| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Composite transformation sketch |
| Difficulty | Standard +0.8 This is a multi-part question requiring: (i) composite transformation of inverse trig function with horizontal translation and vertical stretch, (ii) graphical reasoning about roots, (iii) numerical verification using inverse cosine, and (iv) iterative sequence convergence with algebraic justification. While each individual part uses standard C3 techniques, the combination of transformations, root-finding methods, and connecting iteration to equation-solving requires solid understanding across multiple topics. More demanding than typical C3 questions but not requiring exceptional insight. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x)1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch curve showing (at least) translation in \(x\) direction | M1 | Either positive or negative |
| Show correct sketch with one of 2 and \(3\pi\) indicated | A1 | |
| \(\ldots\) and with other one of 2 and \(3\pi\) indicated | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Draw straight line through \(O\) with positive gradient | B1 | 1 Label and explanation not required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt calculations using 1.8 and 1.9 | M1 | Allow here if degrees used |
| Obtain correct values and indicate change of sign | A1 | 2 Or equiv; \(x=1.8\): LHS \(= 1.93\), diff \(= 0.13\); \(x=1.9\): LHS \(= 1.35\), diff \(= -0.55\); radians needed now |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain correct first iterate 1.79 or 1.78 | B1 | Or greater accuracy |
| Attempt correct process to produce at least 3 iterates | M1 | |
| Obtain 1.82 | A1 | Answer required to exactly 2 d.p.; \(2 \to 1.7859 \to 1.8280 \to 1.8200\); SR: answer 1.82 only - B2 |
| Attempt rearrangement of \(3\cos^{-1}(x-1) = x\) or of \(x = 1 + \cos(\frac{1}{3}x)\) | M1 | Involving at least two steps |
| Obtain required formula or equation respectively | A1 | 5 |
## Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch curve showing (at least) translation in $x$ direction | M1 | Either positive or negative |
| Show correct sketch with one of 2 and $3\pi$ indicated | A1 | |
| $\ldots$ and with other one of 2 and $3\pi$ indicated | A1 | **3** |
## Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Draw straight line through $O$ with positive gradient | B1 | **1** Label and explanation not required |
## Question 7(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt calculations using 1.8 and 1.9 | M1 | Allow here if degrees used |
| Obtain correct values and indicate change of sign | A1 | **2** Or equiv; $x=1.8$: LHS $= 1.93$, diff $= 0.13$; $x=1.9$: LHS $= 1.35$, diff $= -0.55$; radians needed now |
## Question 7(iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain correct first iterate 1.79 or 1.78 | B1 | Or greater accuracy |
| Attempt correct process to produce at least 3 iterates | M1 | |
| Obtain 1.82 | A1 | Answer required to exactly 2 d.p.; $2 \to 1.7859 \to 1.8280 \to 1.8200$; SR: answer 1.82 only - B2 |
| Attempt rearrangement of $3\cos^{-1}(x-1) = x$ or of $x = 1 + \cos(\frac{1}{3}x)$ | M1 | Involving at least two steps |
| Obtain required formula or equation respectively | A1 | **5** |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-3_465_748_1133_717}
The diagram shows the curve with equation $y = \cos ^ { - 1 } x$.\\
(i) Sketch the curve with equation $y = 3 \cos ^ { - 1 } ( x - 1 )$, showing the coordinates of the points where the curve meets the axes.\\
(ii) By drawing an appropriate straight line on your sketch in part (i), show that the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$ has exactly one root.\\
(iii) Show by calculation that the root of the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$ lies between 1.8 and 1.9 .\\
(iv) The sequence defined by
$$x _ { 1 } = 2 , \quad x _ { n + 1 } = 1 + \cos \left( \frac { 1 } { 3 } x _ { n } \right)$$
converges to a number $\alpha$. Find the value of $\alpha$ correct to 2 decimal places and explain why $\alpha$ is the root of the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$.
\hfill \mbox{\textit{OCR C3 2006 Q7 [11]}}