| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and area |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring standard integration techniques. Part (i) involves integrating a simple algebraic function (power of linear expression), and part (ii) applies the standard formula V = π∫y² dx. The algebra is clean with the square eliminating the square root, making this slightly easier than a typical C3 question. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain \(k(4x+1)^{\frac{1}{2}}\) or \(ku^{\frac{1}{2}}\) | *M1 | any constant \(k\) |
| Obtain correct \(\frac{1}{2}\sqrt{3}(4x+1)^{\frac{1}{2}}\) or \(\frac{1}{2}\sqrt{3}u^{\frac{1}{2}}\) | A1 | or exact equiv |
| Apply limits 0 and 20 and attempt subtraction of area of rectangle (or limits 1 and 81 if \(u\) involved) | M1 | dep *M; or equiv such as including term \(-\frac{1}{9}\sqrt{3}\) in the integration or finding \(\int \frac{1}{9}\sqrt{3}\,dx\) separately; allow M1 if decimal values used here. Alternative (region between curve and \(y\)-axis): Obtain equation \(x = \frac{3}{4}y^{-2} - \frac{1}{4}\) B1; Integrate to obtain form \(k_1 y^{-1} + k_2 y\) *M1; Apply limits \(\frac{1}{9}\sqrt{3}\) and \(\sqrt{3}\) the right way round M1 d*M; Obtain \(\frac{6}{\sqrt{3}} - \frac{8}{36}\sqrt{3}\) or better A1 |
| Obtain \(4\sqrt{3} - \frac{20}{9}\sqrt{3}\) and hence \(\frac{16}{9}\sqrt{3}\) | A1 | answer must be exact and a single term; \(\frac{16}{9}\sqrt{3} + c\) as answer is final A0 |
| Total | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State volume is \(\pi\int\dfrac{3}{4x+1}\,dx\) | B1 | no need for limits here; condone absence of \(dx\); condone absence of \(\pi\) here if it appears later in solution. Allow B1 for \(\int\pi y^2\) and \(y^2 = \frac{3}{4x+1}\) stated |
| Obtain integral of form \(k\ln(4x+1)\) | M1 | any constant \(k\) with or without \(\pi\). If brackets missing, and subsequent calculation does not show their 'presence', marks are max B1M1A0M1A0 |
| Obtain \(\frac{3}{4}\pi\ln(4x+1)\) or \(\frac{3}{4}\ln(4x+1)\) | A1 | |
| Apply limits to obtain \(\frac{3}{4}\pi\ln 81\) or \(\frac{3}{4}\ln 81\) | A1 | or exact equiv perhaps with \(\ln 1\) present |
| Attempt to subtract volume of cylinder, using correct radius and 'height' | M1 | with exact volume of cylinder attempted. Do not treat rotation around \(y\)-axis as mis-read: this is 0/6 |
| Obtain \(3\pi\ln 3 - \frac{20}{27}\pi\) or \(\pi\left(\frac{3}{4}\ln 81 - \frac{20}{27}\right)\) | A1 | or exact equiv involving two terms |
| Total | [6] |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain $k(4x+1)^{\frac{1}{2}}$ or $ku^{\frac{1}{2}}$ | *M1 | any constant $k$ |
| Obtain correct $\frac{1}{2}\sqrt{3}(4x+1)^{\frac{1}{2}}$ or $\frac{1}{2}\sqrt{3}u^{\frac{1}{2}}$ | A1 | or exact equiv |
| Apply limits 0 and 20 and attempt subtraction of area of rectangle (or limits 1 and 81 if $u$ involved) | M1 | dep *M; or equiv such as including term $-\frac{1}{9}\sqrt{3}$ in the integration or finding $\int \frac{1}{9}\sqrt{3}\,dx$ separately; allow M1 if decimal values used here. Alternative (region between curve and $y$-axis): Obtain equation $x = \frac{3}{4}y^{-2} - \frac{1}{4}$ B1; Integrate to obtain form $k_1 y^{-1} + k_2 y$ *M1; Apply limits $\frac{1}{9}\sqrt{3}$ and $\sqrt{3}$ the right way round M1 d*M; Obtain $\frac{6}{\sqrt{3}} - \frac{8}{36}\sqrt{3}$ or better A1 |
| Obtain $4\sqrt{3} - \frac{20}{9}\sqrt{3}$ and hence $\frac{16}{9}\sqrt{3}$ | A1 | answer must be exact and a single term; $\frac{16}{9}\sqrt{3} + c$ as answer is final A0 |
| **Total** | **[4]** | |
---
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State volume is $\pi\int\dfrac{3}{4x+1}\,dx$ | B1 | no need for limits here; condone absence of $dx$; condone absence of $\pi$ here if it appears later in solution. Allow B1 for $\int\pi y^2$ and $y^2 = \frac{3}{4x+1}$ stated |
| Obtain integral of form $k\ln(4x+1)$ | M1 | any constant $k$ with or without $\pi$. If brackets missing, and subsequent calculation does not show their 'presence', marks are max B1M1A0M1A0 |
| Obtain $\frac{3}{4}\pi\ln(4x+1)$ or $\frac{3}{4}\ln(4x+1)$ | A1 | |
| Apply limits to obtain $\frac{3}{4}\pi\ln 81$ or $\frac{3}{4}\ln 81$ | A1 | or exact equiv perhaps with $\ln 1$ present |
| Attempt to subtract volume of cylinder, using correct radius and 'height' | M1 | with exact volume of cylinder attempted. Do not treat rotation around $y$-axis as mis-read: this is 0/6 |
| Obtain $3\pi\ln 3 - \frac{20}{27}\pi$ or $\pi\left(\frac{3}{4}\ln 81 - \frac{20}{27}\right)$ | A1 | or exact equiv involving two terms |
| **Total** | **[6]** | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_547_851_1749_605}
The diagram shows the curve $y = \sqrt { \frac { 3 } { 4 x + 1 } }$ for $0 \leqslant x \leqslant 20$. The point $P$ on the curve has coordinates $\left( 20 , \frac { 1 } { 9 } \sqrt { 3 } \right)$. The shaded region $R$ is enclosed by the curve and the lines $x = 0$ and $y = \frac { 1 } { 9 } \sqrt { 3 }$.\\
(i) Find the exact area of $R$.\\
(ii) Find the exact volume of the solid obtained when $R$ is rotated completely about the $x$-axis.
\hfill \mbox{\textit{OCR C3 2014 Q7 [10]}}