and solving a tangent problem. While it has multiple parts and 12 marks total, each step follows standard procedures without requiring problem-solving insight or novel approaches—slightly easier than the typical multi-part A-level question. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(y = \int (3x^2 + 4x + k) \, dx\) | M1 A2 |
| \(y = x^3 + 2x^2 + kx + c\) | B1 | |
| \((0, -2) \therefore c = -2\) | M1 | |
| \((2, 18) : 18 = 8 + 8 + 2k - 2\) | M1 | |
| \(k = 2\) | A1 | |
| \(y = x^3 + 2x^2 + 2x - 2\) | A1 | |
| (b) | \(x^3 + 2x^2 + 2x - 2 = x - 2\) | M1 |
| \(x^3 + 2x^2 + x = 0\) | M1 | |
| \(x(x^2 + 2x + 1) = 0\) | M1 | |
| \(x(x + 1)^2 = 0\) | M1 | |
| repeated root \(\therefore\) tangent | A1 | |
| point of contact where \(x = -1\) | M1 | |
| \(\therefore (-1, -3)\) | A1 | (12) |
(a) | $y = \int (3x^2 + 4x + k) \, dx$ | M1 A2 |
| $y = x^3 + 2x^2 + kx + c$ | B1 |
| $(0, -2) \therefore c = -2$ | M1 |
| $(2, 18) : 18 = 8 + 8 + 2k - 2$ | M1 |
| $k = 2$ | A1 |
| $y = x^3 + 2x^2 + 2x - 2$ | A1 |
(b) | $x^3 + 2x^2 + 2x - 2 = x - 2$ | M1 |
| $x^3 + 2x^2 + x = 0$ | M1 |
| $x(x^2 + 2x + 1) = 0$ | M1 |
| $x(x + 1)^2 = 0$ | M1 |
| repeated root $\therefore$ tangent | A1 |
| point of contact where $x = -1$ | M1 |
| $\therefore (-1, -3)$ | A1 | (12)The curve $C$ with equation $y = \text{f}(x)$ is such that
$$\frac{\text{d}y}{\text{d}x} = 3x^2 + 4x + k,$$
where $k$ is a constant.
Given that $C$ passes through the points $(0, -2)$ and $(2, 18)$,
\begin{enumerate}[label=(\alph*)]
\item show that $k = 2$ and find an equation for $C$, [7]
\item show that the line with equation $y = x - 2$ is a tangent to $C$ and find the coordinates of the point of contact. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q10 [12]}}