| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Parametric or Inverse Function Area |
| Difficulty | Moderate -0.3 This is a straightforward area calculation using integration with respect to y. Part (i) is simple algebraic rearrangement (square both sides, rearrange). Part (ii) requires setting up and evaluating a definite integral ∫(x)dy from y=-1 to y=3, which is a standard technique taught in C2. The bounds are easily identified, and the integration of a quadratic is routine. Slightly easier than average due to the guided structure and standard methods. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x + 4 = (y + 1)^2\) with \(x + 4 = y^2 + 2y + 1\) so \(x = y^2 + 2y - 3\) A.G. | M1 | Attempt to make \(x\) the subject. Allow M1 for \(x = (y + 1)^2 \pm 4\) only. Allow M1 if \((y + 1)^2\) becomes \(y^2 + 1\), but only if clearly attempting to square the entire bracket – squaring term by term is M0. Must be from correct algebra, so M0 if eg \(\sqrt{(x + 4)} = \sqrt{x} + \sqrt{4}\) is used. |
| Verify \(x = y^2 + 2y - 3\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(\int (y^2 + 2y - 3)dy = [\frac{1}{3}y^3 + y^2 - 3y]_1^3 = (9 + 9 - 9) - (\frac{1}{3} + 1 - 3) = (9) - (-1\frac{2}{3}) = 10\frac{2}{3}\) | B1 | State or imply that the required area is given by \(\int_1^3 (y^2 + 2y - 3)dy\). No further work required beyond stating this. Allow if 3x appears in integral. Any further consideration of other areas is B0. |
| Attempt integration | M1 | Increase in power of \(y\) by 1 for at least two of the three terms. Can still get M1 if the -3 disappears, or becomes 3x. Allow M1 for integrating a function of \(y\) that is no longer the given one, eg subtracted from 3, or using their incorrect rearrangement from part (i). |
| Obtain at least two correct terms | A1f1 | Allow for unsimplified coefficients. Allow follow-through on any function of \(y\) as long as at least 2 terms and related to the area required. Condone \(\int, dy\) or \(+c\) present. |
| Attempt F(3) – F(1) for their integral | M1 | Must be correct order and subtraction. This is independent of first M1 so can be given for substituting into any expression other than \(y^2 + 2y - 3\), including \(2y + 2\). If last term is 3x allow M1 for using 3 and 1 throughout integral, but M0 if \(x\) value is used instead. |
| Obtain 10\(\frac{2}{3}\) aef | A1 | 5 |
**(i)** $x + 4 = (y + 1)^2$ with $x + 4 = y^2 + 2y + 1$ so $x = y^2 + 2y - 3$ A.G. | M1 | Attempt to make $x$ the subject. Allow M1 for $x = (y + 1)^2 \pm 4$ only. Allow M1 if $(y + 1)^2$ becomes $y^2 + 1$, but only if clearly attempting to square the entire bracket – squaring term by term is M0. Must be from correct algebra, so M0 if eg $\sqrt{(x + 4)} = \sqrt{x} + \sqrt{4}$ is used.
**Verify $x = y^2 + 2y - 3$** | A1 | 2 | Need to see an extra step from $(y + 1)^2 - 4$ to given answer ie explicit expansion of bracket. No errors seen.
SR B1 for verification, using $y = -1 + \sqrt{(y^2 + 2y - 3 + 4)}$, and confirming relationship convincingly, or for rearranging $x = f(y)$ to obtain given $y = f(x)$.
**(ii)** $\int (y^2 + 2y - 3)dy = [\frac{1}{3}y^3 + y^2 - 3y]_1^3 = (9 + 9 - 9) - (\frac{1}{3} + 1 - 3) = (9) - (-1\frac{2}{3}) = 10\frac{2}{3}$ | B1 | State or imply that the required area is given by $\int_1^3 (y^2 + 2y - 3)dy$. No further work required beyond stating this. Allow if 3x appears in integral. Any further consideration of other areas is B0.
**Attempt integration** | M1 | Increase in power of $y$ by 1 for at least two of the three terms. Can still get M1 if the -3 disappears, or becomes 3x. Allow M1 for integrating a function of $y$ that is no longer the given one, eg subtracted from 3, or using their incorrect rearrangement from part (i).
**Obtain at least two correct terms** | A1f1 | Allow for unsimplified coefficients. Allow follow-through on any function of $y$ as long as at least 2 terms and related to the area required. Condone $\int, dy$ or $+c$ present.
**Attempt F(3) – F(1) for their integral** | M1 | Must be correct order and subtraction. This is independent of first M1 so can be given for substituting into any expression other than $y^2 + 2y - 3$, including $2y + 2$. If last term is 3x allow M1 for using 3 and 1 throughout integral, but M0 if $x$ value is used instead.
**Obtain 10$\frac{2}{3}$ aef** | A1 | 5 | Must be an exact equiv so 10.6 is fine (but $9^6/_{13}$ is A0). 10.7, 10.66... are A0. Must come from correct integral, so A0 if from 3x. Must be given as final answer, so further work eg subtracting another area is A0 rather than ISW. Answer only is 0/5, as no evidence is provided of integration.
SR Finding the shaded area by direct integration with respect to $x$ (ie a C3 technique) can have 5 if done correctly, 4 if non-exact decimal given as final answer but no other partial credit.
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\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-3_588_1136_255_502}
The diagram shows the curve $y = - 1 + \sqrt { x + 4 }$ and the line $y = 3$.\\
(i) Show that $y = - 1 + \sqrt { x + 4 }$ can be rearranged as $x = y ^ { 2 } + 2 y - 3$.\\
(ii) Hence find by integration the exact area of the shaded region enclosed between the curve, the $y$-axis and the line $y = 3$.
\hfill \mbox{\textit{OCR C2 2011 Q4 [7]}}