| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Moderate -0.8 This is a straightforward C2 integration question requiring basic polynomial integration and using a boundary condition to find the constant. Both parts are standard textbook exercises with no problem-solving insight needed—simpler than the average A-level question which typically involves more steps or conceptual challenge. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt integration | M1 | An increase in power by 1 for at least 2 terms. Allow if the \(+5\) disappears. |
| Obtain two correct (algebraic) terms | A1 | Allow if the coefficient of \(x^2\) isn't yet simplified. |
| \(\int(x^2-2x+5)\,dx = \frac{1}{3}x^3 - x^2 + 5x + c\) (allow no \(+c\)) | A1 | A0 if integral sign or \(dx\) still present. A0 if a list of terms rather than an expression. |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(y =\) their integral from (i) | M1* | Must have come from integration attempt. Allow slips when transferring. Can still get M1 if no \(+c\). |
| \(11 = 9 - 9 + 15 + c \Rightarrow c = -4\), attempt to find \(c\) using \((3, 11)\) | M1d* | Need to get as far as attempting \(c\). M0 if no \(+c\) seen or implied. M0 if using \(x=11, y=3\). |
| Hence \(y = \frac{1}{3}x^3 - x^2 + 5x - 4\) | A1 | Coeff of \(x^2\) now needs to be simplified. Must be an equation. Allow aef such as \(3y = x^3 - 3x^2 + 15x - 12\). |
| [3] |
## Question 2:
### Part (i): Integrate $x^2 - 2x + 5$
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt integration | M1 | An increase in power by 1 for at least 2 terms. Allow if the $+5$ disappears. |
| Obtain two correct (algebraic) terms | A1 | Allow if the coefficient of $x^2$ isn't yet simplified. |
| $\int(x^2-2x+5)\,dx = \frac{1}{3}x^3 - x^2 + 5x + c$ (allow no $+c$) | A1 | A0 if integral sign or $dx$ still present. A0 if a list of terms rather than an expression. |
| | **[3]** | |
### Part (ii): Find equation of curve
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $y =$ their integral from (i) | M1* | Must have come from integration attempt. Allow slips when transferring. Can still get M1 if no $+c$. |
| $11 = 9 - 9 + 15 + c \Rightarrow c = -4$, attempt to find $c$ using $(3, 11)$ | M1d* | Need to get as far as attempting $c$. M0 if no $+c$ seen or implied. M0 if using $x=11, y=3$. |
| Hence $y = \frac{1}{3}x^3 - x^2 + 5x - 4$ | A1 | Coeff of $x^2$ now needs to be simplified. Must be an equation. Allow aef such as $3y = x^3 - 3x^2 + 15x - 12$. |
| | **[3]** | |
---2 (i) Find $\int \left( x ^ { 2 } - 2 x + 5 \right) \mathrm { d } x$.\\
(ii) Hence find the equation of the curve for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 2 x + 5$ and which passes through the point $( 3,11 )$.
\hfill \mbox{\textit{OCR C2 2012 Q2 [6]}}