 A-level Sample Test
Paper 3
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The graph of y = x^2 − 9 is shown below. Find the area of the shaded region.

–18
–6
6
18
[1 mark] ? marks

A wooden frame is to be made to support some garden decking. The frame is to be in the shape of a sector of a circle. The sector OAB is shown in the diagram, with a wooden plank AC added to the frame for strength. OA makes an angle of θ with OB. (a)

The exact value of sinθ is given by \frac{4\sqrt{a}}{b}
Find the integer values of a and b

a =

b = [3 marks] ? (b)

Write down the value of θ in radians to 3 significant figures.

[1 mark] ? (c)

Find the area of the garden that will be covered by the decking.

m2 (to 1 d.p.) [2 marks] ? marks

A circular ornamental garden pond, of radius 2 metres, has weed starting to grow and cover its surface.

As the weed grows, it covers an area of A square metres. A simple model assumes that the weed grows so that the rate of increase of its area is proportional to A.

(a)

The area covered by the weed can be modelled by one of the following, where B and k are constants and t is time in days since the weed was first noticed.

A = Bk^t
A = Be^{kt}
A = Bt^{kt}
A = Bt^k
[4 marks] ? (b)

When it was first noticed, the weed covered an area of 0.25 m2. Twenty days later the weed covered an area of 0.5 m2.

(i)

State the value of B.

[1 mark] ? (ii)

The model for the area covered by the weed can be written as

A=2^{\frac{t}{m}-n}

Find the integer values of m and n.

m =

n = [4 marks] ? (iii)

How many days does it take for the weed to cover half of the surface of the pond?

days (to 2 d.p.) [4 marks] ? marks

\int_1^2\ x^3\ln\left(2x\right)dx can be written in the form pln2 + q, where p and q are rational numbers.

Find p and q.

p =

q = [5 marks] ? marks
(a)

The first three terms, in ascending powers of x, in the binomial expansion of (1 + 6x)^{\frac{1}{3}} can be written in the form of a+bx+cx^2

Find a, b and c

a =

b =

c = [3 marks] ? (b)

Use the result from part (a) to obtain an approximation to \sqrt{1.18} giving your answer to 4 decimal places.

[2 marks] ? marks

Find the value of \int_1^2\frac{6x+1}{6x^2-7x+2}dx , expressing your answer in the form mln 2 + nln 3, where m and n are integers.

m =

n = [8 marks] ? marks

The diagram shows part of the graph of y = e^{−x^2} The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.

(a)

Find the values of x for which the graph is concave.

Give answers in surd form: -\frac{\sqrt{a}}{b} < x < \frac{\sqrt{a}}{b} where a, and b are integers

a =

b = [4 marks] ? (b)

The finite region bounded by the x-axis and the lines x = 0.1 and x = 0.5 is shaded. Use the trapezium rule, with 4 strips, to find an estimate for \int_{0.1}^{0.5} e^{−x^2} dx

Give your estimate to four decimal places.

[3 marks] ? marks

Edna wishes to investigate the energy intake from eating out at restaurants for the households in her village.

She wants a sample of 100 households.

She has a list of all 2065 households in the village.

Ralph suggests this selection method.

“Number the households 0000 to 2064. Obtain 100 different four-digit random numbers between 0000 and 2064 and select the corresponding households for inclusion in the investigation.”

(a)

What is the population for this investigation?

Edna and Ralph
The 2065
households
in the village
The energy
intake for the
village from
eating out
The 100
households
selected
[1 mark] ? (b)

What is the sampling method suggested by Ralph?

Opportunity
Random
number
Continuous
random variable
Simple
random
[1 mark] ? marks

A survey has found that, of the 2400 households in Growmore, 1680 eat home-grown fruit and vegetables.

(a)

Using the binomial distribution, find the probability that, out of a random sample of 25 households in Growmore, exactly 22 eat home-grown fruit and vegetables.

P(X = 22) = [2 marks] ? (b)

Circle two reasons why using the Binomial distribution for your calculation in part (a) may not be valid for the 25 households in Gifford Terrace, a residential road in Growmore?

a fixed number of trials.
each trial is independent of the others.
there are only two outcomes.
the probability of each outcome remains constant from trial to trial.
[2 marks] ? marks

Shona calculated four correlation coefficients using data from the Large Data Set.

In each case she calculated the correlation coefficient between the masses of the cars and the CO2 emissions for varying sample sizes.

A summary of these calculations, labelled A to D, are listed in the table below. Shona would like to use calculation A to test whether there is evidence of positive correlation between mass and CO2 emissions.

