June 2023 Paper 1 Q10
10 The diagram shows the graph of \(y = 1.5 + \sin^2 x\) for \(0 \leqslant x \leqslant 2\pi\).
(a) Show that the equation of the graph can be written in the form \(y = a - b\cos 2x\) where \(a\) and \(b\) are constants to be determined. [2]
(b) Write down the period of the function \(1.5 + \sin^2 x\). [1]
(c) Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin^2 x\) with the graph of \(y = 1 + \cos 2x\) in the interval \(0 \leqslant x \leqslant 2\pi\). [3]
| Scheme | Marks | AO |
|---|---|---|
| Use \(\sin^2 x = \frac{1}{2}(1 - \cos 2x)\) So \(1.5 + \sin^2 x = 1.5 + \frac{1}{2}(1 - \cos 2x)\) | M1 | 2.1 |
| So \(y = 2 - 0.5\cos 2x\) | A1 | 2.1 |
| [2] |
Notes
M1: attempt to write \(\sin^2 x\) in terms of \(\cos 2x\)
A1: Allow for \(a = 2,\ b = 0.5\) or fully correct expression
| Scheme | Marks | AO |
|---|---|---|
| [period] \(\pi\) | B1 | 1.2 |
| [1] |
Notes
B1: Cao. Do not accept \(180^\circ\)
| Scheme | Marks | AO |
|---|---|---|
| intersect when \(2 - 0.5\cos 2x = 1 + \cos 2x\) \(\cos 2x = \frac{2}{3}\) | M1 | 3.1a |
| \(x = 0.421,\ 2.72,\ 3.56,\ 5.86\) radians (correct to 3sf) | A1 A1 | 1.1b 1.1b |
| [3] |
Notes
M1: Equate expressions in \(\cos 2x\) and attempt to rearrange
A1: At least 1 correct value
A1: Three other correct values and no others in the interval \(0 \leqslant x \leqslant 2\pi\)
FT their first root
Alternative method
| Scheme | Marks |
|---|---|
| \(1.5 + \sin^2 x = 2 - 2\sin^2 x\) Or \(1.5 + (1 - \cos^2 x) = 1 + 2\cos^2 x - 1\) \(3\sin^2 x = 0.5\) or \(3\cos^2 x = 2.5\) \(\sin x = \pm\sqrt{\frac{1}{6}}\ \left[= \pm\frac{\sqrt{6}}{6}\right]\) or \(\cos x = \pm\sqrt{\frac{5}{6}}\) | M1 |
| \(x = 0.421,\ 2.72,\ 3.56,\ 5.86\) radians (correct to 3sf) | A1 A1 |
M1: Uses correct trig identities to attempt to find a value for \(\sin^2 x\) or \(\cos^2 x\)
A1: At least 1 correct value
A1: Four correct values and no others in the interval \(0 \leqslant x \leqslant 2\pi\)
FT their first root