| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.3 Part (i) is a straightforward 'show that' requiring the double angle formula sin(2x)=2sin(x)cos(x) and converting to reciprocal functions—pure algebraic manipulation with no problem-solving. Part (ii) requires using the identity sec²θ=1+tan²θ to convert to a quadratic in sec(θ), then solving—a standard multi-step C3 exercise but routine once the substitution is recognized. Slightly above average due to the reciprocal function manipulation and multiple solution angles, but still a textbook-style question. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j 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 formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cosec 2x = \frac{1}{\sin 2x}\) | M1 | Uses the identity \(\cosec 2x = \frac{1}{\sin 2x}\) |
| \(= \frac{1}{2\sin x \cos x}\) | M1 | Uses correct identity \(\sin 2x = 2\sin x \cos x\); accept \(\sin 2x = \sin x \cos x + \cos x \sin x\) |
| \(= \frac{1}{2}\cosec x \sec x \Rightarrow \lambda = \frac{1}{2}\) | A1 | \(\lambda = \frac{1}{2}\) following correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3\sec^2\theta + 3\sec\theta = 2\tan^2\theta \Rightarrow 3\sec^2\theta + 3\sec\theta = 2(\sec^2\theta - 1)\) | M1 | Replaces \(\tan^2\theta\) by \(\pm\sec^2\theta \pm 1\) to produce equation in just \(\sec\theta\) |
| \(\sec^2\theta + 3\sec\theta + 2 = 0\) | ||
| \((\sec\theta + 2)(\sec\theta + 1) = 0\) | M1 | Forms 3TQ=0 in \(\sec\theta\) and applies correct method for factorising, or uses formula, or completes the square |
| \(\sec\theta = -2, -1\) | A1 | Correct answers \(\sec\theta = -2, -1\) or \(\cos\theta = -\frac{1}{2}, -1\) |
| \(\cos\theta = -0.5, -1\) | M1 | Uses identity \(\sec\theta = \frac{1}{\cos\theta}\) to find at least one value of \(\theta\) |
| \(\theta = \frac{2\pi}{3}, \frac{4\pi}{3}, \pi\) | A1, A1 | Two correct values (A1); all three correct in terms of \(\pi\) (A1). Accept \(0.6\dot{\pi}, 1.3\dot{\pi}, \pi\). Withhold final mark if further values in range given |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3 \times \frac{1}{\cos^2\theta} + 3 \times \frac{1}{\cos\theta} = 2 \times \frac{\sin^2\theta}{\cos^2\theta}\) leading to \(3 + 3\cos\theta = 2\sin^2\theta\) | M1 | Replaces \(\sec^2\theta\), \(\sec\theta\), \(\tan^2\theta\) with cosine equivalents, multiplies through by \(\cos^2\theta\), and replaces \(\sin^2\theta\) with \(\pm 1 \pm \cos^2\theta\) |
| \(2\cos^2\theta + 3\cos\theta + 1 = 0\) | ||
| \((2\cos\theta + 1)(\cos\theta + 1) = 0 \Rightarrow \cos\theta = -0.5, -1\) | M1, A1 | Forms 3TQ=0 in \(\cos\theta\) and correct method; correct values |
| \(\theta = \frac{2\pi}{3}, \frac{4\pi}{3}, \pi\) | M1, A1, A1 | At least one \(\theta\) value (M1); two correct values (A1); all three correct (A1) |
# Question 6:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cosec 2x = \frac{1}{\sin 2x}$ | M1 | Uses the identity $\cosec 2x = \frac{1}{\sin 2x}$ |
| $= \frac{1}{2\sin x \cos x}$ | M1 | Uses correct identity $\sin 2x = 2\sin x \cos x$; accept $\sin 2x = \sin x \cos x + \cos x \sin x$ |
| $= \frac{1}{2}\cosec x \sec x \Rightarrow \lambda = \frac{1}{2}$ | A1 | $\lambda = \frac{1}{2}$ following correct working |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3\sec^2\theta + 3\sec\theta = 2\tan^2\theta \Rightarrow 3\sec^2\theta + 3\sec\theta = 2(\sec^2\theta - 1)$ | M1 | Replaces $\tan^2\theta$ by $\pm\sec^2\theta \pm 1$ to produce equation in just $\sec\theta$ |
| $\sec^2\theta + 3\sec\theta + 2 = 0$ | | |
| $(\sec\theta + 2)(\sec\theta + 1) = 0$ | M1 | Forms 3TQ=0 in $\sec\theta$ and applies correct method for factorising, or uses formula, or completes the square |
| $\sec\theta = -2, -1$ | A1 | Correct answers $\sec\theta = -2, -1$ or $\cos\theta = -\frac{1}{2}, -1$ |
| $\cos\theta = -0.5, -1$ | M1 | Uses identity $\sec\theta = \frac{1}{\cos\theta}$ to find at least one value of $\theta$ |
| $\theta = \frac{2\pi}{3}, \frac{4\pi}{3}, \pi$ | A1, A1 | Two correct values (A1); all three correct in terms of $\pi$ (A1). Accept $0.6\dot{\pi}, 1.3\dot{\pi}, \pi$. Withhold final mark if further values in range given |
## Part (ii) ALT:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3 \times \frac{1}{\cos^2\theta} + 3 \times \frac{1}{\cos\theta} = 2 \times \frac{\sin^2\theta}{\cos^2\theta}$ leading to $3 + 3\cos\theta = 2\sin^2\theta$ | M1 | Replaces $\sec^2\theta$, $\sec\theta$, $\tan^2\theta$ with cosine equivalents, multiplies through by $\cos^2\theta$, and replaces $\sin^2\theta$ with $\pm 1 \pm \cos^2\theta$ |
| $2\cos^2\theta + 3\cos\theta + 1 = 0$ | | |
| $(2\cos\theta + 1)(\cos\theta + 1) = 0 \Rightarrow \cos\theta = -0.5, -1$ | M1, A1 | Forms 3TQ=0 in $\cos\theta$ and correct method; correct values |
| $\theta = \frac{2\pi}{3}, \frac{4\pi}{3}, \pi$ | M1, A1, A1 | At least one $\theta$ value (M1); two correct values (A1); all three correct (A1) |
---\begin{enumerate}
\item (i) Use an appropriate double angle formula to show that
\end{enumerate}
$$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$
and state the value of the constant $\lambda$.\\
(ii) Solve, for $0 \leqslant \theta < 2 \pi$, the equation
$$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$
You must show all your working. Give your answers in terms of $\pi$.\\
\hfill \mbox{\textit{Edexcel C3 2013 Q6 [9]}}