| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Moderate -0.3 Part (i) requires expanding a binomial, integrating polynomial and root terms using standard rules, then substituting limits—all routine C2 techniques. Part (ii) involves integrating a quadratic, applying two boundary conditions to find k and the constant of integration—straightforward but slightly more involved than typical single-step questions. Both parts are standard textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks |
|---|---|
| \(= \int_1^3 (9 - 6\sqrt{x} + x) \, dx\) | M1 |
| \(= [9x - 4x^{\frac{3}{2}} + \frac{1}{2}x^2]_1^3\) | M1 A2 |
| \(= (27 - 12\sqrt{3} + \frac{9}{2}) - (9 - 4 + \frac{1}{2})\) | M1 |
| \(= 26 - 12\sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x^3 + 2x^2 + kx + c\) | M1 A2 | |
| \((0, -2): c = -2\) | B1 | |
| \((2, 18): 18 = 8 + 8 + 2k - 2\) | M1 | |
| \(k = 2\) | A1 | |
| \(\therefore y = x^3 + 2x^2 + 2x - 2\) | A1 | (13) |
| Answer | Marks |
|---|---|
| Total | (72) |
## Part (i)
$= \int_1^3 (9 - 6\sqrt{x} + x) \, dx$ | M1 |
$= [9x - 4x^{\frac{3}{2}} + \frac{1}{2}x^2]_1^3$ | M1 A2 |
$= (27 - 12\sqrt{3} + \frac{9}{2}) - (9 - 4 + \frac{1}{2})$ | M1 |
$= 26 - 12\sqrt{3}$ | A1 |
## Part (ii)
$y = \int (3x^2 + 4x + k) \, dx$
$y = x^3 + 2x^2 + kx + c$ | M1 A2 |
$(0, -2): c = -2$ | B1 |
$(2, 18): 18 = 8 + 8 + 2k - 2$ | M1 |
$k = 2$ | A1 |
$\therefore y = x^3 + 2x^2 + 2x - 2$ | A1 | (13)
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**Total** | (72)\begin{enumerate}[label=(\roman*)]
\item Evaluate
$$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$
giving your answer in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers. [6]
\item The gradient of a curve is given by
$$\frac{dy}{dx} = 3x^2 + 4x + k,$$
where $k$ is a constant.
Given that the curve passes through the points $(0, -2)$ and $(2, 18)$, show that $k = 2$ and find an equation for the curve. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q9 [13]}}