| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Use series to approximate numerical value |
| Difficulty | Challenging +1.8 This is a substantial Further Maths question requiring arc length integration, surface of revolution about the y-axis (non-standard), radius of curvature formula, centre of curvature calculation, and envelope finding. While each technique is standard for FP3, the combination of five parts testing different advanced calculus topics, plus the non-routine y-axis rotation and envelope work, makes this significantly harder than average A-level questions but still within expected FP3 scope. |
| Spec | 8.06b Arc length and surface area: of revolution, cartesian or parametric |
| Answer | Marks |
|---|---|
| (i) Show arc length for \(1\leq x\leq a\) is \(\frac{1}{2}a^2+\frac{1}{4}\ln a - \frac{1}{2}\) | [5] |
| (ii) Find surface area when arc \(1\leq x\leq 4\) rotated \(2\pi\) about y-axis | [5] |
| (iii) Show radius of curvature at \(x=a\) is \(a\!\left(a+\frac{1}{4a}\right)^2\) | [5] |
| (iv) Find centre of curvature at \(\left(1,\frac{1}{2}\right)\) | [5] |
| (v) Find envelope of family \(y=px^2-p^2\ln x\) | [4] |
## Question 3 (Option 3: Differential geometry)
**(i)** Show arc length for $1\leq x\leq a$ is $\frac{1}{2}a^2+\frac{1}{4}\ln a - \frac{1}{2}$ | [5] |
**(ii)** Find surface area when arc $1\leq x\leq 4$ rotated $2\pi$ about y-axis | [5] |
**(iii)** Show radius of curvature at $x=a$ is $a\!\left(a+\frac{1}{4a}\right)^2$ | [5] |
**(iv)** Find centre of curvature at $\left(1,\frac{1}{2}\right)$ | [5] |
**(v)** Find envelope of family $y=px^2-p^2\ln x$ | [4] |
---3 The curve $C$ has equation $y = \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } \ln x$, and $a$ is a constant with $a \geqslant 1$.\\
(i) Show that the length of the arc of $C$ for which $1 \leqslant x \leqslant a$ is $\frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 4 } \ln a - \frac { 1 } { 2 }$.\\
(ii) Find the area of the surface generated when the arc of $C$ for which $1 \leqslant x \leqslant 4$ is rotated through $2 \pi$ radians about the $\boldsymbol { y }$-axis.\\
(iii) Show that the radius of curvature of $C$ at the point where $x = a$ is $a \left( a + \frac { 1 } { 4 a } \right) ^ { 2 }$.\\
(iv) Find the centre of curvature corresponding to the point $\left( 1 , \frac { 1 } { 2 } \right)$ on $C$.\\
$C$ is one member of the family of curves defined by $y = p x ^ { 2 } - p ^ { 2 } \ln x$, where $p$ is a parameter.\\
(v) Find the envelope of this family of curves.
\hfill \mbox{\textit{OCR MEI FP3 2007 Q3 [24]}}