Practice Paper 2 (DP IB Maths: AI HL)

The cross-sectional profile of a hill is modelled by the function

where  is the altitude above mean sea level, in metres, and  is the horizontal distance, in metres, from a fixed point .

The cross-sectional profile of the hill can be seen in the diagram below.

Point  has coordinates (0, 17.8) correct to 3 significant figures, and point  has exact coordinates (70, 12).

Calculate the altitude at 

A point  is at an altitude of 40 m.

Find the possible values of its horizontal distance from .

Find .

Hence calculate the maximum altitude of the hill.

When  the altitude of the hill is , when, the altitude of the hill is , and when the altitude of the hill is . These points are shown on the diagram as  and  respectively, and the altitudes in each case are given correct to 3 significant figures.

Use the trapezoidal rule with four intervals to estimate the cross-sectional area of the hill

It is believed that the time in minutes, , that a customer spends on hold during a call when calling a customer service line can be modelled by a normal distribution, with 

Using the model find, correct to four decimal places, the probability that during a call a customer chosen at random spends

500 customer service calls are monitored and the length of time that a customer is put on hold for during each call is measured.

By again using the model find, correct to one decimal place, the expected number of the 500 calls in which a customer is put on hold for between

For the 500 monitored calls, the measured lengths of time that a customer was put on hold for during each call are summarised in the following table.

It is decided to perform a  goodness of fit test at the 10% level of significance to decide whether the length of time that a customer is put on hold for during a call can indeed be modelled by a normal distribution, with .

State the null and alternative hypotheses.

Find the -value for the test.

State the conclusion of the test. Give a reason for your answer.

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  and . The cross-section is symmetrical about the -axis as shown in the diagram.  All of the units are in centimetres.

He models the section from  to  as a straight line.

Paul models the section of the bowl that passes through the points  and with a quadratic curve. 

Paul thinks that a quadratic with a minimum at  and passing through the point  is a better option. 

Find the equation of the new model.

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

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

In the town of Petersham, all the residents belong to either one or the other of the town’s two curling clubs – the Slippery Sliders (S) or the Wild Sweepers (W).  Competition between the clubs is fierce, and members tend to be quite loyal.  Still, each year 2% of the members of S switch to W and 3% of the members of W switch to S.  Any other losses or gains of members by the two curling clubs may be ignored.

Write down a transition matrix  representing the movement of members between the two clubs in a particular year.

Diagonalise the matrix by writing it in the form  for appropriate matrices  and .

Initially there are 574 members of  and 621 members of W.

Show that the numbers of members of S and W after n years will be  and , respectively.

Hence write down the number of members that each of the curling clubs can expect to have in the long term.

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

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

The boat leaves its mooring point at time  seconds and 5 minutes later is at the point with coordinates . 

(i)

the values of  and , 

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

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

Find OP.

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

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

where  is the size of the bird population, and  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.

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

Where  and  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.

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

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