Practice Paper 2 (DP IB Maths: AI SL)

Practice Paper Questions

1a
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3 marks

Sharon set up an experiment to investigate the relationship between the mass of a mouse and the time the mouse takes to complete a mini assault course. She conducted the experiment with six mice and recorded her results in the table below.

Mouse mass, x (g) 19.9 18.3 21.1 19.8 17.5 16.3
Time, y (seconds) 18.0 17.7 20.9 18.6 15.0 14.2

i)
Calculate Pearson's product-moment correlation coefficient, r.

ii)
Describe the relationship between the mass of the mice and the time taken to complete the mini assault course.
1b
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2 marks

Write down the equation of the regression line of y on x, in the form space y space equals space m x space plus space c.

1c
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2 marks

Find the coordinates of the point straight M open parentheses x with italic bar on top italic comma y with italic bar on top close parentheses.

1d
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2 marks

Show that the point straight M open parentheses x with italic bar on top italic comma y with italic bar on top close parentheses  lies on the line of regression.

1e
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4 marks

The mass of a seventh mouse is found to be 20.6 g.

i)
Using your line of regression, estimate the time that the seventh mouse will take to complete the mini assault course.

ii)
Justify whether it is valid to use the line of regression to estimate the result for the seventh mouse.
1f
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2 marks

In the actual experiment, it was found that the seventh mouse took 20.7 seconds to complete the mini assault course.

Calculate the percentage error in the estimated value.

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2a
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2 marks

A farm is shown in the diagram below. A motorway runs in a straight line along the edge of the farm from point B to point straight C, and the farmhouse is located at point A. AB and AC form the other two sides of the farm, and the distances from the farmhouse to points straight B and straight C are 222 m and 184 m respectively. Angle straight C straight A with hat on top straight B is 115°, and points A, B and straight C lie in a horizontal plane.

q2a-practice-paper2-setb-ib-dp-ai-sl

Calculate the distance along the motorway from B to straight C.

2b
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2 marks

The cost of fencing in US dollars (USD) is $89.99 per metre.

Calculate the total cost of fencing the whole perimeter of the farm. Give your answer to 2 decimal places. 

2c
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2 marks

Calculate the area of the farm.

2d
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2 marks

Find the sizes of angles straight A straight B with hat on top straight C and straight A straight C with hat on top straight B.

2e
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3 marks

Calculate the shortest distance from the farmhouse to the motorway.

2f
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2 marks

A vertical signpost is located at point straight C, and the top of the signpost is designated as point D. The angle of elevation to the top of the signpost from point B is measured to be 1.4°.

Calculate the distance CD, the vertical height of the signpost.

2g
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2 marks

Calculate the distance between the top of the signpost, D, and point A.

2h
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3 marks

The rate of growth of the grass on the farm, G, in inches per month, can be modelled by the function

G left parenthesis T right parenthesis space equals space minus 0.015 left parenthesis T space long dash space 40 right parenthesis left parenthesis T space long dash space 80 right parenthesis

where T is the temperature in degrees Fahrenheit. 

Find the maximum rate of grass growth on the farm and the temperature required.

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3a
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1 mark

Anna decides she wants to buy a farm and the bank agree to give her a loan provided she makes a 13.9% deposit of $40 000.

Calculate the value of the farm.

3b
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2 marks

She currently has $15 000 saved up and decides to invest it in some high risk high growth shares forecasted to grow at 65% annually. 

Calculate the forecasted number of years it will take for her to be able to afford the deposit.

3c
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3 marks

1.5 years later, the shares outperform their forecasted growth rate and Anna is able to afford the deposit on the farm.

Calculate the percentage error between the forecasted annual growth rate and the actual annual growth rate of the shares.

3d
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1 mark

Anna now takes out the loan from the bank.

Write down the amount of the loan.

3e
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5 marks

The loan is for 25 years, compounded monthly, with equal monthly payments of $1200.

For this loan, find

i)
the amount of interest paid by Anna,

ii)
the annual interest rate of the loan.

3f
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3 marks

After 15 years of paying off this loan, Anna decides to pay the remainder in one final payment.

Find the amount of Anna's final payment.

3g
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3 marks

Find how much money Anna saved by making one final payment after 20 years.

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4a
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3 marks

It is believed that the time in minutes, T, that a customer spends on hold during a call when calling a customer service line can be modelled by a normal distribution, with T tilde N open parentheses 17 comma 2.8 squared close parentheses.

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

i)
less than 15 minutes on hold

ii)
more than 23 minutes on hold.
4b
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3 marks

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

i)
15 and 20 minutes

ii)
20 and 23 minutes.
4c
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2 marks

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.

Length of time on hold, bold italic T Number of calls
 Less than 15 minutes 98
 Between 15 and 20 minutes 309
 Between 20 and 23 minutes 79
 More than 23 minutes 14

It is decided to perform a calligraphic X squared 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 T tilde N left parenthesis 17 comma space 2.8 squared right parenthesis.

State the null and alternative hypotheses.

4d
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3 marks

Find the p-value for the test.

4e
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2 marks

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

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5a
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2 marks

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

h open parentheses x close parentheses space equals 1 over 6 open parentheses 17 x plus 107 minus x squared over 4 close parentheses comma space space space space space space space space space space space 0 less or equal than x less or equal than 70,

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

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

q5a-practice-paper2-setb-ib-dp-ai-sl

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

Calculate the altitude at x space equals space 3.

5b
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3 marks

A point P is at an altitude of 40 m.

Find the possible values of its horizontal distance from straight O.

5c
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2 marks

Find h apostrophe space left parenthesis x right parenthesis.

5d
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2 marks

Hence calculate the maximum altitude of the hill.

5e
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3 marks

When x space equals space 17.5, the altitude of the hill is 54.7 m, when x space equals space 35, the altitude of the hill is 66.0 m, and when space x space equals space 52.5 space the altitude of the hill is 51.7 m. These points are shown on the diagram as straight C, D and straight E 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

5f
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4 marks
i)
Write down the integral which can be used to find the cross-sectional area of the hill.

ii)
Hence find the cross-sectional area of the hill to the nearest square metre.

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