Practice Paper 1 (DP IB Maths: AI HL)

A town is built in a rectangular area bounded by the lines and  as shown on the Voronoi diagram below.

Each horizontal and vertical unit on the grid below represents 1 km.

Points and  represent the locations of first aid responders in the town.

Calculate the area for which the first aid responder at  has responsibility.

A new station is due to open at  and as such the Voronoi diagram will need to be adjusted.

Write down the equation of the perpendicular bisector of . Give your answer in the form .

Write down the equation of the perpendicular bisector of . Give your answer in the form 

Ben and Sam are both cyclists competing in a 22.5 km race at the Herne Hill Velodrome in London, England. One lap of the velodrome is 450 m.

Ben takes a total of 42 minutes to complete the race.

Calculate Ben's mean lap time in seconds.

Given that each of Ben's laps took him 1% longer to complete than the previous one, calculate how long it took him (in seconds) to complete his first and last laps.

Sam completes the first lap in 45 seconds and takes 0.2 seconds longer per lap.

Determine who completed the race the first out of Ben and Sam. Justify your answer.

The temperature, , of a cake, in degrees Celsius, , can be modelled by the function 

where  is a constant and  is the time, in minutes, since the cake was taken out of the oven.

In the context of this model, state what the value of represents.

The cake was  when it was taken out of the oven.

Find the value of .

Find the temperature of the cake half an hour after being taken out of the oven.

is a parallelogram with vertices  and  

Find the vectors and  .

Find the area of the parallelogram.

By finding the scalar product of and , determine if the angle  is acute or obtuse.

The diagram below shows the triangular sail of a windsurfing board, , with a horizontal boom .  = 6.1 m and makes an angle of  to the vertical. = 4.7 m and = 70°.

Find the area of the whole sail.

Calculate the length of the boom .

The volume of a sphere of radius  is given by the formula 

Find .

Find the rate of change of the volume with respect to the radius when .

Give your answer in terms of .

Show that  is an increasing function for all relevant values of  .

Frank plays a game involving a biased six-sided die.
The faces of the die are numbered 1 to 6.
The score of the game, , is the number which lands face up after the die is rolled.
The following table shows the probability distribution for .

Score ,

Calculate the exact value of .

Frank plays the game once.

Calculate the expected score.

Consider the complex numbers  and .   

(i)

(ii)

(iii)

For part (iii) give your answer in the form ,  where  and  are real numbers.

Write down the complex conjugate of  and describe the geometrical relationship between  and .

Let  be the graph below.

Construct the transition matrix for a random walk around .

Determine the probability that a random walk of length 3 starting at will finish at .

The germination rate of a particular seed is 44%. George sows 25 of these seeds selected at random.

Calculate the expected number of seeds that germinate.

Find the probability that more than 7 of these seeds will germinate.

Find the probability that at least 9, but no more than 11 of the seeds germinate.

Scientists are studying a large pond where an invasive plant has been observed growing, and they have begun measuring the area,  m², of the pond's surface that is covered by the plant. According to the scientists' model, the rate of change of the area of the pond covered by the plant at any time, , is proportional to the square root of the area already covered. 

Write down a differential equation to represent the scientists' model.

Solve the differential equation to show that

where  is the constant of proportionality and  is a constant of integration.

At the time when the scientists begin studying the pond the invasive plant covers an area of 100 m². One week later the area has increased to 225 m².

Use this information to determine the values of  and .

A drone travels in a straight line and at a constant speed. It moves from an initial point  to a second point . The person controlling the drone is located at  .

The  and  directions are due east, due north and vertically upwards respectively with all distances in metres.

Write down an equation for the line along which the drone travels.

At some point  on its flight, the drone is vertically level with the person controlling the drone.

Find the coordinates of point .

Find the distance between  and the person controlling the drone.

The weight, , of pumpkins purchased are normally distributed with a mean of 11.3 kg and a standard deviation of 2.1 kg.

In one year, a farmer grows 350 pumpkins on his farm.

Predict the number of pumpkins that weigh between 7.2 kg and 12.5 kg from the farm.

The heaviest 7% of pumpkins are classified as large and are sold at a higher price.

Find the range of weights of pumpkins that can be sold for a higher price.

The diagram below shows the slope field for the differential equation

The graphs of the two solutions to the differential equation that pass through the points  and  are shown.

Explain the relationship that must exist between  and  for =0 to be true.

For the two solutions given, the local minimum points lie on the straight line and the local maximum points lie on the straight line . 

Find the equations of  (i) and  (ii) ,  giving your answers in the form .

Consider the graph G shown below.

Write down the adjacency matrix for the graph G.

Find the total number of walks of length 7 that start at vertex A and finish at vertex D.

Consider the functions  and g defined by  and g , where each function has the largest possible valid domain. 

The graph of can be transformed onto the graph of g by a single translation and a single stretch, both of which are parallel to one of the coordinate axes. 

Describe the sequence of transformations in the case where:

The graph of  can be also transformed onto the graph of g by a single translation using the vector .

Find the exact values of and .