1. 1
   Consider the equation $2\sin x = \sqrt{3}$.

   (a)

   Find all the solutions of the equation for $0 \le x \le 2\pi$.

   [1 mark]

   • A$\frac{\pi}{3}$ and $\frac{2\pi}{3}$
   • B$\frac{\pi}{3}$ only
   • C$\frac{\pi}{3}$ and $\frac{4\pi}{3}$
   • D$\frac{\pi}{6}$ and $\frac{5\pi}{6}$

   (b)

   Find the number of solutions of $2\sin 2x = \sqrt{3}$ for $0 \le x \le 2\pi$.

   [1 mark]

   • A2
   • B8
   • C4
   • D1

   (c)

   Find the largest solution of $2\sin 2x = \sqrt{3}$ for $0 \le x \le 2\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x) = \sqrt{2}\cos x + 1$, where $x$ is measured in degrees.

   (a)

   Which of the following is a solution of $f(x) = 0$ for $-180^\circ \le x \le 180^\circ$?

   [1 mark]

   • A$45^\circ$
   • B$135^\circ$
   • C$225^\circ$
   • D$-45^\circ$

   (b)

   Find the number of solutions of $f(x) = 0$ for $-360^\circ \le x \le 360^\circ$.

   [1 mark]

   • A1
   • B2
   • C3
   • D4

   (c)

   Solve $f(2x) = 0$ for $0^\circ \le x \le 180^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $2\cos^2 x = 3\sin x$, for $0 \le x \le 2\pi$.

   (a)

   Show that the equation can be written as $(2\sin x - 1)(\sin x + 2) = 0$.

   [3 marks]

   (b)

   Hence solve the equation for $0 \le x \le 2\pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of hours of daylight, $D$, in a city on day $t$ of a year is modelled by $D(t) = 12 + 3\sin\left(\frac{2\pi(t - 80)}{365}\right)$, for $1 \le t \le 365$, where $t = 1$ is 1 January. A calculator may be used in this question.

   (a)

   Find the values of $t$ for which $D(t) = 13.5$.

   [6 marks]

   (b)

   A café opens its outdoor terrace only on days with more than 13.5 hours of daylight.
   (i) Find the number of days in the year on which the terrace is open.
   (ii) Find the minimum number of hours of daylight, and the value of $t$ on which it occurs.

   [6 marks]

   Total for question 4: 12 marks

End of questions