| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Two Curves Intersection Area |
| Difficulty | Standard +0.3 This is a standard C2 integration question requiring expansion and integration in part (a), then finding area between two curves in part (b). The curves intersect at a given point, eliminating the need to solve for intersections. Students must set up the correct integral subtracting the lower curve from upper curve, which is routine for this topic. Slightly above average difficulty due to the fractional and negative powers, but still a textbook-style question. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Expand and attempt integration | M1 | Must attempt to expand brackets first. Increase in power by 1 for the majority of their terms. Allow if the constant term disappears |
| Obtain at least two correct (algebraic) terms | A1ft | At least two correct from their expansion. Allow for unsimplified coefficients |
| Obtain fully correct expression, inc \(+ c\) | A1 [3] | All coefficients now simplified. A0 if integral sign or \(dx\) still present in their answer (but allow \(\int = \ldots\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Obtain \(kx^{\frac{5}{2}}\) | M1 | Any exact equiv for the index |
| Obtain \(\frac{12}{5}x^{\frac{5}{2}}\), or any exact equiv | A1 | Including unsimplified coefficient |
| Obtain at least one of \(-8x^{-1}\) and \(-2x\) | M1 | Allow M1 even if \(-2\) disappears. Could be part of a sum or difference; with consistent signs |
| Obtain \(-8x^{-1} - 2x\) | A1 | Allow unsimplified expressions. If subtraction from other curve attempted before integration then allow for \(8x^{-1} + 2x\) |
| State or imply that point of intersection is \((2, 0)\) | B1 | Could imply by using it as a limit |
| Use limits correctly at least once | M1 | Must be using correct \(x\) limits, and subtracting, with the appropriate function (allow implicit use of \(x = 0\)); the only error allowed is an incorrect \((2, 0)\). Allow use in any function other than the original, inc from differentiation |
| Attempt fully correct process to find required area | M1 | Use both pairs of limits correctly (allow an incorrect \((2,0)\)), in appropriate functions and sum the two areas |
| Obtain \(\frac{22}{5}\), or any exact equiv | A1 [8] | Answer only is 0/8, as no evidence is provided of integration |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Obtain \(ky^{\frac{5}{3}}\) | M1 | |
| Obtain \(6^{-\frac{2}{3}} \times \frac{3}{5} \times y^{\frac{5}{3}}\) | A1 | |
| Obtain \(k\sqrt{2+y}\) | M1 | |
| Obtain \(2\sqrt{8}\sqrt{2+y}\) | A1 | |
| Use limits of 6 (and 0) correctly at least once | M1 | |
| Attempt correct method to find required area – correct use of limits required | M1 | |
| Obtain 4.4 | A2 |
## Question 7(a):
**Answer:** $\int(x^3 - 6x^2 + 4x - 24)\,dx = \frac{1}{4}x^4 - 2x^3 + 2x^2 - 24x + c$
| Step | Mark | Guidance |
|------|------|----------|
| Expand and attempt integration | M1 | Must attempt to expand brackets first. Increase in power by 1 for the majority of their terms. Allow if the constant term disappears |
| Obtain at least two correct (algebraic) terms | A1ft | At least two correct from their expansion. Allow for unsimplified coefficients |
| Obtain fully correct expression, inc $+ c$ | A1 [3] | All coefficients now simplified. A0 if integral sign or $dx$ still present in their answer (but allow $\int = \ldots$) |
---
## Question 7(b):
**Answer:** $\int 6x^{\frac{3}{2}}\,dx = \frac{12}{5}x^{\frac{5}{2}}$; $\int(8x^{-2}-2)\,dx = -8x^{-1} - 2x$; total area $= \frac{22}{5}$
| Step | Mark | Guidance |
|------|------|----------|
| Obtain $kx^{\frac{5}{2}}$ | M1 | Any exact equiv for the index |
| Obtain $\frac{12}{5}x^{\frac{5}{2}}$, or any exact equiv | A1 | Including unsimplified coefficient |
| Obtain at least one of $-8x^{-1}$ and $-2x$ | M1 | Allow M1 even if $-2$ disappears. Could be part of a sum or difference; with consistent signs |
| Obtain $-8x^{-1} - 2x$ | A1 | Allow unsimplified expressions. If subtraction from other curve attempted before integration then allow for $8x^{-1} + 2x$ |
| State or imply that point of intersection is $(2, 0)$ | B1 | Could imply by using it as a limit |
| Use limits correctly at least once | M1 | Must be using correct $x$ limits, and subtracting, with the appropriate function (allow implicit use of $x = 0$); the only error allowed is an incorrect $(2, 0)$. Allow use in any function other than the original, inc from differentiation |
| Attempt fully correct process to find required area | M1 | Use both pairs of limits correctly (allow an incorrect $(2,0)$), in appropriate functions and sum the two areas |
| Obtain $\frac{22}{5}$, or any exact equiv | A1 [8] | Answer only is 0/8, as no evidence is provided of integration |
**Alternative scheme** (integrating between curves and $y$-axis):
| Step | Mark | Guidance |
|------|------|----------|
| Obtain $ky^{\frac{5}{3}}$ | M1 | |
| Obtain $6^{-\frac{2}{3}} \times \frac{3}{5} \times y^{\frac{5}{3}}$ | A1 | |
| Obtain $k\sqrt{2+y}$ | M1 | |
| Obtain $2\sqrt{8}\sqrt{2+y}$ | A1 | |
| Use limits of 6 (and 0) correctly at least once | M1 | |
| Attempt correct method to find required area – correct use of limits required | M1 | |
| Obtain 4.4 | A2 | |7
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( x ^ { 2 } + 4 \right) ( x - 6 ) \mathrm { d } x$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-4_449_551_349_758}
The diagram shows the curve $y = 6 x ^ { \frac { 3 } { 2 } }$ and part of the curve $y = \frac { 8 } { x ^ { 2 } } - 2$, which intersect at the point $( 1,6 )$. Use integration to find the area of the shaded region enclosed by the two curves and the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2012 Q7 [11]}}