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

Easy Medium Hard

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 2, calculatorEasy5 marks

A simplified model for the water depth at a harbour is

h(t)=2.4+1.6\sin\left(\frac{\pi}{6}(t-2)\right),

where $h$ is measured in metres and $t$ is the number of hours after midnight.

(a)

State the amplitude and the midline height of the model.

[2]

(b)

Determine the period of the model.

[1]

(c)

Calculate the water depth at $t=5$

[2]

Question 5

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 6

Paper 2, calculatorHard6 marks

The function $f$ is defined by

f(\theta)=a\sin\theta+b\cos\theta,\qquad 0\leq\theta\leq\pi,

where $a,b\in\mathbb R$ Figure 1 shows the graph of $y=f(\theta)$

(a)

Determine the values of $a$ and $b$

[3]

(b)

For $0<\theta<\pi$ $\theta\ne\frac{\pi}{2}$ define

H(\theta)=f(\theta)f(\pi-\theta)+\tan\theta+\tan(\pi-\theta).

Determine whether $H$ attains a maximum. If it does not, state its supremum.

[3]

Question 7

Paper 1, no calculatorHard6 marks

A circuit model represents the normalized voltage at phase $\phi$ by $v(\phi)=\sin\phi$ A controller operates at phase $\theta$ where $0<\theta<\frac{\pi}{2}$ During calibration, the normalized voltage at phase $2\theta$ is $\frac12$

(a)

Starting from the compound-angle identities, derive the double-angle identities for $\sin 2x$ $\cos 2x$ and $\tan 2x$

[3]

(b)

Determine all possible exact values of $\tan\theta$

[3]

Question 8

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]

Question 9

Paper 2, calculatorHard6 marks

For $x,y,z,k\in\mathbb{R}$ consider the planes

P_1:\ x-z=1,\qquad P_2:\ y+z=2,\qquad Q_k:\ x+ky+(1-k)z=3.

(a)

Determine a direction vector of the line of intersection of $P_1$ and $P_2$

[1]

(b)

The acute angle between this line and $Q_k$ is $30^\circ$ Determine all possible values of $k$ and, for each value, the common solution of the three plane equations.

[5]

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