| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Easy -1.2 This is a straightforward C2 integration question testing basic recall of power rule for integration (rewriting 1/x² as x⁻²) and applying initial conditions. Both parts are standard textbook exercises with no problem-solving required—part (i) is direct integration, part (ii) adds one simple substitution step to find the constant. Easier than average A-level questions. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int x^{-2} dx = -x^{-1} + c\) | M1 | For any attempt to integrate \(x^{-2}\) |
| A1 | For correct expression \(-x^{-1}\) (in any form) | |
| B1 | 3 | For adding an arbitrary constant |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = -x^{-1} + c\) passes through \((1, 3)\), so \(3 = -1 + c \Rightarrow c = 4\) | M1 | For attempt to use \((1, 3)\) to evaluate \(c\) |
| A1 | For correct value of \(c\) from their equation | |
| A1 | 3 | For correct equation |
| Hence curve is \(y = -\frac{1}{x} + 4\) | ||
| Total: 6 |
## Part (i)
$\int x^{-2} dx = -x^{-1} + c$ | M1 | For any attempt to integrate $x^{-2}$
| A1 | For correct expression $-x^{-1}$ (in any form)
| B1 | 3 | For adding an arbitrary constant
## Part (ii)
$y = -x^{-1} + c$ passes through $(1, 3)$, so $3 = -1 + c \Rightarrow c = 4$ | M1 | For attempt to use $(1, 3)$ to evaluate $c$
| A1 | For correct value of $c$ from their equation
| A1 | 3 | For correct equation
Hence curve is $y = -\frac{1}{x} + 4$ | |
| **Total: 6** | |\begin{enumerate}[label=(\roman*)]
\item Find $\int \frac{1}{x^2} dx$. [3]
\item The gradient of a curve is given by $\frac{dy}{dx} = \frac{1}{x^2}$. Find the equation of the curve, given that it passes through the point $(1, 3)$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q2 [6]}}