1. 1
   A sector of a circle has centre $O$, radius $9$ cm, and angle $\theta=\dfrac{2\pi}{5}$ radians.

   (a)

   Find the exact arc length of the sector.

   [1 mark]

   • A$\dfrac{18\pi}{5}$ cm
   • B$\dfrac{9\pi}{5}$ cm
   • C$\dfrac{162\pi}{5}$ cm
   • D$18\pi$ cm

   (b)

   Find the exact area of the sector.

   [1 mark]

   • A$\dfrac{18\pi}{5}$ cm$^2$
   • B$\dfrac{81\pi}{5}$ cm$^2$
   • C$\dfrac{162\pi}{5}$ cm$^2$
   • D$\dfrac{9\pi}{5}$ cm$^2$

   (c)

   Hence find the exact perimeter of the sector.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   It is given that $\sin\theta = \dfrac{2}{5}$, where $\theta$ is obtuse.

   (a)

   Find the exact value of $\cos\theta$.

   [1 mark]

   • A$\dfrac{\sqrt{21}}{5}$
   • B$-\dfrac{21}{25}$
   • C$-\dfrac{\sqrt{21}}{5}$
   • D$-\dfrac{\sqrt{21}}{25}$

   (b)

   Find the exact value of $\tan\theta$.

   [1 mark]

   • A$\dfrac{2\sqrt{21}}{21}$
   • B$-\dfrac{5}{2}$
   • C$\dfrac{\sqrt{21}}{2}$
   • D$-\dfrac{2\sqrt{21}}{21}$

   (c)

   Hence find the exact value of $\sin2\theta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A vertical flagpole $OT$ stands at corner $O$ of a horizontal rectangular courtyard $OPQR$, where $OP=24$ m and $OR=10$ m, with $P$ and $R$ adjacent to $O$ and $Q$ diagonally opposite $O$.

   (a)

   The flagpole has height $OT=15$ m. Find the distance from the top of the flagpole, $T$, to the corner $Q$, giving your answer to 3 significant figures.

   [3 marks]

   (b)

   Find the angle of elevation of $T$ from $Q$, giving your answer to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Relative to an origin $O$, points $A$, $B$ and $C$ have position vectors $\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k}$, $\mathbf{b}=\mathbf{i}+4\mathbf{j}-\mathbf{k}$ and $\mathbf{c}=-2\mathbf{i}+\mathbf{j}+5\mathbf{k}$.

   (a)

   (i) Find the vector $\overrightarrow{AB}$. [2]
   (ii) Find $|\overrightarrow{AB}|$, giving your answer in exact form. [2]
   (iii) Find the vector $\overrightarrow{AC}$. [2]

   [6 marks]

   (b)

   (i) Find $\overrightarrow{AB}\cdot\overrightarrow{AC}$. [2]
   (ii) Hence find the angle $B\hat{A}C$, giving your answer to the nearest degree. [4]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $\theta = \dfrac{5\pi}{6}$.

   (a)

   Find the exact value of $\sin\theta$.

   [1 mark]

   • A$\dfrac12$
   • B$-\dfrac12$
   • C$\dfrac{\sqrt3}{2}$
   • D$-\dfrac{\sqrt3}{2}$

   (b)

   Find the exact value of $\cos\theta$.

   [1 mark]

   • A$\dfrac{\sqrt3}{2}$
   • B$\dfrac12$
   • C$-\dfrac12$
   • D$-\dfrac{\sqrt3}{2}$

   (c)

   Hence find the exact value of $\tan\theta$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The angle $\alpha$ satisfies $0<\alpha<\dfrac{\pi}{2}$ and $\sec\alpha=\dfrac{13}{5}$.

   (a)

   Find the exact value of $\cos\alpha$.

   [1 mark]

   • A$\dfrac{5}{13}$
   • B$\dfrac{13}{5}$
   • C$\dfrac{12}{13}$
   • D$\dfrac{5}{12}$

   (b)

   Find the exact value of $\text{cosec}\,\alpha$.

   [1 mark]

   • A$\dfrac{12}{13}$
   • B$\dfrac{13}{5}$
   • C$\dfrac{13}{12}$
   • D$\dfrac{5}{13}$

   (c)

   Hence find the exact value of $\cot\alpha$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   It is given that $\sin A=\dfrac35$ and $\cos B=\dfrac{12}{13}$, where $A$ and $B$ are both acute angles.

   (a)

   Find the exact value of $\sin(A+B)$.

   [3 marks]

   (b)

   Hence, using a symmetry property of the sine function, write down the exact value of $\sin(\pi-(A+B))$. Then find the exact value of $\cos(A+B)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The line $L_1$ has vector equation $\mathbf{r}=\begin{pmatrix}2\\-1\\3\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-1\end{pmatrix}$, and the line $L_2$ has vector equation $\mathbf{r}=\begin{pmatrix}4\\3\\0\end{pmatrix}+\mu\begin{pmatrix}-1\\1\\2\end{pmatrix}$, where $\lambda,\mu\in\mathbb{R}$.

   (a)

   (i) Show that $L_1$ and $L_2$ do not intersect. [4]
   (ii) Explain why $L_1$ and $L_2$ are skew lines. [2]

   [6 marks]

   (b)

   Find the angle between the direction vectors of $L_1$ and $L_2$, giving your answer to the nearest degree, and hence find the acute angle between the lines $L_1$ and $L_2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions