| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after expanding or multiplying out |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring algebraic manipulation (expanding brackets and simplifying indices) followed by standard power rule integration. Part (b) adds routine substitution of limits. The techniques are basic AS-level calculus with no problem-solving insight needed, making it easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int (15x^4 + 11x^{\frac{8}{3}})\ dx\) | M1 | Expands bracket with one term correct |
| A1 | All correct | |
| \(\frac{15x^5}{5} + \frac{11x^{\frac{11}{3}}}{\frac{11}{3}}\ [+c]\) | M1 | Integration with one term correct FT |
| A1 | Both terms correct | |
| \(3x^5 + 3x^{\frac{11}{3}} + c\) oe | A1 | Coefficients must be simplified and must see \(+c\) either here or in previous step |
| Answer | Marks | Guidance |
|---|---|---|
| \(F[8] - F[0] = 3 \times 8^5 + 3 \times 2^{11}\ [-0]\) | M1* | or \(3 \times 2^{15} + 3 \times 2^{11}\ [-0]\) ft (a) |
| e.g. \((3 \times 2^4 + 3) \times 2^{11}\) | *M1 | any correct intermediate step; OR \(104448 \div 2^{11}\) (M2) |
| \(51 \times 2^{11}\) | A1 |
# Question 7:
## Part (a):
$\int (15x^4 + 11x^{\frac{8}{3}})\ dx$ | M1 | Expands bracket with one term correct
| A1 | All correct
$\frac{15x^5}{5} + \frac{11x^{\frac{11}{3}}}{\frac{11}{3}}\ [+c]$ | M1 | Integration with one term correct FT
| A1 | Both terms correct
$3x^5 + 3x^{\frac{11}{3}} + c$ oe | A1 | Coefficients must be simplified and must see $+c$ either here or in previous step
## Part (b):
$F[8] - F[0] = 3 \times 8^5 + 3 \times 2^{11}\ [-0]$ | M1* | or $3 \times 2^{15} + 3 \times 2^{11}\ [-0]$ ft (a)
e.g. $(3 \times 2^4 + 3) \times 2^{11}$ | *M1 | any correct intermediate step; OR $104448 \div 2^{11}$ (M2)
$51 \times 2^{11}$ | A1 |
---7
\begin{enumerate}[label=(\alph*)]
\item Find $\int x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x$.
\item Show that $\int _ { 0 } ^ { 8 } x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x = a \times 2 ^ { 11 }$, where $a$ is a positive integer to be determined.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2019 Q7 [8]}}