IB Maths AA HL Topic 3: Geometry and trigonometry questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

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Question 1

Paper 2, calculatorMedium6 marks

Two survey points, $A$ and $B$ lie on a straight east–west path, with $B$ due east of $A$ The points are $1.20$ km apart. A radio beacon $C$ is north of the path. The bearing of $C$ from $A$ is $035^\circ$ and the bearing of $C$ from $B$ is $310^\circ$ Bearings are measured clockwise from north.

(a)

Construct a labelled diagram showing the positions of $A$ $B$ and $C$ the distance $AB$ north, and the given bearings.

[2]

(b)

Determine $AC$ correct to four significant figures.

[3]

(c)

Hence, using your value of $AC$ determine $BC$ correct to the nearest $0.001$ km.

[1]

Question 2

Paper 2, calculatorMedium12 marks

A survey team locates the ground position $T$ of a radio transmitter using two observation points $A$ and $B$ which lie on level horizontal ground. The distance $AB$ is $3.20\text{ km}$ The bearing of $B$ from $A$ is $090^\circ$ the bearing of $T$ from $A$ is $055^\circ$ and the bearing of $T$ from $B$ is $330^\circ$ Bearings are measured clockwise from north.

The antenna point $P$ is vertically above $T$ The angle of elevation of $P$ from $A$ is $4.0^\circ$

(a)

Draw a labelled plan-view diagram to represent the positions of $A$ $B$ and $T$ Include a north direction and the given bearings.

[2]

(b)

Determine the three interior angles of triangle $ABT$

[3]

(c)

Hence determine $AT$ and $BT$ giving your answers in km correct to 3 significant figures.

[4]

(d)

Hence determine the height of $P$ above the horizontal ground plane through $A$ and $B$ in metres to the nearest metre.

[3]

Question 3

Paper 1, no calculatorMedium6 marks

A survey team uses two points, $A$ and $B$ on level ground to locate the base $P$ of a vertical signal mast. Bearings are measured clockwise from north. Figure 1 is not to scale.

(a)

Use the plan view in Figure 1 to determine $AP$ Give your answer in exact form.

[4]

(b)

Hence, or otherwise, determine the height $PT$ of the mast. Give your answer in exact form.

[2]

Question 4

Paper 1, no calculatorMedium6 marks

In triangle $ABC$ the sides $a$ $b$ and $c$ are opposite the angles $A$ $B$ and $C$ respectively. Angles are measured in radians.

It is given that $A=\frac{\pi}{6}$ $a=6$ and $b=6\sqrt{3}$

(a)

Determine all possible values of $B$

[2]

(b)

Hence determine the corresponding values of $C$

[2]

(c)

For the obtuse value of $B$ determine the exact value of $\tan B$

[2]

Question 5

Paper 2, calculatorMedium6 marks

Figure 1 shows vectors $\mathbf u$ and $\mathbf v$ each drawn from the origin $O$

(a)

Find $\mathbf u\cdot\mathbf v$ and hence determine the angle between $\mathbf u$ and $\mathbf v$ in degrees correct to three significant figures.

[4]

(b)

The vector $\mathbf w=\mathbf u+k\mathbf v$ where $k\in\mathbb R$ Determine the value of $k$ for which $\mathbf w$ is perpendicular to $\mathbf u$

[2]

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