She finds the critical value for a one-tailed test at the 5% level for a sample of size 3827 is 0.027

(a)
(i)

State which option Shona should use for her hypothesis test

H_0: \rho = 0 \text{ and } H_1:\rho < 0
H_0: \rho ≠ 0 \text{ and } H_1:\rho = 0
H_0: \rho > 0 \text{ and } H_1:\rho = 0
H_0: \rho = 0 \text{ and } H_1:\rho > 0
[1 mark] ? (ii)

Determine if there is sufficient evidence to reject the null hypothesis.

Values too close
Yes
No
[1 mark] ? (b)

Shona’s teacher tells her to remove calculation D from the table as it is incorrect.

Give the option that showed why the teachers knew it was incorrect

The correlation coefficient for D was:

an outlier
negative
less than −1
more than −1
[1 mark] ? marks

Terence owns a local shop. His shop has three checkouts, at least one of which is always staffed.

A regular customer observed that the probability distribution for N, the number of checkouts that are staffed at any given time during the spring, is

 P(N = n) = ⎧⎪⎨⎪⎩ \frac{3}{4}\left(\frac{1}{4}\right)^{n-1} for n = 1, 2 k for n = 3
(a)

Find the value of k.

k = [1 mark] ? (b)

Find the probability that a customer, visiting Terence’s shop during the spring, will find at least 2 checkouts staffed.

[2 marks] ? marks

During the 2006 Christmas holiday, John, a maths teacher, realised that he had fallen ill during 65% of the Christmas holidays since he had started teaching.

In January 2007, he increased his weekly exercise to try to improve his health.

For the next 7 years, he only fell ill during 2 Christmas holidays.

(a)

What is the Binomial distribution that should be used

X\sim B\left(a,b\right)

Gives values for a and b.

a =

b = [2 marks] ? (b)

What is the P(≤2)

[1 mark] ? (c)

Which of these options is valid:

There is sufficient evidence to suggest John’s rate of illness has decreased.
There is insufficient evidence to suggest John’s rate of illness has decreased.
[2 marks] ?  marks

In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, £X , per person on food and drink was recorded.

The maximum amount recorded was £40.48 and the minimum amount recorded was £22.00

The results are summarised below, where \overline{x} denotes the sample mean.

\sum_{\ }^{\ } x = 3046.14 \sum_{\ }^{\ } (x−\overline{x})^2=1746.29

(a)
(i)

Find the mean of X

Find the standard deviation of X

Mean =

Standard deviation = [2 marks] ? (ii)

Find the probability that a household in the South West spends less than £25.00 on food and drink per person per week.

P(X < 25) = (to 4 d.p.) [1 mark] ? (b)

For households in the North West of England, the weekly expenditure, £Y, per person on food and drink can be modelled by a normal distribution with mean £29.55

It is known that P(Y < 30) = 0.55

Find the standard deviation of Y, giving your answer to one decimal place.

[3 marks] ? marks

A survey during 2013 investigated mean expenditure on bread and on alcohol.

The 2013 survey obtained information from 12 144 adults.

The survey revealed that the mean expenditure per adult per week on bread was 127p.

For 2012, it is known that the expenditure per adult per week on bread had mean 123p, and a standard deviation of 70p.

(a)
Carry out a hypothesis test, at the 5% significance level, to investigate whether the mean expenditure per adult per week on bread changed from 2012 to 2013. Assume that the survey data is a random sample taken from a normal distribution. State which option should be used for her hypothesis test.
H_0:\ \mu ≠ 123 \text{ and } H_1:\ \mu = 123
H_0:\ \mu = 123 \text{ and } H_1:\ \mu ≠ 123
H_0:\ \mu ≠ 127 \text{ and } H_1:\ \mu = 127
H_0:\ \mu = 127 \text{ and } H_1:\ \mu ≠ 127
[1 mark] ? (b)

What is the value of the test statistic?

(to 1 d.p.) [2 marks] ? (c)

What are the critical values

and (to 2 d.p.) [2 marks] ? (d)

Which of these options is valid:

There is sufficient evidence to suggest the mean expenditure has changed.
There is insufficient evidence to suggest the mean expenditure has changed.
[1 mark] ? (e)

Calculate the greatest and least values for the sample mean expenditure on bread per adult per week for 2013 that would have resulted in acceptance of the null hypothesis for the test you carried out in part (a).

Greatest =

Least = [2 marks] ? marks

A sample of 200 households was obtained from a small town.

Each household was asked to complete a questionnaire about their purchases of takeaway food.

A is the event that a household regularly purchases Indian takeaway food.

B is the event that a household regularly purchases Chinese takeaway food.

It was observed that P(B|A) = 0.25 and P(A|B) = 0.1

Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food.

A household is selected at random from those in the sample.

Find the probability that the household regularly purchases both Indian and Chinese takeaway food.   