Practice Paper 3 (Pure & Statistics) (AQA A Level Maths: Pure)

Given that

find the value of .

Circle your answer.

Determine which one of these graphs represents a function whose inverse function exists.

Tick () one box.

Use the factor theorem to show that is a factor of .

Factorise  completely.

Write down all the real roots of the equation .

In an effort to prevent extinction scientists released some rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

 is the number of birds after  years of being released into the reserve.

Write down the number of birds the scientists released into the nature reserve.

According to this model, how many birds will be in the reserve after 3 years?

How long will it take for the population of birds within the reserve to reach 500?

Write   in the form , giving the value of  to three significant figures.

Write   in the form , giving the value of  to three significant figures.
State, and justify, whether this would represent exponential growth or decay.

The diagram below shows the graph of  The marked point  lies on the graph, and the graph meets the origin at the marked point .

In separate diagrams, sketch the curves with equation

On each diagram, give the coordinates of the images of points  and  under the given transformation.

On the graph of  the image of one of the two marked points has a coordinate of 4. Find the value of .

Show that, for all values of ,

In the expansion of , the coefficient of the  term is 96.
Given that , find the value of .

Given that  find the value of .

A geometric sequence has first term 900 and a common ratio  where .  The 18th term of the sequence is 18.

Show that  satisfies the equation

Hence or otherwise find the value of correct to 3 significant figures.

A geometric series has first term 9, and the sum of the first three terms of the series is 19.  The common ratio of the series is .

Show that .

Find the two possible values of .

Given that the series converges, find the sum to infinity of the series.

Show that

sin sin  sin sin³ 

The probabilities of events  and  are related, as shown in the Venn diagram below.

Find .

Circle your answer.

The table below shows the heights of sunflowers at the start of each week, for 9 weeks.

Calculate the standard deviation of these heights, correct to 1 decimal place.

Circle your answer.

The cumulative frequency diagram below shows completion times for 100 competitors at the 2019 Rubik’s cube championships.  The quickest completion time was 9.8 seconds and the slowest time was 52.4 seconds.

The grid below shows a box plot of the 2020 championship data.  Draw a box plot on the grid to represent the 2019 championship data.

240 students are surveyed regarding their belief in supernatural creatures.  144 say they believe in unicorns  .  75 say they believe in vampires .  Of those who believe in vampires, 27 also believe in unicorns.

Draw a two-way table to show this information.

One student is chosen at random.  Find:

Adrenaline is a new rollercoaster at a theme park. It is known that the time a customer spends in the queue follows a normal distribution with a variance of 52 minutes².  The mean time spent in a queue for other rollercoasters is 41 minutes.  The manager of the theme park wants to use a hypothesis test to investigate whether the mean time in the queue for Adrenaline is different to the mean time for the other rollercoasters.  She takes a sample of 10 customers over a period of several days and records their times spent in the queue for Adrenaline.

Find the critical region for the test at the 10% level of significance. State your hypotheses clearly.

The queuing times for the 10 people in the sample are:

38        49        40        39        49

39        59        32        55        41

State the conclusion of the test in context.

A company wants to survey 15% of its staff to find out whether employees would like to continue working from home after the Covid-19 pandemic. The company's 580 members of staff are grouped by job as follows: 295 engineers, 11 managers, 154 office staff and 120 apprentices.

Suggest a suitable sampling method and explain how the company can use this method to obtain its sample.

A teacher, Ms Pearman, claims that there is a positive correlation between the number of hours spent studying for a test and the percentage scored on it.

Write down suitable null and alternative hypotheses to test Ms Pearman's claim.

Ms Pearman takes a random sample of 25 students and gives them a week to prepare for a test. She records the percentage they score in the test,  %, and the amount of revision they did,  hours.

Ms Pearson calculates the product moment correlation coefficient for these data as .

Given that the p-value for the test statistic  is , test at the 5% level of significance whether Ms Pearman's claim is justified.

A machine is used to fill bags of potatoes for a supermarket chain.  The weight, kg, of potatoes in the bags is normally distributed with mean 3 kg and standard deviation kg. 

Given that 7% of the bags contain a weight of potatoes that is at least 50 g more than  the mean, find:

Twelve of the bags of potatoes are chosen at random.

Find the probability that not more than one of the bags will contain less than 2.96 kg of potatoes.

In the town of Wooster, Ohio, it is known that 90% of the residents prefer the locally produced Woostershire brand sauce when preparing a Caesar salad.  The other 10% of residents prefer another well-known brand.

 30 residents are chosen at random by a pollster.  Let the random variable  represent the number of those 30 residents that prefer Woostershire brand sauce.

 Suggest a suitable distribution for  and comment on any necessary assumptions.

Find the probability that

The pollster knows that there is a greater than 97% chance of at least  of the 30 residents preferring Woostershire brand sauce, where  is the largest possible value that makes that statement true.

Find the value of .

Vauxhall state that 8.5% of their cars produce less than 100 g/km of carbon dioxide emissions.

Elaine would like to investigate this claim using the large data set by testing the following hypotheses:

There are 1069 Vauxhall cars in the large data set. The critical regions are stated to be  and .

Calculate the actual level of significance based on these critical regions.

73 Vauxhall cars out 1069 in the large data set have less than 100 g/km of carbon dioxide emissions.

State a conclusion to the hypothesis test for this value, giving a reason for your answer.

With reference to the large data set, state one limitation of your conclusion.