| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Improper integral evaluation |
| Difficulty | Standard +0.3 Part (a) is routine polynomial integration. Part (b)(i) is straightforward power rule with negative exponent. Part (b)(ii) introduces an improper integral but only requires evaluating the antiderivative at limits and solving a simple equation—slightly above average due to the infinity limit concept, but still a standard C2 exercise with no novel problem-solving required. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{5}{4}x^4 - 3x^2 + x + c\) | M1 | Attempt integration. Increase in power by 1 for at least two of the three terms. Allow M1 if the \(+1\) disappears |
| A1 | Obtain at least 2 correct (algebraic) terms. Integral must be of form \(ax^4 + bx^2 + cx\). Allow for unsimplified \(\frac{6}{2}x^2\) and/or \(1x\) | |
| A1 | Obtain a fully correct integral, including \(+c\). Coeff of \(x^2\) must now be simplified, as well as \(x\) not \(1x\). A0 if integral sign or \(dx\) still present in final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-12x^{-2} + c\) | M1 | Obtain integral of form \(kx^{-2}\). Any \(k\), including unsimplified |
| A1 | Obtain fully correct integral, including \(+c\). Coeff must now be simplified. A0 if integral sign or \(dx\) still present |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0) - (-12a^{-2}) = 3\) | M1* | Attempt \(F(\infty) - F(a)\) and use or imply that \(F(\infty) = 0\). Must be subtraction and correct order. \(0 - 12a^{-2}\), with no other supporting method, is M0 |
| \(a^2 = 4\) | M1d* | Equate to 3 and attempt to find \(a\). Dependent on first M1 soi. Allow muddle with fractions eg \(a^2 = \frac{1}{4}\) |
| \(a = 2\) | A1 | Obtain \(a = 2\) only. A0 if \(-2\) still present as well. NB watch for \(a=2\) as a result of solving \(24a^{-3} = 3\), which gets no credit |
# Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{5}{4}x^4 - 3x^2 + x + c$ | M1 | Attempt integration. Increase in power by 1 for at least two of the three terms. Allow M1 if the $+1$ disappears |
| | A1 | Obtain at least 2 correct (algebraic) terms. Integral must be of form $ax^4 + bx^2 + cx$. Allow for unsimplified $\frac{6}{2}x^2$ and/or $1x$ |
| | A1 | Obtain a fully correct integral, including $+c$. Coeff of $x^2$ must now be simplified, as well as $x$ not $1x$. A0 if integral sign or $dx$ still present in final answer |
---
# Question 4(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-12x^{-2} + c$ | M1 | Obtain integral of form $kx^{-2}$. Any $k$, including unsimplified |
| | A1 | Obtain fully correct integral, including $+c$. Coeff must now be simplified. A0 if integral sign or $dx$ still present |
---
# Question 4(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0) - (-12a^{-2}) = 3$ | M1* | Attempt $F(\infty) - F(a)$ and use or imply that $F(\infty) = 0$. Must be subtraction and correct order. $0 - 12a^{-2}$, with no other supporting method, is M0 |
| $a^2 = 4$ | M1d* | Equate to 3 and attempt to find $a$. Dependent on first M1 soi. Allow muddle with fractions eg $a^2 = \frac{1}{4}$ |
| $a = 2$ | A1 | Obtain $a = 2$ only. A0 if $-2$ still present as well. **NB** watch for $a=2$ as a result of solving $24a^{-3} = 3$, which gets no credit |
---4
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( 5 x ^ { 3 } - 6 x + 1 \right) \mathrm { d } x$.
\item \begin{enumerate}[label=(\roman*)]
\item Find $\int 24 x ^ { - 3 } \mathrm {~d} x$.
\item Given that $\int _ { a } ^ { \infty } 24 x ^ { - 3 } \mathrm {~d} x = 3$, find the value of the positive constant $a$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C2 2013 Q4 [8]}}