| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard C3 techniques: integration with substitution/reverse chain rule, applying Simpson's rule formula with given strip count, and algebraic manipulation. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08h Integration by substitution1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain integral of form \(k(7x+1)^{\frac{4}{3}}\) | \*M1 | Any non-zero constant \(k\) |
| Obtain \(\frac{3}{28}(7x+1)^{\frac{4}{3}}\) | A1 | Or unsimplified equiv |
| Apply limits correctly and attempt exact evaluation | M1 | Dep \*M; substitution of limits to be seen |
| Obtain \(\frac{180}{7}\) | A1 | Or exact equiv such as \(\frac{720}{28}\) or \(25\frac{5}{7}\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt expression of form \(k(y_0+4y_1+y_2)\) | M1 | Any constant \(k\); attempting exact \(y\) values corresponding to \(x\) values \(1, 5, 9\). Missing brackets which are not implied by subsequent calculation and which lead to \(ky_0+4y_1+y_2\) earn M0 |
| Obtain \(\frac{4}{3}(\sqrt[3]{8}+4\times\sqrt[3]{36}+\sqrt[3]{64})\) | A1 | |
| Obtain \(8+\frac{16}{3}\sqrt[3]{36}\) | A1 | No need for \(m\) and \(n\) to be stated separately |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Equate answers to parts (i) and (ii) and carry out complete correct relevant rearrangement | M1 | Provided \(\sqrt[3]{36}\) is involved. Correct answer only seen: M1A1. Answer only seen: if follows correctly from their parts (i) and (ii): M1A0 |
| Obtain \(\frac{93}{28}\) or \(\frac{372}{112}\) | A1 | Or equiv of requested form |
| [2] |
# Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain integral of form $k(7x+1)^{\frac{4}{3}}$ | \*M1 | Any non-zero constant $k$ |
| Obtain $\frac{3}{28}(7x+1)^{\frac{4}{3}}$ | A1 | Or unsimplified equiv |
| Apply limits correctly and attempt exact evaluation | M1 | Dep \*M; substitution of limits to be seen |
| Obtain $\frac{180}{7}$ | A1 | Or exact equiv such as $\frac{720}{28}$ or $25\frac{5}{7}$ |
| **[4]** | | |
---
# Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt expression of form $k(y_0+4y_1+y_2)$ | M1 | Any constant $k$; attempting exact $y$ values corresponding to $x$ values $1, 5, 9$. Missing brackets which are not implied by subsequent calculation and which lead to $ky_0+4y_1+y_2$ earn M0 |
| Obtain $\frac{4}{3}(\sqrt[3]{8}+4\times\sqrt[3]{36}+\sqrt[3]{64})$ | A1 | |
| Obtain $8+\frac{16}{3}\sqrt[3]{36}$ | A1 | No need for $m$ and $n$ to be stated separately |
| **[3]** | | |
---
# Question 7(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Equate answers to parts (i) and (ii) and carry out complete correct relevant rearrangement | M1 | Provided $\sqrt[3]{36}$ is involved. Correct answer only seen: M1A1. Answer only seen: if follows correctly from their parts (i) and (ii): M1A0 |
| Obtain $\frac{93}{28}$ or $\frac{372}{112}$ | A1 | Or equiv of requested form |
| **[2]** | | |
---7 (i) Find the exact value of $\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$.\\
(ii) Use Simpson's rule with two strips to show that an approximate value of $\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$ can be expressed in the form $m + n \sqrt [ 3 ] { 36 }$, where the values of the constants $m$ and $n$ are to be stated.\\
(iii) Use the results from parts (i) and (ii) to find an approximate value of $\sqrt [ 3 ] { 36 }$, giving your answer in the form $\frac { p } { q }$ where $p$ and $q$ are integers.
\section*{Question 8 begins on page 4.}
\hfill \mbox{\textit{OCR C3 2015 Q7 [9]}}