Practice Paper 2 (DP IB Maths: AI SL)

Practice Paper Questions

1a
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1 mark

A laboratory is testing the effectiveness of a new indicator for the presence of a particular mineral in water. This mineral occurs naturally in approximately 8% of water sources. 

The laboratory collected samples of water from several sources in the country. The country has been divided into different areas based on their geology and the number of samples taken from each area is proportional to the size of the area.

State the sampling method being used.

1b
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4 marks

It is intended that if the mineral is present in the water sample the test result will be "positive" and if the mineral is not present in the sample the test result will be "negative".

There is a certain amount of unreliability in the results of the test. 4% of samples that have the mineral present test negative. 89% of the samples that do not have the mineral present test negative.

The tree diagram shows some of this information.

q1-practice-paper2-setc-ib-dp-ai-sl

Write down the value of

i)
a

ii)
b

iii)
c

iv)
d.
1c
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8 marks

Use the tree diagram to find the probability that a water sample selected at random

i)
will not have the mineral present and will test negative,

ii)
will test positive,

iii)
will have the mineral given that it tested positive.
1d
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3 marks

The scientists from the laboratory collected additional information about each sample: if the sample was taken from an area where livestock was present (L), if the sample was within 100 miles of the laboratory (D) and if the sample was tested in the laboratory within 24 hours of being collected (T).

  • 39 samples were taken from an area with livestock;
  • 33 samples were taken within 100 miles of the laboratory;
  • 36 samples were tested in the laboratory within 24 hours of being collected;
  • 5 samples were taken from an area with livestock, within 100 miles of the laboratory and tested within 24 hours;
  • 2 samples were taken from an area with livestock and within 100 miles of the laboratory but were not tested within 24 hours;
  • 11 samples were taken within 100 miles of the laboratory and tested within 24 hours but were not taken from an area with livestock;
  • 12 samples were tested within 24 hours but were not from an area with livestock nor were they taken from an area within 100 miles of the laboratory;
  • 3 samples were taken from an area more than 100 miles from the laboratory without livestock and they were not tested within 24 hours of being collected.

Draw a Venn diagram to represent this information making sure that the diagram is labelled with all relevant information.

1e
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2 marks

Find the total number of water samples that were tested by the laboratory.

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2a
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4 marks

The diagram below shows a triangular field on a farm. AB space equals space 17 space straight mAC space equals space 45 space straight m and angle space straight B straight A with hat on top straight C space equals space 38 degree.

X is a point on AC, such that AX space colon space XC is 1 space colon space 4.

q2-practice-paper2-setc-ib-dp-ai-sl

The field is going to be used for livestock, so a fence is to be installed around its perimeter.

Calculate the total length of fencing required.

2b
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2 marks

The owner of the field had estimated the length of fence required to be 98 m.

Calculate the percentage error in her estimation.

2c
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4 marks

The field is to be divided into two parts by installing a new fence connecting B to straight X.

Calculate the area of BXC.

2d
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6 marks

The farmer decides that fieldspace ABX spaceis too small and wishes instead to divide the original field by adjusting the position of X such that angle straight A straight B with hat on top straight X space equals space 32 degree.

Determine how much less fencing is required for space BX spacegiven the new position of straight X.

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3a
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2 marks

The number of seats a row has at a comedy festival follows a regular pattern where the first row has u subscript 1 seats and the number of seats in each successive row increases by d seats. In the fourth row there are 25 seats and in the sixteenth row there are 49 seats.

Write down an equation, in terms of u subscript 1 and d, for the number of seats

i)
in the fourth row

ii)
in the sixteenth row.
3b
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2 marks

Find the value of u subscript 1 and the value of d.

3c
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3 marks

The festival has 18 rows of seats in total.

Calculate the total number of seats.

3d
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8 marks

The price for a seat in the first row is $22 and the price decreases by 5% each successive row.

i)
Find the row in which the price of a seat first falls below $10.

ii)
Find the total revenue the comedy festival generates if 22 tickets are sold for every row. Give your answer rounded to the nearest dollar.

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4a
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2 marks

The table below shows the distribution of the number of baskets scored by 150 netball players during a weekly game.

Number of baskets 0 1 2 3 4 5 6
Frequency 41 17 34 31 10 15 2

Calculate

i)
the mean number of baskets scored by a player

ii)
the standard deviation.
4b
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1 mark

Find the median number of baskets scored.

4c
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2 marks

Find the interquartile range.

4d
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2 marks

Determine if a player who scored 8 baskets would be considered an outlier.

4e
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4 marks

Two players are randomly chosen.

Given that the first player scored 2 or less baskets, find the probability that both players scored exactly 1 basket.

4f
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3 marks

The number of hours each player trains each week is normally distributed with a mean of 5 hours and standard deviation of 0.8 hours.

i)
Calculate the probability that a player trains less than 6 hours a week.

ii)
Calculate the probability that a player trains less than 4 hours a week.

iii)
Calculate the expected number of players that train between 4 and 6 hours a week.

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5a
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1 mark

Chun-hee is creating some packaging in the shape of a square based pyramid where the base has length x cm and the perpendicular height of the pyramid is h cm. Chun-hee wants to keep the distance from the apex of the pyramid to the midpoint of the base edge fixed at 7 cm.

Write down an equation for the volume, V, of the packaging in terms of x and h.

5b
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3 marks

Show that V can be expressed by 196 over 3 h space minus space 4 over 3 h cubed.

5c
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2 marks

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

5d
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2 marks

Find the value of h for which the volume of the pyramid is maximised.

5e
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2 marks

Find the value of x when the volume of the pyramid is maximised.

5f
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4 marks

Chun-hee decides to make the packaging using the dimensions required to maximise the volume. The material for the packaging costs 4 KRW / cm2

Calculate the number of units that Chun-hee can make given that she has 90, 000 KRW.

5g
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3 marks

Chun-hee takes out a 3 year loan for 90,000 KRW at a nominal annual interest rate of 2.3% compounded monthly. Repayments are made at the end of each month.

Find the value of the repayments that Chun-hee must make to pay off the loan.

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