Practice Paper 2 (Pure & Statistics) (OCR A Level Maths: Pure)

Simplify fully     

Given that  is small such that and higher powers of  can be ignored show that

Find the percentage error between your calculator answer and the approximation in part (a) when , giving your answer to one decimal place.

For which values of  is the approximation in part (a) valid?

Show that    satisfies the equation  .

Hence solve the equation cos³  cos² cos   for .

Fully factorise .

Prove that, if is odd, is odd and that if is even,  is even.

Show that

The point A lies on the circle with equation .  A has position vector , where k is a constant.

Find the value of k , and hence determine the coordinates of A.

Explain why a line passing through O and A must be a tangent to the circle.

A large weather balloon is being inflated.

The rate of change of its volume, , where V m³ is the volume of the balloon t minutes after inflation commenced, is inversely proportional to its volume.

Show that the general solution to the differential equation found in part (a) is

where c is a constant.

The table below shows the population figures in Northumberland from the 2011 census results. The age of a person, in years, is treated as a continuous variable.

Age (years)	Frequency
0 – 17	7 631
18 – 29	39 403
30 – 44	56 156
45 – 59	71 135
60 – 74	58 899
75+	26 251

A person was 29 years and 10 months old when the census was taken, which class would this person belong to?

The total amount of time cleaners spent dealing with unplanned incidents in a supermarket was recorded each day.  Data collected over 49 days is summarised in the table below.

On the grid below, draw a histogram to represent this data.

Estimate how often cleaners spent longer than 3 hours dealing with incidents.

The existing treatment for a disease is known to be effective in 73% of cases.  Dr Sabir develops a new treatment which she claims is more effective than the existing one.  To test her claim she uses the new treatment on a sample of 60 patients with the disease and uses a binomial distribution to model the number of them who are cured.

Explain two assumptions that Dr Sabir has made when using a binomial distribution to model the number of patients cured by the vaccine.

Dr Sabir notes that her treatment was effective for 51 out of the 60 patients used in the sample.

Test, at the 1% level of significance, the validity of Dr Sabir’s claim that her treatment is more effective than the existing one. State your hypotheses clearly.

State the conclusion you would have reached if a 5% level of significance had been used for this test.

The number of goals scored by the 24 teams that played in the first 44 games of the UEFA Euro cup 2020 can be summarised in the table below.

Goals scored	0-1	2-3	4-5	6-7	8-9	10-11
Frequency	3	5	5	6	4	1

Estimate the mean number of goals scored by each team.

Find the standard deviation of the number of goals scored by each team.

The test scores, , of a group of RAF recruits in an aptitude test are modelled as a normal distribution with

Find the values of  and  such that  and 

Those who score in the top 30% on the test move on to the next stage of training.

One of the recruits, Amelia, achieves a score of 231. Determine whether Amelia will move on to the next stage of training.

The IQ of a student at Calculus High can be modelled as a random variable with the distribution . The headteacher decides to play classical music during lunchtimes and suspects that this has caused a change in the average IQ of the students.

Write suitable null and alternative hypotheses to test the headteacher’s suspicion.

The headteacher selects 10 students and asks them to complete an IQ test.  Their scores are:

127, 127, 129, 130, 130, 132, 132, 132, 133, 138

Test, at the 5% level of significance, whether there is evidence to support the headteacher’s suspicion.

It was later discovered that the 10 students used in the sample were all in the same advanced classes.

Comment on the validity of the conclusion of the test based on this information.

A bag contains 15 blue tokens and 27 yellow tokens.  A token is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second token is then taken from the bag and its colour is recorded.

 Draw a tree diagram to represent this information.

Find the probability that:

A teacher took 19 students on an international trip. The incomplete box plot below shows part of the summary of the weights, in kg, of the luggage brought by each student. Each student's luggage weighed a different value.

The median weight is 4 kg more than the lower quartile. The range of weights is three times the interquartile range of weights.

Use the information above to complete the box plot.

Calculate the proportion of weights which were less than the 20 kg.

A discrete random variable  has the probability distribution shown in the following table:

	-1	1	2

Find the value of .

X  is sampled twice such that the results of the two experiments are independent of each other, and the outcomes of the two experiments are recorded.  A new random variable,Y, is defined as the sum of the two outcomes.

Complete the following probability distribution table for Y:

	-2	0	1	2	3	4