Extended Questions (Paper 2, HL) (DP IB Maths: AI HL)

Exam Questions

1 hour6 questions
1a
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2 marks

Paul finds an unusually shaped bowl when excavating his garden. It appears to be made out of bronze, and Paul decides to model the shape in order to work out its volume.  

By uploading a photograph of the object onto some graphing software, Paul identifies that the cross-section of the bowl goes through the points open parentheses negative 4 comma 0 close parentheses comma space open parentheses negative 6 comma 6 close parentheses comma open parentheses negative 5 comma 4 close parentheses comma open parentheses negative 3 comma 1.5 close parentheses and open parentheses 0 comma 1 close parentheses. The cross-section is symmetrical about the y-axis as shown in the diagram.  All of the units are in centimetres.

mi-q4a-pp2-set-b-ai-hl-maths-dig

He models the section from open parentheses negative 4 comma 0 close parentheses to open parentheses negative 6 comma 6 close parentheses as a straight line.

Find the equation of the line passing through these two points.

1b
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1 mark

Paul models the section of the bowl that passes through the points open parentheses negative 6 comma 6 close parentheses comma open parentheses negative 5 comma 4 close parentheses comma open parentheses negative 3 comma 1.5 close parentheses and open parentheses 0 comma 1 close parentheseswith a quadratic curve. 

(i)
Find the equation of the least squares quadratic curve for these four points.

(ii)

By considering the gradient of this curve when x equals negative 0.5,  explain why it may not be a good model.

1c
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1 mark

Paul thinks that a quadratic with a minimum at open parentheses 0 comma 1 close parentheses and passing through the point open parentheses negative 6 comma 6 close parentheses is a better option. 

Find the equation of the new model.

1d
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1 mark

Believing this to be a better model for the bowl, Paul finds the volume of revolution about the y-axis to estimate the volume of the bowl. 

Re-arrange the answers to parts (a) and (c) to make xa function of y.

1e
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1 mark
(i)
Write down an expression for Paul’s estimate of the volume as the difference of two integrals.

(ii)
Hence find the value of Paul’s estimate.

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2a
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4 marks

A boat is moving such that its position vector when viewed from above at time t  seconds can be modelled by

 r equals open parentheses table row cell 10 minus a space sin open parentheses πt over 600 close parentheses end cell row cell b open parentheses 1 minus cos open parentheses πt over 600 close parentheses close parentheses end cell end table close parentheses 

with respect to a rectangular coordinate system from a point O, where the non-zero constants a  and b can be determined. All distances are given in metres. 

The boat leaves its mooring point at time t equals 0 seconds and 5 minutes later is at the point with coordinates open parentheses negative 20 comma space 40 close parentheses. 

Find

(i)
the values of a and b, 

(ii)
the displacement of the boat from its mooring point.
2b
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2 marks

Find the velocity vector of the boat at time t seconds.

2c
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6 marks

After setting off, the boat reaches a point P where it is moving parallel to the x-axis.

Find OP.

2d
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3 marks

Find the time that the boat returns to its mooring point and the acceleration of the boat at this moment.

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3a
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5 marks

On a particular island, a particular species of bird was initially recorded as having a population of 80 at the start of a programme of observations.  Over time, the scientists conducting the programme determined that the growth rate of the bird population could be modelled by the following differential equation

fraction numerator d x over denominator d t end fraction equals 7 over 5 x 

where x is the size of the bird population, and t is the length of time in years since the start of the programme. 

Find the population of the bird species two years after the start of the programme.

3b
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6 marks

When the population of the bird species reaches 2000, a new reptile species is introduced to the island in order to control the bird population. Initially 280 reptiles are introduced to the island. Based on their research the scientists believe that the interaction between the two species after the introduction of the reptiles can be modelled by the system of coupled differential equations

 fraction numerator d x over denominator d t end fraction equals open parentheses 3 minus 0.012 y close parentheses x

fraction numerator d x over denominator d t end fraction equals open parentheses 0.0007 x minus 1 close parentheses y 

Where x and y represent the size of the bird and reptile populations respectively.

