| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| 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 testing standard calculus techniques: differentiation of simple functions (power rule), finding stationary points, second derivative test, and volume of revolution using a standard formula. All parts follow routine procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx}=-8x^{-2}+2\) | B1 | unsimplified ok |
| \(\frac{d^2y}{dx^2}=16x^{-3}\) | B1 | unsimplified ok |
| \(\int y^2\,dx = -64x^{-1}+32x+\frac{4x^3}{3}\) oe \((+c)\) | 3×B1 [5] | B1 for each term – unsimplified ok |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(\frac{dy}{dx}\) to \(0 \rightarrow x=\pm2\) | M1 | Sets to \(0\) and attempts to solve |
| \(\rightarrow M(2,8)\) | A1 | Any pair of correct values |
| Other turning point is \((-2,-8)\) | A1 | Second pair of values |
| If \(x=-2\), \(\frac{d^2y}{dx^2}<0\) | M1 | Using their \(\frac{d^2y}{dx^2}\) if \(kx^{-3}\) and \(x<0\) |
| \(\therefore\) Maximum | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Vol \(=\pi\times[\text{part (i)}]\) from \(1\) to \(2\) | M1 | Evidence of using limits \(1\) & \(2\) in their integral of \(y^2\) (ignore \(\pi\)) |
| \(\frac{220\pi}{3}\), \(73.3\pi\), \(230\) | A1 [2] |
# Question 10(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx}=-8x^{-2}+2$ | **B1** | unsimplified ok |
| $\frac{d^2y}{dx^2}=16x^{-3}$ | **B1** | unsimplified ok |
| $\int y^2\,dx = -64x^{-1}+32x+\frac{4x^3}{3}$ oe $(+c)$ | **3×B1** [5] | B1 for each term – unsimplified ok |
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# Question 10(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $\frac{dy}{dx}$ to $0 \rightarrow x=\pm2$ | **M1** | Sets to $0$ and attempts to solve |
| $\rightarrow M(2,8)$ | **A1** | Any pair of correct values |
| Other turning point is $(-2,-8)$ | **A1** | Second pair of values |
| If $x=-2$, $\frac{d^2y}{dx^2}<0$ | **M1** | Using their $\frac{d^2y}{dx^2}$ if $kx^{-3}$ and $x<0$ |
| $\therefore$ Maximum | **A1** [5] | |
---
# Question 10(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Vol $=\pi\times[\text{part (i)}]$ from $1$ to $2$ | **M1** | Evidence of using limits $1$ & $2$ in their integral of $y^2$ (ignore $\pi$) |
| $\frac{220\pi}{3}$, $73.3\pi$, $230$ | **A1** [2] | |
---10\\
\includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826}
The diagram shows the part of the curve $y = \frac { 8 } { x } + 2 x$ for $x > 0$, and the minimum point $M$.\\
(i) Find expressions for $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\int y ^ { 2 } \mathrm {~d} x$.\\
(ii) Find the coordinates of $M$ and determine the coordinates and nature of the stationary point on the part of the curve for which $x < 0$.\\
(iii) Find the volume obtained when the region bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 2$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
\hfill \mbox{\textit{CAIE P1 2016 Q10 [12]}}