Practice Paper 2

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

Horizontal distance ( m)	0	20	60
Vertical distance (m)	0	16	24

This information was used to create Model A, where  is a function of .

Model A:    where

  At a horizontal distancem, Model A can be represented by the equation   

Write down a second equation to represent Model, when the horizontal distance is m.
   

Find the values of and .

Find the coordinates of the vertex on the graph and describe what this point represents in the context of the question.

Sketch the graph of for and , clearly showing the vertex.

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

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

Model : 

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

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

Calculate the percentage error in the estimate in part .

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  m from ground level.

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

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.

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

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.

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:

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  goodness of fit test. The test is carried out at a 5 % significance level.

Write down the null hypothesis for this test.

Copy and complete the following table in your answer booklet.

Write down the number of degrees of freedom.

Find the value for the test.

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

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.

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  % 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, .

State which option Natasha should choose. Justify your answer.

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.

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()	7.5	9.5	6.5	8	6.5	7	5.5	9	4	7
UK judge()	7.5	9	7	8.5	6	7.5	5	8	5	7.5

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

Using the value of , interpret the relationship between the US judge's scores and the UK judge's scores.

Write down the equation of the regression line  on.

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()		1				5.5				5.5
UK judge()		1					9.5		9.5

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

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