Practice Paper 1 (DP IB Maths: AI SL)

Practice Paper Questions

1a
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2 marks

Let Q space equals space fraction numerator 30 space sin space 2 a over denominator 8 b end fraction , wherespace a space equals space 45 degree spaceand b space equals space 2.

Calculate the exact value of Q.

1b
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2 marks

Give your answer from part (a) correct to

i)
two decimal places

ii)
two significant figures.
1c
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2 marks

Nina estimates the value of Q to be 2.

Calculate the percentage error in Nina's estimate.

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2
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4 marks

A pharmacy sells face masks in a variety of sizes. Their sales over a week are recorded in the table below:

  Kids Adults
Size  Small Large S M L XL
Frequency f  29 4 8 24 15 4

i)
Write down the mode for this data.

ii)
Explain why, in this case, the mode from part (i) would not be particularly helpful to the shop owner when reordering masks.

iii)
Given that the shop is open every day of the week, calculate the mean number of masks sold per day.

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3a
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2 marks

A coffee scoop is made out of 2 mm thick stainless steel. It is in the shape of a hemisphere and has an outer diameter of 3.8 cm.

Show that the outer surface area of the scoop is 361 over 50 straight pi space cm squared.

3b
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3 marks

Calculate the volume of coffee that the scoop can hold, in cm3.

3c
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4 marks

The density of the coffee when compacted in the scoop is 0.825 g/cm3. A single bag of ground coffee beans contains 350 g and costs $16.

Calculate the cost per scoop of ground coffee.

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4a
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1 mark

The total cost, C, in New Zealand dollars (NZD), of a premium gym membership at Cityfitness can be modelled by the function

C space equals space 16.99 t space plus space 49 comma space t greater or equal than 0

where t is the time in weeks.

Calculate the cost of the gym membership for a year. Give your answer correct to 2 decimal places.

4b
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2 marks

Find the number of weeks it takes for the total cost to exceed 2000 NZD.

4c
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1 mark

At Les Mills the initial payment is 20 NZD lower than Cityfitness, however the weekly cost is 8.51 NZD higher than Cityfitness

Write a cost function for a gym membership at Les Mills using an appropriate model.

4d
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3 marks

Calculate how many weeks it will take for the cost of a Les Mills gym membership be more than the cost of a Cityfitness gym membership.

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5a
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3 marks

A town is built in a rectangular area bounded by the lines y=-2, y=10, x = —4 and x = 10 as shown on the Voronoi diagram below.

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

Points A(—2, 0), B(4, 8) and C(8, 0) represent the locations of first aid responders in the town.

q5a-practice-paper1-setb-ib-dp-ai-sl

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

5b
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3 marks

A new station is due to open at D(8, 2) and as such the Voronoi diagram will need to be adjusted.

i)
Write down the equation of the perpendicular bisector of [BD]. Give your answer in the form y space equals space m x space plus space c.

ii)
Write down the equation of the perpendicular bisector of [CD]. Give your answer in the form y space equals space m x space plus space c.

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6a
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2 marks

An environmental organisation is trying to establish if altitude affects the growth of pine needles. A number of needles have been taken from trees at both high and low altitudes and their lengths, in inches, recorded. The results are shown in the table below.

Low altitude 6.1 8.2 7.7 8.0 11.9 6.9 7.5 7.1 8.1
High altitude 7.4 7.9 8.3 6.6 9.5 7.9 8.2 8.1 8.5

Perform a t-test to compare the mean lengths of the pine needles.

Write down the null and alternative hypotheses.

6b
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1 mark

State whether this is a one-tailed test or a two-tailed test.

6c
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2 marks

Perform a t-test at the 10% significance level. Write down the p-value.

6d
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2 marks

Write down the conclusion of the test. Give a reason for your answer.

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7a
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1 mark

The temperature, T, of a cake, in degrees Celsius, degree straight C, can be modelled by the function

T open parentheses t close parentheses space equals space a cross times 1.17 to the power of negative t over 4 end exponent plus 18 comma space space space space space space space space t greater or equal than 0 comma

where a is a constant and t 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 18 represents.

7b
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2 marks

The cake was 180°C when it was taken out of the oven.

Find the value of a.

7c
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2 marks

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

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8a
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2 marks

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.

8b
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3 marks

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.

8c
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3 marks

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.

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9a
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4 marks

The diagram below shows the triangular sail of a windsurfing board, ABC, with a horizontal boom PC. space AB space equals space 6.1 spacem and makes an angle of 18° to the vertical. space BC space equals space 4.7 spacem and space straight B straight C with hat on top straight P space equals space 70 degree.

q9a-practice-paper1-setb-ib-dp-ai-sl

Find the area of the whole sail.

9b
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2 marks

Calculate the length of the boom PC.

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10a
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2 marks

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

Score , bold italic x 1 2 3 4 5 6
bold P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 1 over 6 1 half p 1 over 8 3 over 2 p
1 over 12 3 p

Calculate the exact value of p.

10b
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3 marks

Frank plays the game once.

Calculate the expected score.

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11a
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3 marks

Consider the function g open parentheses x close parentheses equals square root of 4 minus x end root.

Sketch the graph of the function g left parenthesis x right parenthesis, labelling the x and y intercepts.

q11a-practice-paper1-setb-ib-dp-ai-sl

11b
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2 marks

Find

i)
g open parentheses negative 5 close parentheses

ii)
x when g open parentheses x close parentheses equals space 1 half.

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12a
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1 mark

The volume of a sphere of radius r is given by the formula V space equals space 4 over 3 pi r cubed.

Find fraction numerator d V over denominator d r end fraction.

12b
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2 marks

Find the rate of change of the volume with respect to the radius when r space equals space 5.

Give your answer in terms of straight pi.

12c
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3 marks

Show that fraction numerator d V over denominator d r end fraction  is an increasing function for all relevant values of r.

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13a
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4 marks

The following diagram shows an arch that is 4.5 m tall and 3 m wide. The arch crosses the x-axis at the origin, O, and at point straight P, and its vertex is at point straight V. The arch may be represented by a curve with an equation of the form y space equals space x left parenthesis a x space plus space 6 right parenthesis, where all units are measured in metres.

q13a-practice-paper1-setb-ib-dp-ai-sl

Find

i)
the coordinates of straight P

ii)
the coordinates of V

iii)
the value of a.
13b
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2 marks

Find the cross-sectional area under the arch.

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