| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.3 Part (i) is a routine trapezium rule application with straightforward function evaluations. Part (ii) requires binomial expansion for small x (standard P3 technique) followed by term-by-term integration. Both parts are standard textbook exercises with no novel problem-solving required, making this slightly easier than average but not trivial due to the two-part structure and algebraic manipulation needed. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply correct ordinates \(1, 0.94259\ldots, 0.79719\ldots, 0.62000\ldots\) | B1 | |
| Use correct formula or equivalent with \(h = 0.1\) and four \(y\) values | M1 | |
| Obtain \(0.255\) with no errors seen | A1 | [3] |
| Answer | Marks |
|---|---|
| Obtain or imply \(a = -6\) | B1 |
| Obtain \(x^4\) term including correct attempt at coefficient | M1 |
| Obtain or imply \(b = 27\) | A1 |
| Answer | Marks |
|---|---|
| Integrate to obtain \(x - 2x^3 + \frac{27}{5}x^5\), following their values of \(a\) and \(b\) | B1♦ |
| Obtain \(0.259\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Use correct trapezium rule with at least 3 ordinates | M1 | |
| Obtain \(0.259\) (from 4) | A1 | [5] |
**(i)**
State or imply correct ordinates $1, 0.94259\ldots, 0.79719\ldots, 0.62000\ldots$ | B1 |
Use correct formula or equivalent with $h = 0.1$ and four $y$ values | M1 |
Obtain $0.255$ with no errors seen | A1 | [3]
**(ii)**
Obtain or imply $a = -6$ | B1 |
Obtain $x^4$ term including correct attempt at coefficient | M1 |
Obtain or imply $b = 27$ | A1 |
**Either**
Integrate to obtain $x - 2x^3 + \frac{27}{5}x^5$, following their values of $a$ and $b$ | B1♦ |
Obtain $0.259$ | B1 |
**Or**
Use correct trapezium rule with at least 3 ordinates | M1 |
Obtain $0.259$ (from 4) | A1 | [5]6 It is given that $I = \int _ { 0 } ^ { 0.3 } \left( 1 + 3 x ^ { 2 } \right) ^ { - 2 } \mathrm {~d} x$.\\
(i) Use the trapezium rule with 3 intervals to find an approximation to $I$, giving the answer correct to 3 decimal places.\\
(ii) For small values of $x , \left( 1 + 3 x ^ { 2 } \right) ^ { - 2 } \approx 1 + a x ^ { 2 } + b x ^ { 4 }$. Find the values of the constants $a$ and $b$. Hence, by evaluating $\int _ { 0 } ^ { 0.3 } \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \mathrm { d } x$, find a second approximation to $I$, giving the answer correct to 3 decimal places.
\hfill \mbox{\textit{CAIE P3 2014 Q6 [8]}}