The graph of y = x^2 − 9 is shown below.

Find the area of the shaded region.

Circle your answer.

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*.

The exact value of sin*θ* is given by \frac{4\sqrt{a}}{b}

Find the integer values of *a* and *b*

*a* =

*b*=

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

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

^{2}(to 1 d.p.)

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*.

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.

Circle the correct answer.

When it was first noticed, the weed covered an area of 0.25 m^{2}. Twenty days later the
weed covered an area of 0.5 m^{2}.

State the value of *B*.

Give your answer as a fraction in its simplest form.

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*=

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

\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*=

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*=

Use the result from part (a) to obtain an approximation to \sqrt[3]{1.18} giving your answer to 4 decimal places.

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*=

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.

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*=

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.

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.”

What is the population for this investigation?

Circle your answer.

households

in the village

intake for the

village from

eating out

households

selected

What is the sampling method suggested by Ralph?

Circle your answer.

number

random variable

random

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

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) =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?

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 CO_{2} 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 CO_{2} emissions.

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

State which option Shona should use for her hypothesis test

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

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:

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 |

Find the value of *k*.

*k*=

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

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.

What is the Binomial distribution that should be used

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

Gives values for *a* and *b*.

*a* =

*b*=

What is the P(≤2)

Which of these options is valid:

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

Find the mean of *X*

Find the standard deviation of *X*

Give your answers to two decimal places

Mean =

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) =
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.

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.

What is the value of the test statistic?

What are the critical values

Which of these options is valid:

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).

Give your answers to two decimal places.

Greatest =

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.

Give your answer as a decimal.