Practice Paper 2 (DP IB Maths: AI SL)

Practice Paper Questions

1a
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2 marks

The Amazing Box Company manufactures rectangular gift boxes.

Their largest box size has length 120 cm, width 70 cm and height 50 cm as shown in the diagram.

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

Customers often ask for other measurements relating to the gift boxes.

Calculate the length GB.

1b
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2 marks

Calculate the surface area of the box in straight m squared.

1c
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3 marks

Each month, The Amazing Box Company expects to sell x hundred gift boxes.
It is known that fraction numerator d P over denominator d x end fraction equals negative 2 x plus 480 comma space x greater or equal than 0, where P is the monthly profit, in euros, from the sale of x hundred boxes.

Find the number of boxes that should be sold each month to maximize the profit.

1d
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5 marks

It is also known that the company makes a profit of €960 in a month where it sells 15000 boxes.

Find P open parentheses x close parentheses.

1e
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3 marks

Find the least number of boxes which must be sold each month in order to make a profit.

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2a
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4 marks

A catapult is designed to project a boulder from ground level. The distance travelled horizontally by the boulder, x m, and the height of the boulder above ground level, h spacem  were recorded. The results are summarized in the following table.

Horizontal distance (x m) 0 20 60
Vertical distance (h spacem) 0 16 24

This information was used to create Model A, where h is a function of x comma space x space greater or equal than 0.

Model A: h open parentheses x close parentheses space equals space p x plus q x squared   where p comma q space element of space straight integer numbers

  At a horizontal distancespace 20 spacem, Model A can be represented by the equationspace p space plus space 20 q space equals space 0.8. space   

i)
Write down a second equation to represent Modelspace straight A, when the horizontal distance isspace 60 space m.
   
ii)
Find the values of space p spaceand q.
2b
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2 marks

Find the coordinates of the vertex on the graphspace y space equals space h left parenthesis x right parenthesis space and describe what this point represents in the context of the question.

2c
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3 marks

Sketch the graph of y equals h open parentheses x close parentheses for 0 space less or equal than space x space less or equal than space 120 and negative 30 space less or equal than space y space less or equal than space 30, clearly showing the vertex.

2d
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1 mark

Hence identify why Model A may not be appropriate once the boulder has covered a horizontal distance of 100 m.

2e
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2 marks

Amongst other findings, it was found that the catapult was releasing boulders from a height 2 m above ground level. In light of these findings, a revised model, Model straight B comma for the position of the boulder was created.

Model Bh open parentheses x close parentheses equals 0.84 x minus 0.012 x squared plus 2

Use Model B to calculate an estimate for the height of the boulder at the point when it has covered  22m horizontally.

2f
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2 marks

The actual height of a boulder that has covered 22 m horizontally is 14.7 m.

Calculate the percentage error in the estimate in part open parentheses e close parentheses.

2g
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3 marks

The catapult is used to recreate a famous Roman battle. In order to cause as much damage to the castle as possible with a single shot, the boulder should hit the castle at a point no lower than 8 m from ground level.

q2g-practice-paper2-seta-ib-dp-ai-sl

Using Model B find the maximum distance from the castle that the catapult should be located.

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3a
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2 marks

In an online game there are various 'power-ups' that players can discover while progressing through the game. The power-ups come in two sizes, major and minor, and they all belong to one of two categories, either combat or survival.

It is known that 85 % of the power-ups in the game are minor power-ups while the other 15 % are major power-ups. Of the major power-ups it is known that 60 % are combat power-ups, while of the minor power-ups it is known that 30 % are combat power-ups.

Using the given information, copy and complete the following tree diagram.

q3a-practice-paper2-seta-ib-dp-ai-sl

3b
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4 marks

Power-ups that are discovered in the game occur at random.

i)
Find the probability that when a power-up is discovered it will be a survival power-up.

ii)
Given that a power-up is a combat power-up, find the probability that it is a major power-up.
3c
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1 mark

In the online game it is possible to purchase 'mystery bundles' containing random assortments of power-ups. According to the game's website, the power-ups in each mystery bundle should be distributed as follows.

