| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a structured, multi-part question on fixed point iteration that guides students through each step: algebraic manipulation using the double angle formula, verification of a root's location, applying a given iterative formula, and solving trigonometric equations. While it requires several techniques (trigonometric identities, iteration, inverse trig), each part is scaffolded and follows standard procedures with no novel insight required. Slightly easier than average due to the extensive guidance provided. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct identity for \(\tan 2x\) and obtain \(at^4 + bt^3 + ct^2 + dt = 0\), where \(b\) may be zero | M1 | |
| Obtain correct horizontal equation, e.g. \(4t + 5t^2 - 5t^4 = 0\) | A1 | |
| Obtain \(kt(t^3 + et + f) = 0\) or equivalent | M1 | |
| Confirm given results \(t = 0\) and \(t = \sqrt[3]{t + 0.8}\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Consider sign of \(t - \sqrt[3]{t + 0.8}\) at 1.2 and 1.3 or equivalent | M1 | |
| Justify the given statement with correct calculations (\(-0.06\) and \(0.02\)) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the iterative formula correctly at least once with \(1.2 < t_n < 1.3\) | M1 | |
| Obtain final answer 1.276 | A1 | |
| Show sufficient iterations to justify answer or show there is a change of sign in interval \((1.2755, 1.2765)\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Evaluate \(\tan^{-1}\) (answer from part (iii)) to obtain at least one value | M1 | |
| Obtain \(-2.24\) and \(0.906\) | A1 | |
| State \(-\pi\), \(0\) and \(\pi\) | B1 | [3] |
| If A0, B0, allow B1 for any 3 roots | SR |
# Question 10(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct identity for $\tan 2x$ and obtain $at^4 + bt^3 + ct^2 + dt = 0$, where $b$ may be zero | M1 | |
| Obtain correct horizontal equation, e.g. $4t + 5t^2 - 5t^4 = 0$ | A1 | |
| Obtain $kt(t^3 + et + f) = 0$ or equivalent | M1 | |
| Confirm given results $t = 0$ and $t = \sqrt[3]{t + 0.8}$ | A1 | [4] |
# Question 10(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider sign of $t - \sqrt[3]{t + 0.8}$ at 1.2 and 1.3 or equivalent | M1 | |
| Justify the given statement with correct calculations ($-0.06$ and $0.02$) | A1 | [2] |
# Question 10(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once with $1.2 < t_n < 1.3$ | M1 | |
| Obtain final answer 1.276 | A1 | |
| Show sufficient iterations to justify answer or show there is a change of sign in interval $(1.2755, 1.2765)$ | A1 | [3] |
# Question 10(iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Evaluate $\tan^{-1}$ (answer from part (iii)) to obtain at least one value | M1 | |
| Obtain $-2.24$ and $0.906$ | A1 | |
| State $-\pi$, $0$ and $\pi$ | B1 | [3] |
| If A0, B0, allow B1 for any 3 roots | | SR |10 (i) It is given that $2 \tan 2 x + 5 \tan ^ { 2 } x = 0$. Denoting $\tan x$ by $t$, form an equation in $t$ and hence show that either $t = 0$ or $t = \sqrt [ 3 ] { } ( t + 0.8 )$.\\
(ii) It is given that there is exactly one real value of $t$ satisfying the equation $t = \sqrt [ 3 ] { } ( t + 0.8 )$. Verify by calculation that this value lies between 1.2 and 1.3 .\\
(iii) Use the iterative formula $t _ { n + 1 } = \sqrt [ 3 ] { } \left( t _ { n } + 0.8 \right)$ to find the value of $t$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
(iv) Using the values of $t$ found in previous parts of the question, solve the equation
$$2 \tan 2 x + 5 \tan ^ { 2 } x = 0$$
for $- \pi \leqslant x \leqslant \pi$.
\hfill \mbox{\textit{CAIE P3 2012 Q10 [12]}}