| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Iterative formula with graphical justification |
| Difficulty | Standard +0.3 This is a slightly above-average C3 question requiring curve sketching of y=(x-2)^4 and y=x+16, identifying intersections, and applying a given iterative formula. The sketching is straightforward (quartic and linear), the smaller root is obvious by inspection (x=0), and the iteration is routine application of a provided formula. No novel insight or complex problem-solving required. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Draw sketch of \(y = (x - 2)^4\) | *B1 | touching positive \(x\)-axis and extending at least as far as the \(y\)-axis; no need for 2 or 16 to be marked; ignore wrong intercepts at least in first quadrant and reaching positive \(y\)-axis; assess the two graphs independently of each other |
| Draw straight line with positive gradient | *B1 | |
| Indicate two roots | B1 | 3 marks: AG; dep *B *B and two correct graphs which meet on the \(y\)-axis; indicated in words or by marks on sketch |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) State \(0\) or \(x = 0\) | B1 | 1 mark: not merely for coordinates (0, 16) |
| (iii) Obtain correct first iterate | B1 | to at least 3 dp; any starting value (> -16) producing at least 3 iterates in all; may be implied by plausible converging values |
| Show correct iteration process | M1 | |
| Obtain at least 3 correct iterates | A1 | allowing recovery after error; iterates given to only 3 d.p. acceptable; values may be rounded or truncated |
| Obtain 4.118 | A1 | 4 marks: answer required to exactly 3 dp; A0 here if number of iterates is not enough to justify 4.118; attempt consisting of answer only earns 0/4 |
**(i)** Draw sketch of $y = (x - 2)^4$ | *B1 | touching positive $x$-axis and extending at least as far as the $y$-axis; no need for 2 or 16 to be marked; ignore wrong intercepts at least in first quadrant and reaching positive $y$-axis; assess the two graphs independently of each other
Draw straight line with positive gradient | *B1 |
Indicate two roots | B1 | 3 marks: AG; dep *B *B and two correct graphs which meet on the $y$-axis; indicated in words or by marks on sketch
[SC: Draw sketch of $y = (x-2)^4 - x - 16$ and indicate the two roots: B1 (i.e. max 1 mark)]
**(ii)** State $0$ or $x = 0$ | B1 | 1 mark: not merely for coordinates (0, 16)
**(iii)** Obtain correct first iterate | B1 | to at least 3 dp; any starting value (> -16) producing at least 3 iterates in all; may be implied by plausible converging values
Show correct iteration process | M1 |
Obtain at least 3 correct iterates | A1 | allowing recovery after error; iterates given to only 3 d.p. acceptable; values may be rounded or truncated
Obtain 4.118 | A1 | 4 marks: answer required to exactly 3 dp; A0 here if number of iterates is not enough to justify 4.118; attempt consisting of answer only earns 0/4
Iteration table provided: $[0 \to 4 \to 4.114743 \to 4.117769 \to 4.117849; 1 \to 4.030543 \to 4.115549 \to 4.117790 \to 4.117849; 2 \to 4.059767 \to 4.116321 \to 4.117811 \to 4.117850; 3 \to 4.087798 \to 4.117060 \to 4.117830 \to 4.117850; 4 \to 4.114743 \to 4.117769 \to 4.117849 \to 4.117851; 5 \to 4.140695 \to 4.118452 \to 4.117867 \to 4.117851]$
---4 (i) Show by means of suitable sketch graphs that the equation
$$( x - 2 ) ^ { 4 } = x + 16$$
has exactly 2 real roots.\\
(ii) State the value of the smaller root.\\
(iii) Use the iterative formula
$$x _ { n + 1 } = 2 + \sqrt [ 4 ] { x _ { n } + 16 }$$
with a suitable starting value, to find the larger root correct to 3 decimal places.
\hfill \mbox{\textit{OCR C3 2011 Q4 [8]}}