   Type        Sword         Lasgun        Armour        Rations      Medical    Other Gear
Percentage(%)            15             10            15            30           10          20

Dylan and his teammates have joined together to purchase a 'mega mystery bundle' containing 60 power-ups. After going through the bundle, they find it contains the following frequencies of each type of power-up:

   Type        Sword         Lasgun        Armour        Rations      Medical    Other Gear
Observed frequency            4             2            5            21           9          19

Dylan and his friends are upset that they did not receive more weapons and armour. Some of them think that they were simply unlucky this time around, while others insist that the website's claims about the contents of the bundles are misleading.

Dylan is also studying Mathematics as part of his IB course, so he decides to investigate if
the sample is consistent with the website's claims by conducting a  straight chi squared goodness of fit test. The test is carried out at a 5 % significance level.

Write down the null hypothesis for this test.

3d
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2 marks

Copy and complete the following table in your answer booklet.

Type Sword Lasgun Armour Rations Medical Other Gear
Expected frequency            

3e
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1 mark

Write down the number of degrees of freedom.

3f
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2 marks

Find the p minusvalue for the test.

3g
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2 marks

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

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4a
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7 marks

In this question, give all answers to two decimal places.

Natasha needs to purchase new sound equipment for her band, the Oscillots. The sound equipment costs £12000, and as she does not have enough money to pay the full amount up front, she is considering two different methods of financing the purchase.

Option 1:

Natasha is eligible to receive a personal loan from her bank for the full amount of £12000. It will be a 5 year loan at a nominal annual interest rate of 3.9 % compounded monthly. Repayments are made each month.

i)
Find the repayment made each month.

ii)
Find the total amount paid for the sound equipment.

iii)
Find the interest paid on the loan.
4b
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5 marks

Option 2:

The music shop from which Natasha is buying the sound equipment offers its own financing scheme. This scheme offers her a 5 year loan at a nominal annual interest rate of r % compounded quarterly. The terms of the loan require a 10 % payment up front and quarterly repayments of £600.

(i)  Find the amount to be borrowed for this option.

(ii) Find the annual interest rate, r.

4c
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2 marks

State which option Natasha should choose. Justify your answer.

4d
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3 marks

The sound equipment Natasha is buying depreciates at an annual rate of 15 % per year.

Find the value of the sound equipment five years after it is purchased.

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5a
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4 marks

In a dressage competition judges award scores for each test attempted by competitors. For the "rhythm test" 10 competitors were awarded scores on a scale from 0 to 10 (10 being the highest) by a US and a UK judge. The scores are collated in the table below.

Competitor A B C D E F G H I J
US judge(bold italic x) 7.5 9.5 6.5 8 6.5 7 5.5 9 4 7
UK judge(bold italic y) 7.5 9 7 8.5 6 7.5 5 8 5 7.5

i)
Write down the value of the Pearson's product—moment correlation coefficient, r.

ii)
Using the value of r, interpret the relationship between the US judge's scores and the UK judge's scores.
5b
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2 marks

Write down the equation of the regression line y onspace x.

5c
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4 marks
i)
Use your regression equation from part (b) to estimate the US judge's score when the UK judge awards a score of 5. Give your answer to the nearest half mark.

ii)
Briefly explain whether or not the regression equation from part (b) predicts both judges awarding a score of 0 to the same competitor.
5d
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2 marks

Following complaints from competitors about the judges' scores, officials for the competition ran an investigation that, amongst other things, involved finding the Spearman's rank correlation coefficient.

Copy and complete the information in the following table.

Competitor A B C D E F G H I J
US judge(bold italic x)   1       5.5       5.5
UK judge(bold italic y)   1         9.5   9.5  
5e
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4 marks
i)
Briefly explain why there is no rank of 10 in the UK judge's rankings.

ii)
Find the value of the Spearman's rank correlation coefficient, r subscript s.
5f
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1 mark

Following their investigation, officials decreased the UK judge's score for competitor B from 9 to 8.5.

Explain, giving a reason, whether or not you would expect the value of the Spearman's rank correlation coefficient, r subscript s, to change.

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