AQA A Level Maths Paper 3 June 2025 question paper 7357/3

A student states that the product of two irrational numbers is always irrational.

One of the options is a counter example that shows the student’s statement is incorrect.

Identify the counter example.

Circle your answer.

The diagram shows the graph of y = arctan x

The graph has horizontal asymptotes at y = –a and y = a

State the value of a

Circle your answer.

The function f is defined by

Identify which one of the following statements describes the function f

Tick () one box.

Decreasing and concave

Decreasing and convex

Increasing and concave

Increasing and convex

State the value of

where a > 0

Find the value of

where a > 0

The diagram shows the graphs of y = 2 x and y = (x – 3)(x + 4).

The region R is defined by the inequalities

Shade the region R on the diagram.

The first four terms in ascending powers of x of the binomial expansion of

are given by

where p and q are constants.

Find the value of p and the value of q

Hence find an approximation for 1.94⁵

Give your answer to five decimal places.

Solve the equation

to find the value of a

The expression

can be written in the form

Find the value of A and the value of B

Use your answer to part (a) to show that

where q is a rational number to be found.

Describe a sequence of two transformations which maps the graph with equation

onto the graph with equation

State the equation of the vertical asymptote of the graph with equation

Express 2 sin x + 5 cos x in the form R sin (x + α)

where R > 0 and 0° ≤ α ≤ 90°

In 2010, the water temperature at a beach could be modelled by the formula

T = 14.2 – (2 sin d ° + 5 cos d °)

where: T is the temperature of the water in °C d is the number of days after 1 January 2010

Find the value of d when the temperature of the water reached its lowest value in 2010

Find the maximum temperature of the water predicted by the model in 2010

Give your answer to one decimal place.

Find the number of weeks in 2010 for which the temperature of the water was higher than 15 °C

Give your answer to the nearest whole number.

A block of ice is melting.

At time t minutes, the block is in the shape of a cuboid with dimensions of 4x, 2x and x, as shown in the diagram.

All measurements are in centimetres.

The volume, V cm³, of the block of ice decreases at a rate which is proportional to its surface area.

When x = 4 the volume of the block of ice is decreasing at a rate of 7 cm³ per minute.

Show that

Using the results from parts (a) and (b), find

Interpret, in context, your answer to part (c)(i).

The variables A and B have a product moment correlation coefficient of 0.47

Identify which one of the scatter diagrams shows the relationship between A and B.

Tick () one box.

2 14 It is given that X ~ N(9, 1.5) and P(a ≤ X ≤ b) ≈ 95%

Identify the correct pair of values of a and b.

Tick () one box.

a = 4.5 and b = 13.5

a = 6 and b = 12

a = 7 and b = 11

a = 7.5 and b = 10.5

The proportion of diners at a restaurant who are vegan is 20%

The number of diners at the restaurant who are vegan can be modelled by a binomial distribution.

In a random sample of 25 diners:

find the probability that no diner is a vegan.

find the probability that fewer than 6 diners are vegan.

find the probability that at least 9 diners are vegan.

State one assumption necessary for the binomial distribution to be valid in the context of this question.

The distance, in kilometres, travelled from home to work by 340 individuals is shown in the histogram.

Estimate the number of individuals who travel between 3 km and 5 km from home to work.

Fully justify your answer.

Give one reason why your answer to part (a)(i) is only an estimate.

The histogram shows that the data is skewed.

State the type of skewness.

Explain why frequency density is used in this histogram.

A maths teacher holds a revision session every Wednesday.

In a random sample of eight Wednesdays, the number of students who attended is listed below.

2 4 3 1 4 5 2 3

Find the mean of these eight values.

Find the variance of these eight values.

The teacher believes that the number of students who attended a revision session each Wednesday can be modelled by the binomial distribution B(30, 0.1).

Comment on whether the mean and variance found in parts (a) and (b) support the teacher’s belief.

Fully justify your answer.

The teacher wanted to decide if a Saturday morning revision session was worth doing.

The teacher asked the first 10 students who entered the classroom if they would attend a Saturday session.

Name this method of sampling.

Describe one advantage of this method of sampling.

The cholesterol level, X, of an adult can be modelled by a normal distribution with mean 5.7 mmol/l and standard deviation 1.2 mmol/l.

The lower quartile of X is a and the upper quartile is b

It is given that the P(X ≤ a) = 0.25 and P(X ≥ b) = 0.25

Find the interquartile range of X

A group of scientists believe that a new dietary supplement can reduce the mean cholesterol level.

A random sample of 160 adults were given the new dietary supplement for a trial period of one year.

After the trial, their cholesterol level was found to have a mean of 5.6 mmol/l.

Carry out a hypothesis test at the 5% significance level to investigate whether the mean cholesterol level has reduced after taking the dietary supplement.

It can be assumed that the cholesterol level after taking the dietary supplement follows a normal distribution with an unchanged variance.

Three events X, Y and Z are such that

P(X) = P(Y) = 0.38 P(X ∪ Z) = 0.5 P [ (X ∪ Y) ' ] = 0.35 X and Z are mutually exclusive.

Find P(Z)

Complete the Venn diagram.

X Y Z

0.27........

0.3

Find P[(X ∩ Y) ∪ (Y ∩ Z)]

Find P(Y' | Z')

Show that the events Y and Z are not independent of each other.

The oxides of nitrogen emissions, F g/km, of a car registered in 2000 can be modelled by a normal distribution with mean 0.41 and standard deviation 0.07

Find P(F < 0.39)

Find P(0.3 < F < 0.5)

Find P(F > 0.6)

Explain why P(F ≥ 0.6) = P(F > 0.6) in this model.

The oxides of nitrogen emissions, F g/km, of a car in 2010 can be modelled by a normal distribution with mean 0.36 and standard deviation 0.09

Compare the oxides of nitrogen emissions in 2000 with those in 2010.

A researcher collected data on the oxides of nitrogen emissions from the Large Data Set for cars registered in 2002 and 2016.

The researcher cleaned the data by removing the details for some cars.

Using your knowledge of the Large Data Set, give one reason why the researcher cleaned the data.

The researcher wished to make a comparison between the oxides of nitrogen emissions for a sample of Nissan cars in their local town from 2016 with the data from all cars registered in the same year in the Large Data Set.

Using your knowledge of the Large Data Set, give one reason why a meaningful comparison could not be made.