| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring: (i) testing integer values to find a root, (ii) algebraic rearrangement of a polynomial equation, and (iii) applying a given iterative formula. All steps are routine and mechanical with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(b = 3\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Commence division by \(x - b\) and reach partial quotient \(x^3 + kx^2\) | M1 | |
| Obtain quotient \(x^3 + x^2 + 3x + 2\) | A1 | There being no remainder |
| Equate quotient to zero and rearrange to make the subject \(a\) | M1 | |
| Obtain the given equation | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula \(a_{n+1} = -\frac{1}{3}(2 + a_n^2 + a_n^3)\) correctly at least once | M1 | |
| Obtain final answer \(-0.715\) | A1 | |
| Show sufficient iterations to 5 d.p. to justify \(-0.715\) to 3 d.p., or show there is a sign change in the interval \((-0.7145, -0.7155)\) | A1 | |
| Total: 3 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $b = 3$ | B1 | |
| **Total: 1** | | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Commence division by $x - b$ and reach partial quotient $x^3 + kx^2$ | M1 | |
| Obtain quotient $x^3 + x^2 + 3x + 2$ | A1 | There being no remainder |
| Equate quotient to zero and rearrange to make the subject $a$ | M1 | |
| Obtain the given equation | A1 | |
| **Total: 4** | | |
## Question 6(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula $a_{n+1} = -\frac{1}{3}(2 + a_n^2 + a_n^3)$ correctly at least once | M1 | |
| Obtain final answer $-0.715$ | A1 | |
| Show sufficient iterations to 5 d.p. to justify $-0.715$ to 3 d.p., or show there is a sign change in the interval $(-0.7145, -0.7155)$ | A1 | |
| **Total: 3** | | |6\\
\includegraphics[max width=\textwidth, alt={}, center]{87392b1c-3683-45b4-8d55-36760b5f0cc1-10_547_531_260_806}
The diagram shows the curve $y = x ^ { 4 } - 2 x ^ { 3 } - 7 x - 6$. The curve intersects the $x$-axis at the points $( a , 0 )$ and $( b , 0 )$, where $a < b$. It is given that $b$ is an integer.\\
(i) Find the value of $b$.\\
(ii) Hence show that $a$ satisfies the equation $a = - \frac { 1 } { 3 } \left( 2 + a ^ { 2 } + a ^ { 3 } \right)$.\\
(iii) Use an iterative formula based on the equation in part (ii) to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2019 Q6 [8]}}