A2 June 2023 Paper 1 Q7
7 The function \(\mathrm{f}\) is defined by
\[\mathrm{f}(x) = \left|\sin x + \frac{1}{2}\right| \qquad (0 \leqslant x \leqslant 2\pi)\]Find the set of values of \(x\) for which
\[\mathrm{f}(x) \geqslant \frac{1}{2}\]Give your answer in set notation. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Sketches graph of \(y = \sin x + \tfrac{1}{2}\) PI by graph of \(y = \left|\sin x + \tfrac{1}{2}\right|\) or considers one equation or inequality without modulus sign eg \(\sin x + \tfrac{1}{2} = \tfrac{1}{2}\) or \(\left(\sin x + \tfrac{1}{2}\right)^2 = \tfrac{1}{4}\) | M1 | 3.1a |
| Obtains the set of values \(0 \leqslant x \leqslant \pi\) Condone \(0 \lt x \lt \pi\) | A1 | 2.2a |
| Obtains a graph of \(y = \left|\sin x + \tfrac{1}{2}\right|\) with the correct shape or Obtains the other equation or inequality without modulus sign eg \(\sin x + \tfrac{1}{2} = -\tfrac{1}{2}\) or Obtains two critical values from a quadratic in \(\sin x\) | M1 | 1.1a |
| Obtains \(3\pi/2\) | A1 | 1.1b |
| Obtains a completely correct answer, and expresses it using set notation. eg \([0, \pi] \cup \left\{\dfrac{3\pi}{2}, 2\pi\right\}\) \(\{x : 0 \leqslant x \leqslant \pi\} \cup \left\{x : x = \dfrac{3\pi}{2}\right\} \cup \{x : x = 2\pi\}\) Condone \(\left\{x : 0 \leqslant x \leqslant \pi, \dfrac{3\pi}{2}, 2\pi\right\}\) | A1 | 2.5 |
| (5 marks) |
Typical solution
