| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and tangent/normal |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining standard differentiation (chain rule for finding tangent), simple algebraic manipulation to show an equation and solve it, and a routine volume of revolution calculation using the standard formula. All techniques are textbook exercises with no novel insight required, though it requires competent execution across multiple steps. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = \frac{1}{2}(x4 + 4x + 4)^{-\frac{1}{2}} \times [4x^3 + 4]\) | B1B1 | |
| At \(x = 0\), \(\frac{dy}{dx} = \frac{1}{2} \times \frac{1}{2} \times 4 = (1)\) | M1 | Sub \(x = 0\) and attempt eqn of line following differentiation. |
| Equation is \(y = 2 - x\) | A1 | |
| [4] | ||
| (ii) \(x + 2 = \sqrt{x^4 + 4x + 4} \Rightarrow (x + 2)^2\) | B1 | AG www |
| \(= x^4 + 4x + 4\) | B1 | |
| \(x^2 - x^4 = 0\) oe | B1 | |
| \(x = 0, \pm 1\) | B2,1,0 | |
| [4] | ||
| (iii) \((\pi)\left[\frac{x^5}{5} + 2x^2 + 4x\right]\) | M1A1 | Attempt to integrate \(y^2\) |
| \((\pi)\left[0 - \left(\frac{-1}{5} + 2 - 4\right)\right]\) | DM1 | |
| \(\frac{11\pi}{5}(6.91)\) oe | A1 | Apply limits \(-1 \to 0\) |
| [4] |
**(i)** $\frac{dy}{dx} = \frac{1}{2}(x4 + 4x + 4)^{-\frac{1}{2}} \times [4x^3 + 4]$ | B1B1 |
At $x = 0$, $\frac{dy}{dx} = \frac{1}{2} \times \frac{1}{2} \times 4 = (1)$ | M1 | Sub $x = 0$ and attempt eqn of line following differentiation.
Equation is $y = 2 - x$ | A1 |
| | [4]
**(ii)** $x + 2 = \sqrt{x^4 + 4x + 4} \Rightarrow (x + 2)^2$ | B1 | AG www
$= x^4 + 4x + 4$ | B1 |
$x^2 - x^4 = 0$ oe | B1 |
$x = 0, \pm 1$ | B2,1,0 |
| | [4]
**(iii)** $(\pi)\left[\frac{x^5}{5} + 2x^2 + 4x\right]$ | M1A1 | Attempt to integrate $y^2$
$(\pi)\left[0 - \left(\frac{-1}{5} + 2 - 4\right)\right]$ | DM1 |
$\frac{11\pi}{5}(6.91)$ oe | A1 | Apply limits $-1 \to 0$
| | [4]11\\
\includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-4_547_1057_255_543}
The diagram shows the curve $y = \sqrt { } \left( x ^ { 4 } + 4 x + 4 \right)$.\\
(i) Find the equation of the tangent to the curve at the point ( 0,2 ).\\
(ii) Show that the $x$-coordinates of the points of intersection of the line $y = x + 2$ and the curve are given by the equation $( x + 2 ) ^ { 2 } = x ^ { 4 } + 4 x + 4$. Hence find these $x$-coordinates.\\
(iii) The region shaded in the diagram is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the volume of revolution.
\hfill \mbox{\textit{CAIE P1 2013 Q11 [12]}}