| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Inverse function graphs and properties |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on inverse trig functions requiring standard knowledge of arctan properties (asymptotes at ±π/2), basic transformations, solving a simple equation by equating functions, and applying Simpson's rule with clear instructions. All parts are routine C3 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(a = \frac{1}{2}\pi\) | B1 | For correct exact value stated |
| (ii) \(x = \tan(\frac{1}{4}a) = 1\) | M1, A1✓ | For use of \(x = \tan(\frac{1}{4}a)\); For correct answer, following their \(a\) |
| (iii) For x-translation of (approx) +1 For y-stretch with (approx) factor 2 Asymptotes are \(y = \pm 2a\) | B1, B1, B1 | For x-translation of (approx) +1; For y-stretch with (approx) factor 2; For correct statement of asymptotes |
| (iv) | \(x\) | \(\tan^{-1}x\) |
| 1.535 | 0.993 | 0.983 |
| 1.545 | 0.996 | 0.998 |
| Hence graphs cross between 1.535 and 1.545 | M1, A1 | For relevant evaluations at 1.535, 1.545; For correct details and explanation |
| (v) Relevant values of \((\tan^{-1}x)^2\) are (approximately) 0, 0.0600, 0.2150, 0.4141, 0.6169 \(\frac{1}{12}[0 + 4(0.0600 + 0.4141) + 2 \times 0.2150 + 0.6169]\) Hence required approximation is 0.245 | M1, M1, A1 | For the relevant function values seen or implied; must be radians, not degrees; For use of correct formula with \(h = \frac{1}{4}\); For correct (2 or 3sf) answer |
**(i)** $a = \frac{1}{2}\pi$ | B1 | For correct exact value stated
**(ii)** $x = \tan(\frac{1}{4}a) = 1$ | M1, A1✓ | For use of $x = \tan(\frac{1}{4}a)$; For correct answer, following their $a$
**(iii)** For x-translation of (approx) +1 For y-stretch with (approx) factor 2 Asymptotes are $y = \pm 2a$ | B1, B1, B1 | For x-translation of (approx) +1; For y-stretch with (approx) factor 2; For correct statement of asymptotes
**(iv)** | $x$ | $\tan^{-1}x$ | $2\tan^{-1}(x-1)$ |
| --- | --- | --- |
| 1.535 | 0.993 | 0.983 |
| 1.545 | 0.996 | 0.998 |
Hence graphs cross between 1.535 and 1.545 | M1, A1 | For relevant evaluations at 1.535, 1.545; For correct details and explanation
**(v)** Relevant values of $(\tan^{-1}x)^2$ are (approximately) 0, 0.0600, 0.2150, 0.4141, 0.6169 $\frac{1}{12}[0 + 4(0.0600 + 0.4141) + 2 \times 0.2150 + 0.6169]$ Hence required approximation is 0.245 | M1, M1, A1 | For the relevant function values seen or implied; must be radians, not degrees; For use of correct formula with $h = \frac{1}{4}$; For correct (2 or 3sf) answer
**Total for Question 9: 4 marks**9\\
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_424_707_1260_724}
The diagram shows the curve $y = \tan ^ { - 1 } x$ and its asymptotes $y = \pm a$.\\
(i) State the exact value of $a$.\\
(ii) Find the value of $x$ for which $\tan ^ { - 1 } x = \frac { 1 } { 2 } a$.
The equation of another curve is $y = 2 \tan ^ { - 1 } ( x - 1 )$.\\
(iii) Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of $a$.\\
(iv) Verify by calculation that the value of $x$ at the point of intersection of the two curves is 1.54 , correct to 2 decimal places.
Another curve (which you are not asked to sketch) has equation $y = \left( \tan ^ { - 1 } x \right) ^ { 2 }$.\\
(v) Use Simpson's rule, with 4 strips, to find an approximate value for $\int _ { 0 } ^ { 1 } \left( \tan ^ { - 1 } x \right) ^ { 2 } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C3 Q9 [11]}}