| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration inequality bounds |
| Difficulty | Challenging +1.8 This FP2 question requires understanding of Riemann sums and geometric series to establish integral bounds, then numerical evaluation. While the geometric series manipulation is non-trivial and the proof structure requires insight into upper/lower rectangle sums, the individual steps are methodical once the approach is identified. The multi-part structure and need to connect geometric reasoning with algebraic manipulation places it well above average difficulty. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.08g Integration as limit of sum: Riemann sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to sum areas of rectangles | M1 | \((h.3^h + h.3^{2h} + ... + h.3^{(n-1)h})\) |
| Use G.P. on \(h(1+3^h+3^{2h}+...+3^{(n-1)h})\) | M1 | All terms not required, but last term needed (or \(3^{1-h}\)); or specify \(a, r\) and \(n\) for a G.P. |
| Simplify to A.G. | A1 | Clearly use \(nh = 1\) |
| (ii) Attempt to find sum areas of different rect. | M1 | Different from (i) |
| Use G.P. on \(h(3^h+3^{2h}+...+3^{nh})\) | M1 | All terms not required, but last term needed; G.P. specified as in (i), or deduced from (i) |
| Simplify to A.G. | A1 | |
| (iii) Get \(1.8194(8), 1.8214(8)\) correct | B1, B1 | Allow \(1.81 \leq A \leq 1.83\) |
**(i)** Attempt to sum areas of rectangles | M1 | $(h.3^h + h.3^{2h} + ... + h.3^{(n-1)h})$
Use G.P. on $h(1+3^h+3^{2h}+...+3^{(n-1)h})$ | M1 | All terms not required, but last term needed (or $3^{1-h}$); or specify $a, r$ and $n$ for a G.P.
Simplify to A.G. | A1 | Clearly use $nh = 1$
**(ii)** Attempt to find sum areas of different rect. | M1 | Different from (i)
Use G.P. on $h(3^h+3^{2h}+...+3^{nh})$ | M1 | All terms not required, but last term needed; G.P. specified as in (i), or deduced from (i)
Simplify to A.G. | A1 |
**(iii)** Get $1.8194(8), 1.8214(8)$ correct | B1, B1 | Allow $1.81 \leq A \leq 1.83$6\\
\includegraphics[max width=\textwidth, alt={}, center]{52b43f20-e0e6-4ddd-9518-bea9782982bf-3_623_1354_262_392}
The diagram shows the curve with equation $y = 3 ^ { x }$ for $0 \leqslant x \leqslant 1$. The area $A$ under the curve between these limits is divided into $n$ strips, each of width $h$ where $n h = 1$.\\
(i) By using the set of rectangles indicated on the diagram, show that $A > \frac { 2 h } { 3 ^ { h } - 1 }$.\\
(ii) By considering another set of rectangles, show that $A < \frac { ( 2 h ) 3 ^ { h } } { 3 ^ { h } - 1 }$.\\
(iii) Given that $h = 0.001$, use these inequalities to find values between which $A$ lies.
\hfill \mbox{\textit{OCR FP2 2006 Q6 [8]}}