| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and stationary points |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining standard differentiation (finding stationary points) with volume of revolution. Part (i) requires routine chain rule differentiation and verification of a given equation. Part (ii) involves expanding the squared function and integrating term-by-term using standard power rules. While it requires careful algebraic manipulation, all techniques are standard A-level procedures with no novel problem-solving required. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07n Stationary points: find maxima, minima using derivatives4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 2(x+1) - (x+1)^{-2}\) | B1 | |
| Set \(= 0\) and obtain \(2(x+1)^3 = 1\) convincingly www AG | B1 | |
| \(\frac{d^2y}{dx^2} = 2 + 2(x+1)^{-3}\) www | B1 | |
| Sub, e.g., \((x+1)^{-3} = 2\) OE or \(x = \left(\frac{1}{2}\right)^{\frac{1}{3}} - 1\) | M1 | Requires exact method – otherwise scores M0 |
| \(\frac{d^2y}{dx^2} = 6\) CAO www | A1 | and exact answer – otherwise scores A0 |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y^2 = (x+1)^4 + (x+1)^{-2} + 2(x+1)\) SOI | B1 | OR \(y^2 = (x^4 + 4x^3 + 6x^2 + 4x + 1) + (2x+2) + (x+1)^{-2}\) |
| \((\pi)\int y^2\,dx = (\pi)\left[\dfrac{(x+1)^5}{5}\right] + \left[\dfrac{(x+1)^{-1}}{-1}\right] + \left[\dfrac{2(x+1)^2}{2}\right]\) | B1B1B1 | Attempt to integrate \(y^2\). Last term might appear as \((x^2 + 2x)\) |
| OR \((\pi)\left[\dfrac{x^5}{5} + x^4 + 2x^3 + 2x^2 + x\right] + \left[x^2 + 2x\right] + \left[-\dfrac{1}{x+1}\right]\) | ||
| \((\pi)\left[\dfrac{32}{5} - \dfrac{1}{2} + 4 - \left(\dfrac{1}{5} - 1 + 1\right)\right]\) | M1 | Substitute limits \(0 \to 1\) into an attempted integration of \(y^2\). Do not condone omission of value when \(x = 0\) |
| \(9.7\pi\) or \(30.5\) | A1 | Note: omission of \(2(x+1)\) in first line \(\to 6.7\pi\) scores 3/6. Ignore initially an extra volume, e.g. \((\pi)\int(4\tfrac{1}{2})^2\). Only take into account for the final answer |
| 6 |
## Question 11(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 2(x+1) - (x+1)^{-2}$ | B1 | |
| Set $= 0$ and obtain $2(x+1)^3 = 1$ convincingly www AG | B1 | |
| $\frac{d^2y}{dx^2} = 2 + 2(x+1)^{-3}$ www | B1 | |
| Sub, e.g., $(x+1)^{-3} = 2$ OE or $x = \left(\frac{1}{2}\right)^{\frac{1}{3}} - 1$ | M1 | Requires exact method – otherwise scores M0 |
| $\frac{d^2y}{dx^2} = 6$ CAO www | A1 | and exact answer – otherwise scores A0 |
| | **5** | |
## Question 11(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y^2 = (x+1)^4 + (x+1)^{-2} + 2(x+1)$ SOI | **B1** | OR $y^2 = (x^4 + 4x^3 + 6x^2 + 4x + 1) + (2x+2) + (x+1)^{-2}$ |
| $(\pi)\int y^2\,dx = (\pi)\left[\dfrac{(x+1)^5}{5}\right] + \left[\dfrac{(x+1)^{-1}}{-1}\right] + \left[\dfrac{2(x+1)^2}{2}\right]$ | **B1B1B1** | Attempt to integrate $y^2$. Last term might appear as $(x^2 + 2x)$ |
| OR $(\pi)\left[\dfrac{x^5}{5} + x^4 + 2x^3 + 2x^2 + x\right] + \left[x^2 + 2x\right] + \left[-\dfrac{1}{x+1}\right]$ | | |
| $(\pi)\left[\dfrac{32}{5} - \dfrac{1}{2} + 4 - \left(\dfrac{1}{5} - 1 + 1\right)\right]$ | **M1** | Substitute limits $0 \to 1$ into an attempted integration of $y^2$. Do not condone omission of value when $x = 0$ |
| $9.7\pi$ or $30.5$ | **A1** | Note: omission of $2(x+1)$ in first line $\to 6.7\pi$ scores 3/6. Ignore initially an extra volume, e.g. $(\pi)\int(4\tfrac{1}{2})^2$. Only take into account for the final answer |
| | **6** | |11\\
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-18_645_723_258_573}
The diagram shows part of the curve $y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }$ and the line $x = 1$. The point $A$ is the minimum point on the curve.\\
(i) Show that the $x$-coordinate of $A$ satisfies the equation $2 ( x + 1 ) ^ { 3 } = 1$ and find the exact value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.\\
(ii) Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P1 2018 Q11 [11]}}