Practice Paper 1 (Pure) (OCR A Level Maths: Pure)

On the same diagram sketch the following position vectors

The position vector is given by .

Find the magnitude of , giving your answer in the form where and are integers to be found.

Find the angle between and the positive x-axis, giving your answer in degrees to one decimal place.

State the range for the following functions based on the given domains.

               

       

               

The graph of  is shown below.

Use the graph to write down the domain and range of .

On the diagram above sketch the graph of 

Lloyd is to start training in order to run a marathon.

For the first week of training he will run a total of 2 miles.

Each subsequent week he’ll increase the total number of miles run by 3 miles.

He intends training for 15 weeks.

Calculate how far Lloyd will run during his eighth week of training

Work out how much further Lloyd will run in his last week of training compared to his first.

Find the total number of miles Lloyd will run across all 15 weeks of his training schedule.

Show that the point (0 , π) lies on the curve with equation

Find an expression for .

Find the gradient at the point (0 ,π).

Hence find an equation of the tangent to the curve at the point (0, π).

Prove, from first principles, that the derivative of is .

A geometric series has first term , and its terms are connected by the relationship  for all .

Given that all the terms of the series are positive, show that the sum of the first twelve terms of the series may be written in the form

where  and  are positive integers and  is a surd.

A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.

Show that can be written in the form where and .

Sketch the graph of sin cos for .
Label any points where the graph intercepts the coordinate axes.

The diagram below shows the sector of a circle .

Given that the area of triangle cm², find the area of the shaded segment. Give your answer correct to 3 significant figures.

Find the perimeter of the sector , giving your answer correct to 3 significant figures.

Show that the equation can be rewritten as

Starting with , use the iterative formula

to find values for and , giving each to four decimal places where appropriate.

Use the substitution  to show that

where c is the constant of integration.

Use integration by parts to find an expression for

Use a suitable substitution to find the following

Show that

where a and k are constants.

Find the coordinates of the stationary points, and their nature, on the graph with equation  .

Find the coordinates of any points of inflection, determine if these are also stationary points and find the intervals in which the graph is convex and concave.

Give values to three significant figures.