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

Part of the graph of is shown below, where  is measured in radians.

Explain why the change of sign rule would fail if attempting to locate a root of the function using the values of θ = 1.55 and θ = 1.65.

The sum of the first ten terms in an arithmetic series is 40.  The sum of the first twenty terms in the same series is 280.  Find the first term, , and the common difference, , of the series.

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide.
At a certain point along the tank the height of water is measured relative to the calm water level which has a height of .
The height of water in the tank is modelled by the function

where cm is the height of water and  seconds is the time after the peak of the first wave passes the measuring point.

Sketch a graph of against for .

Comment on the suitability of using this model to simulate actual sea waves.

The line L is perpendicular to the line with equation , and passes through the origin.

Find the equation of the line L.

The points and lie on a circle.

Show that ∠ACB=90°.

Deduce a geometrical property of the line segment AB.

Hence find the equation of the circle.

The energy company Power^X operate a wind turbine.
Engineers from Power^X model the power output, P kW (kiloWatts), of the wind turbine over a twelve-hour period according to the function

where t is time measured in hours.  The graph of is shown below.

Using the trapezium rule, with six strips of width 1, estimate the total power generated by the wind turbine in the first six hours.
You may use the table below to help.

Briefly explain why it is difficult for the engineers from Power^X to determine whether the total amount of power found, using the method in part (a), is an over- or under- estimate.

Given that

Show that

Hence prove that .

Given that , where  and are constants, find the values of  and .

The diagram below shows a cube whose vertices are O, A, B, C, D, E, F and G.

a, b and c are the vectors and respectively.

Find vectors and

Let P be a point on OE, and let Q be a point on AG.

Explain why the vectors and can be expressed in the forms 

where  and   are constants with and  

By solving the equation , using your results from (a) and (b), show that the diagonals OE and AG intersect each other, and determine the ratios into which they are cut by the point of intersection.

An estate agent, Terry, claims that there is a correlation between the value of a house (in £1000s) and the distance between that house and the nearest nightclub (miles).

Terry has a database containing over 100 houses and he takes a random sample of seven houses to investigate his claim. The scatter graph below shows the results:

Terry calculates the product moment correlation coefficient as . Using the scatter graph, explain how you know Terry's PMCC value is incorrect.

Terry corrects his mistake and calculates the correct PMCC as .
The table below gives the critical values, for different significance levels, of the product moment correlation coefficient, , for a sample of size 7.

State, giving a reason, whether the conclusion to the test would be different if a 1% level of significance had been used.

Suggest one way in which Terry could improve his investigation.

N’Oréal, a hair and beauty company, release an advertising campaign for its new product called ‘Face Amazifier’. The small print at the bottom of the advert says the following

‘55% of 25 existing customers agree that the product makes your face feel more amazing.’

The Advertising Standards Agency get complaints that the advert is not representative.

Explain the meaning of ‘bias’ in relation to the choice of sample N’Oréal has used.

Suggest two ways N’Oréal could improve their sample to make it more representative.

The mean time that teenagers in the UK spend on social media is 132 minutes per day and the standard deviation is known to be 24 minutes.  Mr Headnovel, a teacher in the UK, claims that the students at his school spend more time on social media than the country’s average.  He takes a random sample of 15 students and calculates the mean time spent on social media to be 144 minutes.

Stating your hypotheses clearly, test Mr Headnovel’s claim using a 5% level of significance.

State two assumptions you had to make about the times that teenagers in Mr Headnovel’s school spend on social media?

A bag contains 12 orange marbles, 8 purple marbles and 5 red marbles.  A marble is taken from the bag and its colour is recorded, but it is not replaced in the bag.  A second marble is then taken from the bag and its colour is recorded.

Draw a tree diagram to represent this information.

Find the probability that:

The table below shows an extract from the large data set for the year 2011.
The figures shown are the number of people travelling to work by train in 7 randomly selected local authorities in the East of England.

Find the median, the upper and lower quartiles, and interquartile range.

An outlier is defined as any data value that falls either more than
1.5 (interquartile range) above the upper quartile or less than
1.5 (interquartile range) below the lower quartile.

Find the boundaries (fences) at which outliers are defined.

Explain why, in this case, there are no outliers.

Leonardo has constructed a biased spinner with six sectors labelled 0,1, 1, 2, 3 and 5.  The probability of the spinner landing on each of the six sectors is shown in the following table:

number on sector	0	1	1	2	3	5
probability

Find the value of .

Leonardo is playing a game with his biased spinner.  The score for the game, , is the number which the spinner lands on after being spun.

 Leonardo plays the game twice and adds the two scores together. Find the probability that Leonardo has a total score of 5.

Complete the following cumulative probability function table for :

Score	0	1	2	3	5
					1

Find the probability that X is

The independent random variables X  and Y  have probability distributions

where  and  are constants.

 Find  .