| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Identify/describe sequence of transformations between two given equations |
| Difficulty | Standard +0.3 This is a straightforward function transformations question requiring identification of standard transformations (translation and reflection), sketching with modulus, and solving a routine equation. The inverse sine function adds slight complexity, but the transformations themselves are standard C3 material requiring no novel insight—slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02w Graph transformations: simple transformations of f(x)1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either: Refer to translation and reflection | B1 | in either order; allow clear equivs |
| State translation by 1 in negative \(x\)-direction | B1 | or equiv but now using correct terminology |
| Or: Refer to translation and reflection | B1 | in either order; allow clear equivs |
| State reflection in \(x\)-axis | B1 3 | using correct terminology |
| State translation by 1 in positive \(x\)-direction | B1 (3) | with order reflection then translation clearly intended |
| (ii) Show sketch with attempt at reflection of 'negative' part in \(x\)-axis | M1 | and curve for \(0 |
| Show (more or less) correct sketch | A1 2 | with correct curvature |
| (iii) Attempt correct process for finding at least one value | M1 | as far as \(x = \ldots\); accept decimal equivs (degrees or radians) or expressions involving \(\sin(\frac{1}{4}\pi)\) |
| Obtain \(1 - \frac{1}{2}\sqrt{3}\) | A1 | |
| Obtain \(1 + \frac{1}{2}\sqrt{3}\) | A1 3 | or exact equiv; give A1A0 if extra incorrect solution(s) provided |
**(i)** Either: Refer to translation and reflection | B1 | in either order; allow clear equivs
State translation by 1 in negative $x$-direction | B1 | or equiv but now using correct terminology
Or: Refer to translation and reflection | B1 | in either order; allow clear equivs
State reflection in $x$-axis | B1 3 | using correct terminology
State translation by 1 in positive $x$-direction | B1 (3) | with order reflection then translation clearly intended
**(ii)** Show sketch with attempt at reflection of 'negative' part in $x$-axis | M1 | and curve for $0<x<1$ unchanged
Show (more or less) correct sketch | A1 2 | with correct curvature
**(iii)** Attempt correct process for finding at least one value | M1 | as far as $x = \ldots$; accept decimal equivs (degrees or radians) or expressions involving $\sin(\frac{1}{4}\pi)$
Obtain $1 - \frac{1}{2}\sqrt{3}$ | A1 |
Obtain $1 + \frac{1}{2}\sqrt{3}$ | A1 3 | or exact equiv; give A1A0 if extra incorrect solution(s) provided
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\includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_641_837_1306_657}
The diagram shows the graph of $y = - \sin ^ { - 1 } ( x - 1 )$.\\
(i) Give details of the pair of geometrical transformations which transforms the graph of $y = - \sin ^ { - 1 } ( x - 1 )$ to the graph of $y = \sin ^ { - 1 } x$.\\
(ii) Sketch the graph of $y = \left| - \sin ^ { - 1 } ( x - 1 ) \right|$.\\
(iii) Find the exact solutions of the equation $\left| - \sin ^ { - 1 } ( x - 1 ) \right| = \frac { 1 } { 3 } \pi$.
\hfill \mbox{\textit{OCR C3 2008 Q6 [8]}}