| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2002 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Iterative formula with graphical justification |
| Difficulty | Standard +0.3 This is a standard multi-part question involving curve sketching to show root existence, verification by substitution, algebraic manipulation to derive an equivalent form, and applying a given iterative formula. All techniques are routine for P2 level with no novel insight required—slightly easier than average due to the structured guidance through each step. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make recognisable sketches over the given range of a suitable pair of graphs e.g. \(y = \sin x\) and \(y = \frac{1}{x^2}\) | B1 | |
| State or imply connection between intersections and roots and justify given statement | B1 | 2 |
| (ii) Calculate values (or signs) of \(\sin x - \frac{1}{x^2}\) at \(x = 1\) and \(x = 1.5\) | M1 | |
| Derive given result correctly | A1 | 2 |
| (iii) Rearrange \(\sin x = \frac{1}{x^2}\) and obtain given answer | B1 | 1 |
| (iv) Use the iterative formula correctly with \(1 \leq x_n \leq 1.5\) | M1 | |
| Obtain final answer 1.07 | A1 | |
| Show sufficient iterations to justify its accuracy to 3d.p., or show there is a sign change in the interval (1.065, 1.075) | A1 | 3 |
**(i)** Make recognisable sketches over the given range of a suitable pair of graphs e.g. $y = \sin x$ and $y = \frac{1}{x^2}$ | B1 |
State or imply connection between intersections and roots and justify given statement | B1 | 2 |
**(ii)** Calculate values (or signs) of $\sin x - \frac{1}{x^2}$ at $x = 1$ and $x = 1.5$ | M1 |
Derive given result correctly | A1 | 2 |
**(iii)** Rearrange $\sin x = \frac{1}{x^2}$ and obtain given answer | B1 | 1 |
**(iv)** Use the iterative formula correctly with $1 \leq x_n \leq 1.5$ | M1 |
Obtain final answer 1.07 | A1 |
Show sufficient iterations to justify its accuracy to 3d.p., or show there is a sign change in the interval (1.065, 1.075) | A1 | 3 |
---4 (i) By sketching a suitable pair of graphs, show that there is only one value of $x$ in the interval $0 < x < \frac { 1 } { 2 } \pi$ that is a root of the equation
$$\sin x = \frac { 1 } { x ^ { 2 } }$$
(ii) Verify by calculation that this root lies between 1 and 1.5.\\
(iii) Show that this value of $x$ is also a root of the equation
$$x = \sqrt { } ( \operatorname { cosec } x )$$
(iv) Use the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( \operatorname { cosec } x _ { n } \right)$$
to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate.
\hfill \mbox{\textit{CAIE P2 2002 Q4 [8]}}