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

Use the substitution to find

.

Find the first three terms, in ascending powers of , in the binomial expansion of

State the values of  for which your expansion in part (a) is valid.

Using a suitable value of , use your expansion from part (a) to estimate , giving your answer to 3 significant figures.

A curve is defined by the parametric equations

By changing these equations into Cartesian form, show that this is the equation of a circle, and determine its centre and radius.

A gardener wants to model the number of hours of daylight his allotment receives at different times of the year using a function of the form

where  is the number of hours of daylight on a given day, is the time measured in whole days, and  and  are positive constants.  Note that  corresponds to the first day of the model.

Given that the model needs to predict a maximum of 17 hours daylight and a minimum of 7 hours daylight find the values of  and .

Explain the significance of the value  in the model.

A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:

The base of each triangle is 2x metres, and the equal sides are each y metres in length.

Although x and y can vary, the total amount of fencing to be used is fixed at P metres.

Explain why .

Show that

where A is the total area of the garden bed.

Using your answer to (b) find, in terms of P, the maximum possible area of the garden bed.

Describe the shape of the bed when the area has its maximum value.

Two rational numbers, a and b are such that   and   , where  are integers with no common factors and .

Find expressions for  and .

Deduce whether or not  and  are rational or irrational.

Expand and simplify .

A cuboid has a length of  units, a width of  units, and a height of units.  Find an expression for the volume of the cuboid in terms of  and .

Simon, an economist, is investigating the trends in employment rates in London and Wales. The large data set for 2001 does not show the number of people that are not in employment.

Below is an extract from the large data set from 2001:

Using your knowledge of the large data set, explain why there is not enough information in the table above to calculate the number of people that are not in employment.

Simon wants to compare unemployment between the two regions for each year.

Explain why Simon should use the proportions of unemployed people in each local authority instead of the number of unemployed people.

The lengths of unicorn horns are measured in cm. For a group of adult unicorns, the lower quartile was 87 cm and the upper quartile was 123 cm. For a group of adolescent unicorns, the lower quartile was 33 cm and the upper quartile was 55 cm.

An outlier is an observation that falls either more than 1.5  (interquartile range) above the upper quartile or less than 1.5  (interquartile range) below the lower quartile.

Which of the following adult unicorn horn lengths would be considered outliers?

32 cm                96 cm             123 cm             188 cm

A machine is used to fill cans of a particular brand of soft drink.  The volume,  ml, of soft drink in the cans is normally distributed with mean 330 ml and standard deviation  ml.  Given that 15% of the cans contain more than 333.4 ml of soft drink, find:

the value of 

Six cans of the soft drink are chosen at random.

Find the probability that all of the cans contain less than 329 ml of soft drink.

Carol is a new employee at a company and wishes to investigate whether there is a difference in pay based on gender, but she does not have access to information for all the employees.  It is known that the average salary of a male employee is £32500, and it can be assumed the salary of a female employee follows a normal distribution with a standard deviation of £6100.  Carol forms a sample using 20 randomly selected female employees.

Write suitable null and alternative hypotheses to test whether the average salary of a female employee is different to the average salary of a male employee.

Using a 5% level of significance, find the critical regions for the test.

The total of the salaries of the 20 employees used in the sample is .

Use this information to state a conclusion for Carol’s investigation into pay differences based on gender.

Would the outcome of the test have been different if a 10% level of significance had been used?

During a zombie attack, Richard suspects that the number of flies in the area, , is dependent on the number of zombies, .

Richard is trying to decide whether the correlation is linear or non-linear, so he uses a graphical software package to plot two scatter graphs. Figure 1 shows the graph of  plotted against , and Figure 2 show the graph of log plotted against log .

Richard calculates the product moment correlation coefficient for each graph. One value is found to be 0.847 while the other is 0.985.

State, with a reason, which PMCC value corresponds with Figure 2.

Test, using a 5% level of significance, whether there is positive linear correlation in the graph shown in Figure 2. State your hypotheses clearly.
You are given that the critical value for this test is 0.1654.

State, with a reason, whether the relationship between number of zombies and number of flies is better represented as linear or non-linear.

Write suitable null and alternative hypotheses for each of the following situations.

The developer of a new cryptocurrency tests, at the 5% level of significance, for any correlation between the new cryptocurrency's net value and that of a more popular cryptocurrency. She calculates the product moment correlation coefficient between the two cryptocurrency's net values to be .

The random variable  X has the probability function

Find the value of .

Construct a table giving the probability distribution of X.

Find .

A student claims that a random variable  has a probability distribution defined by the following probability mass function:

Explain how you know that the student’s function does not describe a probability distribution.

Given that the correct probability mass function is of the form

where  is a constant,

 Find the exact value of .

Find 

State, with a reason, whether or not  is a discrete random variable.

For bars of a particular brand of chocolate labelled as weighing 300 g, the actual weight of the bars varies. Although the company's quality control assures that the mean weight of the bars remains at 300 g, it is known from experience that the probability of any particular bar of the chocolate weighing between 297 g and 303 g is 0.9596. For bars outside that range, the proportion of underweight bars is equal to the proportion of overweight bars.

Millie buys 25 bars of this chocolate to hand out as snacks at her weekly Chocophiles club meeting. It may be assumed that those 25 bars represent a random sample. Let  represent the number of bars out of those 25 that weigh less than 297 g.

Write down the probability distribution that describes .

The random variable Find:

the largest value of such that  

the largest value of  such that 

the smallest value of  such that  .