OCR A Level Maths A Pure Mathematics and Statistics June 2022 question paper H240/02

In this question you must show detailed reasoning.

Solve the following equations.

The points A and B have position vectors 3 i + 2 j and 4 i + 2 j - 5 k respectively.

Find the length of AB.

Point P has position vector p i - 3 k, where p is a constant. P lies on the circumference of a circle of which AB is a diameter.

Find the two possible values of p.

Amaya and Ben integrated (1 + x)², with respect to x, using different methods, as follows.

Charlie said that, because these answers are different, at least one of them must be wrong.

Explain whether you agree with Charlie’s statement.

You are given that a is a constant greater than 1.

In this question you must show detailed reasoning.

An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is 5% less than the height of the step immediately below it.

In the painting the total height of the steps is 205 cm, correct to the nearest centimetre.

Determine the number of steps in the design.

In this question you must show detailed reasoning.

A curve has equation y = x³ - 3x² + 4x.

Show that the curve has no stationary points.

Show that the curve has exactly one point of inflection.

It is given that any integer can be expressed in the form 3 m + r, where m is an integer and r is 0, 1 or 2.

Use this fact to answer the following.

By considering the different values of r, prove that the square of any integer cannot be

expressed in the form 3 n + 2, where n is an integer.

Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different.

By considering the different values of r, determine the probability that the sum of these three

integers is divisible by 3.

The diagram shows a water tank which is shaped as an inverted cone with semi-vertical angle 30° and height 50 cm. Initially the tank is full, and the depth of the water is 50 cm.

Water flows out of a small hole at the bottom of the tank. The rate at which the water flows out is

Determine the time taken for the tank to become empty.

[For a cone with base radius r and height h the volume V is given by r²h.]

The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.

Frequency

One of the 150 plants is chosen at random, and its height, X cm, is noted.

Show that P(20 < X < 30) = 0.147, correct to 3 significant figures.

Sam suggests that the distribution of X can be well modelled by the distribution N(40, 100).

Give a brief justification for the use of the normal distribution in this context.

Give a brief justification for the choice of the parameter values 40 and 100.

Use Sam’s model to find P (20 < X < 30).

Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, m and s, for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N(m, s²) as her model.

Use Nina’s model to find P (20 < X < 30).

Complete the table in the Printed Answer Booklet to show the probabilities obtained

from Sam’s model and Nina’s model.

By considering the different ranges of values of X given in the table, discuss how well

the two models fit the original distribution.

The table shows the age structure of usual residents of 18 Local Authorities (LAs) in the North West region of the UK in 2011.

Without reference to any other columns, explain how you would use only the columns for the age ranges 0 to 17 and 18 to 24 to decide whether an LA might be one of the following.

An LA that includes a university

An LA that attracts young couples to live

An LA that attracts retired people to live

Using your answers to part (a), identify the following.

Four LAs that might include a university

Three LAs that might be attractive to retired people

Explain why your answer to part (b)(ii), based only on the columns for the age ranges 0 to 17

and 18 to 24, may not be reliable.

The lower quartile, median and upper quartile of the percentages in the column “Age 65 and over” are 14.56%, 15.99% and 16.76% respectively.

Use this information to comment on your answers to part (b)(ii) and part (c).

In a magazine article, a councillor plans to describe a typical LA in the North West region. He wants to quote the average percentage of residents aged 65 or over.

The mean of the percentages in the column “Age 65 and over” is 16.90%.

Use this information, and the information given in part (d), to explain whether the median or

the mean better represents the data in the column “Age 65 and over”.

In the past the masses of new-born babies in a certain country were normally distributed with mean 3300 g. Last year a publicity campaign was held to encourage pregnant women to improve their diet.

Following this campaign, it is required to test whether the mean mass of new-born babies has increased. A random sample of 200 new-born babies is chosen, and it is found that their mean mass is 3360 g. It is given that the standard deviation of the masses of new-born babies is 450 g.

Carry out the test at the 2.5% significance level.

A firm claims that no more than 2% of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than 2%. In order to test the firm’s claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the 5% significance level, using the null hypothesis p = 0.02.

Given that the researcher’s null hypothesis is correct, determine the probability that the

researcher will conclude that the firm’s claim is incorrect.

The researcher finds that 18 out of the 600 packets are underweight. A colleague says

“18 out of 600 is 3%, so there is evidence that the actual proportion of underweight bags is greater than 2%.”

Criticise this statement.

There are 25 students in a class.

• The number of students who study both History and English is 3.

• The number of students who study neither History nor English is 14.

• The number of students who study History but not English is three times the number who study English but not History.

• Show this information on a Venn diagram.

• Determine the probability that a student selected at random studies English.

Two different students from the class are chosen at random.

Given that exactly one of the two students studies English, determine the probability that

exactly one of the two students studies History.