| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find coordinate from gradient condition |
| Difficulty | Standard +0.3 This is a standard iterative methods question requiring differentiation with quotient rule, solving dy/dx = 1/2, algebraic rearrangement to show the given form, interval verification by substitution, and applying fixed-point iteration. While it involves multiple steps, each component is routine A-level technique with clear guidance provided in the question structure. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07q Product and quotient rules: differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use quotient rule or equivalent | *M1 | |
| Obtain \(\frac{6x(x^2 + 4) - 6x^3}{(x^2+4)^2}\) or equivalent | A1 | |
| Equate first derivative to \(\frac{1}{4}\) and remove algebraic denominators dep on *M1 | DM1 | |
| Obtain \(48p = p^4 + 8p^2 + 16\) or \(48x = x^4 + 8x^2 + 16\) or equivalent | A1 | |
| Confirm given result \(p = \sqrt{\frac{48p - 16}{p^2 + 8}}\) | A1 | [5] |
| (ii) Consider sign of \(p - \sqrt{\frac{48p-16}{p^2+8}}\) at 2 and 3 or equivalent | M1 | |
| Complete argument correctly with appropriate calculations | A1 | [2] |
| (iii) Carry out iteration process correctly at least once | M1 | |
| Obtain final answer 2.728 | A1 | |
| Show sufficient iterations to justify accuracy to 4 sf or show sign change in interval (2.7275, 2.7285) | B1 | [3] |
(i) Use quotient rule or equivalent | *M1 |
Obtain $\frac{6x(x^2 + 4) - 6x^3}{(x^2+4)^2}$ or equivalent | A1 |
Equate first derivative to $\frac{1}{4}$ and remove algebraic denominators dep on *M1 | DM1 |
Obtain $48p = p^4 + 8p^2 + 16$ or $48x = x^4 + 8x^2 + 16$ or equivalent | A1 |
Confirm given result $p = \sqrt{\frac{48p - 16}{p^2 + 8}}$ | A1 | [5]
(ii) Consider sign of $p - \sqrt{\frac{48p-16}{p^2+8}}$ at 2 and 3 or equivalent | M1 |
Complete argument correctly with appropriate calculations | A1 | [2]
(iii) Carry out iteration process correctly at least once | M1 |
Obtain final answer 2.728 | A1 |
Show sufficient iterations to justify accuracy to 4 sf or show sign change in interval (2.7275, 2.7285) | B1 | [3]6 The equation of a curve is $y = \frac { 3 x ^ { 2 } } { x ^ { 2 } + 4 }$. At the point on the curve with positive $x$-coordinate $p$, the gradient of the curve is $\frac { 1 } { 2 }$.\\
(i) Show that $p = \sqrt { } \left( \frac { 48 p - 16 } { p ^ { 2 } + 8 } \right)$.\\
(ii) Show by calculation that $2 < p < 3$.\\
(iii) Use an iterative formula based on the equation in part (i) to find the value of $p$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
\hfill \mbox{\textit{CAIE P2 2016 Q6 [10]}}