| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve trigonometric equation via iteration |
| Difficulty | Standard +0.3 This is a standard C3 iteration question with routine curve sketching, straightforward application of a given iterative formula, and basic transformations. While it has multiple parts, each component uses well-practiced techniques without requiring novel insight or complex problem-solving beyond typical A-level expectations. |
| 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 graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Draw more or less correct sketch of \(y = \cos^{-1} x\) existing in first and second quadrants | *B1 | Ignore any curve outside \(0 \le y \le \pi\); condone no or wrong intercepts on axes |
| Draw U-shaped parabola passing through origin and showing minimum point | *B1 | Curve must exist in first and third quadrants |
| Indicate one intersection in first quadrant by blob or reference in words or ... | B1 | Dep *B *B |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain correct first iterate showing at least 4 s.f. rounded or truncated | B1 | |
| Show iterative process to produce at least three iterates in all showing at least 3 s.f. | M1 | Implied by incorrect values apparently converging |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain at least four correct iterates in all showing at least 4 s.f. | A1 | Allowing recovery after error |
| Conclude with value 0.242 | A1 | Answer to be clearly indicated by underlining final value in sequence or by separate statement; answer required to precisely 3 s.f.; allow final A1 even if iterates have been shown to only 3 s.f.; answer only earns 0/4 |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| State \(y = -\cos^{-1}(-x)\) or \(y = \cos^{-1} x - \pi\) | B1 | |
| State \(y = x(-2x + 5)\) or equiv | B1 | Allow \(y = --x(2(-x) + 5)\) or similar; condone missing \(y =\) in each case |
| State \(-0.242\) for x-coordinate | B1 FT | Following their answer to (ii); allow greater accuracy here |
| State \(-1.33\) for y-coordinate | B1 | Allow value rounding to \(-1.33\) |
| [4] |
## Part i
Draw more or less correct sketch of $y = \cos^{-1} x$ existing in first and second quadrants | *B1 | Ignore any curve outside $0 \le y \le \pi$; condone no or wrong intercepts on axes
Draw U-shaped parabola passing through origin and showing minimum point | *B1 | Curve must exist in first and third quadrants
Indicate one intersection in first quadrant by blob or reference in words or ... | B1 | Dep *B *B
| [3] |
## Part ii
Obtain correct first iterate showing at least 4 s.f. rounded or truncated | B1 |
Show iterative process to produce at least three iterates in all showing at least 3 s.f. | M1 | Implied by incorrect values apparently converging | 0.25
0.23965...
0.24250...
0.24172...
0.24193...
Obtain at least four correct iterates in all showing at least 4 s.f. | A1 | Allowing recovery after error
Conclude with value 0.242 | A1 | Answer to be clearly indicated by underlining final value in sequence or by separate statement; answer required to precisely 3 s.f.; allow final A1 even if iterates have been shown to only 3 s.f.; answer only earns 0/4
| [4] |
## Part iii
State $y = -\cos^{-1}(-x)$ or $y = \cos^{-1} x - \pi$ | B1 |
State $y = x(-2x + 5)$ or equiv | B1 | Allow $y = --x(2(-x) + 5)$ or similar; condone missing $y =$ in each case
State $-0.242$ for x-coordinate | B1 FT | Following their answer to (ii); allow greater accuracy here
State $-1.33$ for y-coordinate | B1 | Allow value rounding to $-1.33$
| [4] |7 (i) By sketching the curves $y = x ( 2 x + 5 )$ and $y = \cos ^ { - 1 } x$ (where $y$ is in radians) in a single diagram, show that the equation $x ( 2 x + 5 ) = \cos ^ { - 1 } x$ has exactly one real root.\\
(ii) Use the iterative formula
$$x _ { n + 1 } = \frac { \cos ^ { - 1 } x _ { n } } { 2 x _ { n } + 5 } \text { with } x _ { 1 } = 0.25$$
to find the root correct to 3 significant figures. Show the result of each iteration correct to at least 4 significant figures.\\
(iii) Two new curves are obtained by transforming each of the curves $y = x ( 2 x + 5 )$ and $y = \cos ^ { - 1 } x$ by the pair of transformations:\\
reflection in the $x$-axis followed by reflection in the $y$-axis.\\
State an equation of each of the new curves and determine the coordinates of their point of intersection, giving each coordinate correct to 3 significant figures.
\hfill \mbox{\textit{OCR C3 2016 Q7 [11]}}