Using the Euler method with a step size of 0.5, find an estimate for

(i)
the bird population 2 years after the reptiles were introduced 

(ii)
the reptile population 2 years after the reptiles were introduced. 
3c
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1 mark

Explain how the approximation in part (b) could be improved.

3d
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3 marks

Show that the origin is an equilibrium point for the system, and determine the coordinates of the other equilibrium point.

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4a
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1 mark

In a game, enemies appear independently and randomly at an average rate of 2.5 enemies every minute. 

Find the probability that exactly 3 enemies will appear during one particular minute.

4b
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2 marks

Find the probability that exactly 10 enemies will appear in a five-minute period.

4c
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2 marks

Find the probability that at least 3 enemies will appear in a 90-second period.

4d
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2 marks

The probability that at least one enemy appears in k seconds is 0.999. Find the value of k  correct to 3 significant figures.

4e
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4 marks

A 10-minute interval is divided into ten 1-minute periods (first minute, second minute, third minute, etc.). Find the probability that there will be exactly two of those 1-minute periods in which no enemies appear.

4f
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4 marks

On the next level of the game, there is a boss enemy and a number of additional henchmen to fight against. 

The number of times that the boss enemy appears in a one-minute period can be modelled by a Poisson distribution with a mean of 1.1. 

The number of times that an individual henchman appears in a one-minute period can be modelled by a Poisson distribution with a mean of 0.6. 

It may be assumed that the boss enemy and the henchmen each appear randomly and independently of one another. 

Each time that the boss enemy or any particular henchman appears, it is counted as one ‘enemy appearance’. 

Determine the least number of henchmen required in order that the probability of 40 or more ‘enemy appearances’ occurring in a 3-minute period is greater than 0.38. You may assume that neither the boss enemy nor any of the henchmen are able to be totally eliminated from the game during this 3-minute period.

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5a
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1 mark

James throws a throws ball to his friend Mia. The height, h, in metres, of the ball above the ground is modelled by the function

h open parentheses t close parentheses equals negative 1.05 t squared plus 3.84 t plus 1.97 comma space space space space space space space space space t greater or equal than 0

where t is the time, in seconds, from the moment that James releases the ball.

Write down the height of the ball when James releases it.

5b
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2 marks

After 4 seconds the ball is at a height of metres above the ground.

Find the value of q.

5c
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2 marks

Find h apostrophe open parentheses t close parentheses

5d
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3 marks

Find the maximum height reached by the ball and write down the corresponding time t.

5e
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4 marks

James then drives a remote-controlled car in a straight horizontal line from a starting position right in front of his feet.  The velocity of the remote-controlled car in ms to the power of negative 1 end exponent is given by the equation

 v open parentheses t close parentheses equals 5 over 4 t squared minus 19 over 2 t squared plus 18 t minus 2 

Find an expression for the horizontal displacement of the remote-controlled car from its starting position at time t seconds.

5f
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3 marks

Find the total horizontal distance that the remote-controlled car has travelled in the first 5 seconds.

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6a
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6 marks

Consider the following system of differential equations:

                   fraction numerator straight d x over denominator straight d t end fraction equals x plus 2 y 

                  fraction numerator straight d y over denominator straight d t end fraction equals negative 3 x minus 4 y

Find the eigenvalues and corresponding eigenvectors of the matrix  open parentheses table row 1 2 row cell negative 3 end cell cell negative 4 end cell end table close parentheses.

6b
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2 marks

Hence write down the general solution of the system.

6c
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3 marks

When  t equals 0, x equals 2 and y equals 4.

Use the given initial condition to determine the exact solution of the system.

6d
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3 marks
(i)
Find the value of fraction numerator d y over denominator d x end fraction when t equals 0.

(ii)
Find the values of x comma y and fraction numerator d y over denominator d x end fractionwhen t equals ln 9 over 7.
6e
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3 marks

Hence sketch the solution trajectory of the system for t greater or equal than 0.

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