June 2024 Paper 1 Q15
15
(a) Show that the expression\[\sin 2\theta \operatorname{cosec}\theta + \cos 2\theta \sec\theta\]can be written as\[4\cos\theta - \sec\theta\]where \(\sin\theta \neq 0\) and \(\cos\theta \neq 0\) [4 marks]
(b) A student is attempting to solve the equation\[\sin 2\theta \operatorname{cosec}\theta + \cos 2\theta \sec\theta = 3 \quad \text{for } 0^\circ \leqslant \theta \leqslant 360^\circ\]They use the result from part (a), and write the following incorrect solution:\[\sin 2\theta \operatorname{cosec}\theta + \cos 2\theta \sec\theta = 3\]
| Step 1 | \(4\cos\theta - \sec\theta = 3\) |
| Step 2 | \(4\cos\theta - \dfrac{1}{\cos\theta} - 3 = 0\) |
| Step 3 | \(4\cos^2\theta - 3\cos\theta - 1 = 0\) |
| Step 4 | \(\cos\theta = 1\) or \(\cos\theta = -0.25\) |
| Step 5 | \(\theta = 0^\circ,\ 104.5^\circ,\ 255.5^\circ,\ 360^\circ\) |
(i) Explain why the student should reject one of their values for \(\cos\theta\) in Step 4. [1 mark]
(ii) State the correct solutions to the equation\[\sin 2\theta \operatorname{cosec}\theta + \cos 2\theta \sec\theta = 3 \quad \text{for } 0^\circ \leqslant \theta \leqslant 360^\circ\] [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| Recalls the identity for \(\sin 2\theta = 2\sin\theta\cos\theta\) or an identity for \(\cos 2\theta\) eg \(\cos 2\theta = 2\cos^2\theta - 1\) \(\cos 2\theta = 1 - 2\sin^2\theta\) \(\cos 2\theta = \cos^2\theta - \sin^2\theta\) This mark could be scored later if compound angle formula is used with a completely correct argument. | B1 | 1.2 |
| Substitutes \(A\sin\theta\cos\theta\) and a correct identity for \(\cos 2\theta\) OR Substitutes \(2\sin\theta\cos\theta\) and an identity for \(\cos 2\theta\) with sign errors condoned provided \(\cos 2\theta\) is not replaced with an expression equivalent to a constant. | M1 | 3.1a |
| Simplifies \(B\sin\theta\operatorname{cosec}\theta\) to \(B\) Or \(D\cos^2\theta\sec\theta\) to \(D\cos\theta\) | M1 | 1.1a |
| Completes reasoned argument to obtain \(4\cos\theta - \sec\theta\) | R1 | 2.1 |
| (4) |
Typical solution
\[\sin 2\theta\operatorname{cosec}\theta + \cos 2\theta\sec\theta\]\[= 2\sin\theta\cos\theta\operatorname{cosec}\theta + \left(2\cos^2\theta - 1\right)\sec\theta\]\[= 2\sin\theta\cos\theta\operatorname{cosec}\theta + 2\cos^2\theta\sec\theta - \sec\theta\]\[= 2\cos\theta + 2\cos\theta - \sec\theta\]\[= 4\cos\theta - \sec\theta\]| Scheme | Marks | AO |
|---|---|---|
| (i) Explains that \(\operatorname{cosec}\theta\) is undefined when \(\cos\theta = 1\) Or explains \(\cos\theta = 1\) would mean that \(\sin\theta = 0\) or uses \(\sin\theta \neq 0\) to show that \(\cos\theta = 1\) should be rejected | E1 | 2.4 |
| (1) | ||
| (ii) Obtains 104.5, 255.5 CAO | B1 | 2.2a |
| (1) | ||
| (6 marks) |
Typical solution
(i)
\(\cos\theta \neq 1\) as \(\operatorname{cosec}\theta\) is undefined
(ii)
\[\therefore \theta = 104.5^\circ,\ 255.5^\circ\]