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

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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 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 4

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 5

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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