| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Combined stretch and translation |
| Difficulty | Standard +0.8 This question combines transformations, curve sketching, and iterative sequences converging to roots. Part (i) requires careful algebraic manipulation to decompose a composite transformation. Part (ii) involves sketching and reasoning about intersection points. Part (iii) connects iteration formulas to equation roots, requiring substitution and numerical work. The multi-step nature, the need to link different concepts (transformations, iterations, root-finding), and the algebraic manipulation of the iteration formula make this moderately harder than a typical C3 question, though each individual component uses standard techniques. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State scale factor is \(\sqrt{2}\) | B1 | allow 1.4 |
| State translation is in negative x-direction \(\ldots\) \(\ldots\) by \(\frac{3}{2}\) units | B1 | or clear equiv |
| B1 | 3 | |
| (ii) Draw (more or less) correct sketch of \(y=\sqrt{2x+3}\) | B1 | 'starting' at point on negative x-axis |
| Draw (more or less) correct sketch of \(y=\frac{N}{x^3}\) | B1 | showing both branches |
| Indicate one point of intersection | B1 | 3 with both sketches correct |
| [SC: if neither sketch complete or correct but diagram correct for both in first quadrant | B1] | |
| (iii) (a) Substitute 1.9037 into \(x=N^{\frac{1}{3}}(2x+3)^{-\frac{1}{4}}\) | M1 | or into equation \(\sqrt{2x+3}=\frac{N}{x^3}\); or equiv |
| Obtain 18 or value rounding to 18 | A1 | 2 with no error seen |
| (b) State or imply \(2.6282=N^{\frac{1}{3}}(2 \times 2.6022+3)^{-\frac{1}{4}}\) | B1 | |
| Attempt solution for N | M1 | using correct process |
| Obtain 52 | A1 | 3 concluding with integer value |
(i) State scale factor is $\sqrt{2}$ | B1 | allow 1.4
State translation is in negative x-direction $\ldots$ $\ldots$ by $\frac{3}{2}$ units | B1 | or clear equiv
| B1 | 3
(ii) Draw (more or less) correct sketch of $y=\sqrt{2x+3}$ | B1 | 'starting' at point on negative x-axis
Draw (more or less) correct sketch of $y=\frac{N}{x^3}$ | B1 | showing both branches
Indicate one point of intersection | B1 | 3 with both sketches correct
[SC: if neither sketch complete or correct but diagram correct for both in first quadrant | B1]
(iii) (a) Substitute 1.9037 into $x=N^{\frac{1}{3}}(2x+3)^{-\frac{1}{4}}$ | M1 | or into equation $\sqrt{2x+3}=\frac{N}{x^3}$; or equiv
Obtain 18 or value rounding to 18 | A1 | 2 with no error seen
(b) State or imply $2.6282=N^{\frac{1}{3}}(2 \times 2.6022+3)^{-\frac{1}{4}}$ | B1
Attempt solution for N | M1 | using correct process
Obtain 52 | A1 | 3 concluding with integer value
**Total: 11**
---\begin{enumerate}[label=(\roman*)]
\item The curve $y = \sqrt{x}$ can be transformed to the curve $y = \sqrt{2x + 3}$ by means of a stretch parallel to the $y$-axis followed by a translation. State the scale factor of the stretch and give details of the translation. [3]
\item It is given that $N$ is a positive integer. By sketching on a single diagram the graphs of $y = \sqrt{2x + 3}$ and $y = \frac{N}{x}$, show that the equation
$$\sqrt{2x + 3} = \frac{N}{x}$$
has exactly one real root. [3]
\item A sequence $x_1, x_2, x_3, \ldots$ has the property that
$$x_{n+1} = N^{\frac{1}{2}}(2x_n + 3)^{-\frac{1}{4}}.$$
For certain values of $x_1$ and $N$, it is given that the sequence converges to the root of the equation
$$\sqrt{2x + 3} = \frac{N}{x}.$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of the integer $N$ for which the sequence converges to the value 1.9037 (correct to 4 decimal places). [2]
\item Find the value of the integer $N$ for which, correct to 4 decimal places, $x_3 = 2.6022$ and $x_4 = 2.6282$. [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2010 Q8 [11]}}