Practice Paper 2 (Pure) (Edexcel A Level Maths: Pure)

The drilling rig AlphaBeta began leaking oil into the North Sea following a technical fault. It took 14 hours for engineers to trace and repair the fault.

During that time the rate of oil leaking, measured in tonnes per hour, was recorded every 2 hours.  The results are shown in the table below.

For safety reasons oil rigs are required to shut down and stop all operations until an inspection is carried out should the total amount of oil leaked during any incident exceed 250 tonnes.

Use all the data in the table to decide if the AlphaBeta rig should be shut down or not.

A rainbow-shaped logo is formed from two semicircles as shown below.
The radius of the smaller semicircle is 1 cm less than the radius of the larger semicircle.

Find the area of the logo.

The graph of the ellipse E shown below is defined by the parametric equations

Find an expression for    in terms of θ.

Find the equation of the tangent to E, at the point where  , giving your answer in the form , where a and b are real numbers that should be given in exact form.

The diagram below shows a sketch of the curve with equation 

The point   lies on the curve.

The shaded rectangle shown has width  δx and height y.

Calculate

The function f(x) is defined as

Explain why the inverse of  f(x) does not exist.

Suggest an adaption to the domain of f(x) so the following conditions are met:

• the inverse of f(x) exists,
• the graph of y = f(x) lies in the first quadrant only, and,
• the domain of f(x) is as large as possible.

State the range for your adapted f(x).

The domain of f(x) is changed to .
Find an expression for    and state its domain and range.

The leakage rate of water from a pipe,  (litres per second), is directly proportional to flow rate,  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leakage rate was  when the flow rate was .

Show that the constant of proportionality is 0.5 and hence write down an equation connecting and .

Find the leakage rate when the flow rate is .

The flow rate is reduced should the leakage rate exceed .
Find the maximum possible flow rate before it is reduced.

Given that ,

Show that 

For this value of , calculate .

An exponential model of the form    is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream,  hours after the drug was administered by injection.   and  are constants.

The graph below shows values of   plotted against  with a line of best fit drawn.

(i)        Use the graph and line of best fit to estimate at time .

(ii)       Work out the gradient of the line of best fit.

Use your answers to part (a) to write down an equation for the line of best fit in the form ,  where and  are constants.

Show that  can be rearranged to give 

Hence find estimates for the constants and .

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

In triangle ABC, point F lies on AB and point G lies on BC.  

F divides AB in the ratio m:n.  

The line segment FG is parallel to AC.

Explain why for some constant , where .

Given that   and ,   show that

Using your result from (b), prove that G divides BC in the ratio n:m.

The function, is defined by

Show that the equation can be written in the form

On the same diagram sketch the graphs of and .

The equation has a root, α, close to .
The iterative formula     x_n+1=e^-x_n+1      with x₀=2 is to be used to find correct to three significant figures.

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root .

Use an identity for cos to derive an identity for cos , in terms of cos .

Hence, or otherwise, solve the equation

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.

Find

Show that the general solution to the differential equation

can be written in the form

where A is a constant.

Find the particular solution to the following differential equation, using the given boundary condition