| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Given one function find others |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: finding tan α from cot α is immediate recall, solving the sec²β equation uses the identity sec²β = 1 + tan²β to form a quadratic, and the compound angle formula for tan(α+β) is direct application. While it involves multiple steps, each is routine for C3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(\tan\alpha = 2\) | B1 | Ignoring subsequent work to find angle |
| Use identity \(\sec^2\beta = 1 + \tan^2\beta\) | B1 | |
| Attempt solution of quadratic equation for \(\tan\beta\) | M1 | 3-term quad eqn; using reasonable attempt at factorisation or quadratic formula (with no more than one slip) |
| Obtain \(\tan\beta = 5\) | A1 | Ignoring subsequent work; value 5 must be obtained legitimately |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute their values of \(\tan\alpha\) and \(\tan\beta\) in formula | M1 | Of form \(\frac{\pm\tan\alpha \pm \tan\beta}{\pm 1 \pm \tan\alpha\tan\beta}\) |
| Obtain \(\frac{2+5}{1-2\times5}\) | A1ft | Following their values from part (i) |
| Obtain \(-\frac{7}{9}\) | A1 | Or correct simplified exact equiv including \(\frac{7}{-9}\); A0 if \(\tan\beta=5\) obtained incorrectly in part (i) |
# Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $\tan\alpha = 2$ | B1 | Ignoring subsequent work to find angle |
| Use identity $\sec^2\beta = 1 + \tan^2\beta$ | B1 | |
| Attempt solution of quadratic equation for $\tan\beta$ | M1 | 3-term quad eqn; using reasonable attempt at factorisation or quadratic formula (with no more than one slip) |
| Obtain $\tan\beta = 5$ | A1 | Ignoring subsequent work; value 5 must be obtained legitimately |
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# Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute their values of $\tan\alpha$ and $\tan\beta$ in formula | M1 | Of form $\frac{\pm\tan\alpha \pm \tan\beta}{\pm 1 \pm \tan\alpha\tan\beta}$ |
| Obtain $\frac{2+5}{1-2\times5}$ | A1ft | Following their values from part (i) |
| Obtain $-\frac{7}{9}$ | A1 | Or correct simplified exact equiv including $\frac{7}{-9}$; A0 if $\tan\beta=5$ obtained incorrectly in part (i) |
---4 The acute angles $\alpha$ and $\beta$ are such that
$$2 \cot \alpha = 1 \text { and } 24 + \sec ^ { 2 } \beta = 10 \tan \beta \text {. }$$
(i) State the value of $\tan \alpha$ and determine the value of $\tan \beta$.\\
(ii) Hence find the exact value of $\tan ( \alpha + \beta )$.
\hfill \mbox{\textit{OCR C3 2012 Q4 [7]}}