IB Maths AI 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 Paper 2

Difficulty

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Question 1

Paper 1Medium6 marks

A mapping team approximates a wetland by triangle $ABC$ The horizontal distance between survey stations $A$ and $B$ is $1.80\text{ km}$ The measured interior angles are $\angle BAC=52.6^\circ$ and $\angle ABC=68.3^\circ$

The wetland qualifies for a reserve if its mapped area is at least $1.20\text{ km}^2$

(a)

Find $\angle ACB$

[1]

(b)

Show that $AC=1.95\text{ km}$ correct to three significant figures.

[2]

(c)

Hence, using your answer to part (b), calculate the area of triangle $ABC$ correct to three significant figures, and state whether the wetland qualifies for a reserve.

[3]

Question 2

Paper 1Medium6 marks

A surveyor uses triangulation to estimate the area of a triangular wetland $ABC$ Points $A$ and $B$ lie on an accessible straight bank, and $AB=d$ metres. The measured interior angles are $\angle BAC=48^\circ$ and $\angle ABC=67^\circ$

(a)

Find $\angle ACB$

[1]

(b)

Show that the area $T$ in $\text{m}^2$ is approximately given by $T=0.377d^2$

[2]

(c)

The surveyor wants to map a region with an area of at least $6000\text{ m}^2$ Determine the required value of $d$ to 3 significant figures. Hence state whether a baseline of $125\text{ m}$ is sufficient.

[3]

Question 3

Paper 1Hard6 marks

A flat map uses Cartesian coordinates to locate radio beacons. The segment $AB$ lies on the line $3x-4y-10=0$ and its midpoint is $M(2,-1)$

(a)

Determine the equation of the perpendicular bisector of $AB$ giving your answer in the form $ax+by+c=0$

[2]

(b)

Beacons $C$ and $D$ have coordinates $C(-2,4)$ and $D(4,2)$ An emergency receiver $Q$ must be equidistant from $A$ and $B$ and also equidistant from $C$ and $D$

Using your answer to (a), determine the exact coordinates of $Q$

[4]

Question 4

Paper 1Medium6 marks

Two survey boats, A and B, move at constant velocities in a horizontal plane. Coordinates are measured in metres, with the $x$ -axis pointing east and the $y$ -axis pointing north. At $t=0$ their position vectors are

\mathbf r_A=(120,30), \qquad \mathbf r_B=(20,100),

and their velocity vectors are

\mathbf v_A=(3,1), \qquad \mathbf v_B=(5,-2),

in $\text{m s}^{-1}$ The time $t$ is measured in seconds.

(a)

Find an expression, in terms of $t$ for the relative position vector of B from A, $\overrightarrow{AB}$

[2]

(b)

Using your answer to part (a), determine the time $t\geq 0$ at which the boats are closest and their minimum separation. Give both answers to 3 significant figures.

[4]

Question 5

Paper 1Medium8 marks

A courier moves between three depots, $A$ $B$ and $C$ The directed routes leaving each depot are shown below.

Depot	Depots reachable in one leg
$A$	$B, C$
$B$	$A, C$
$C$	$A$

At each depot, the courier chooses uniformly from the routes leaving that depot. Use the convention that the entry in row $i$ column $j$ of a transition matrix is the probability of moving from depot $i$ to depot $j$

(a)

Write down the adjacency matrix $A$ for this network, using the order $A,B,C$ for both rows and columns.

[2]

(b)

Find the number of two-leg walks from $A$ back to $A$ using the appropriate entry of $A^2$

[2]

(c)

Construct the transition matrix $T$ for the courier’s movements, using the order $A,B,C$ for both rows and columns.

[2]

(d)

Using your transition matrix, find the probability that the courier is at $A$ after exactly two legs, starting from $A$

[2]

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