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

The functions f(x) and  g(x)  are defined as follows

Find

Write down f⁻¹(x) and state its domain and range.

Solve the equation

giving your answers correct to 3 significant figures.

Given that  is small, write an approximation in terms of  for

                                             sin cos tan² 

A curve has the equation .

Find expressions forand .

Determine the coordinates of the local minimum of the curve.

The diagram below shows the sector of a circle with centre .  The radii  and  are each equal to cm, and the angle at the centre, , is equal to  radians.  The line  is perpendicular to the line .

Given that  , show that the area of the shaded shape  is given by

cm².
                         

The curve C has equation .  The point P lies on C.

Find an expression for .

Show that an equation of the normal to C at point P is .

The diagram below shows part of the graph of

Find the area of the shaded region labelled R.

Show that the derivative function of the curve given by 

is given by

Find the equation of the normal to the curve given in part (a) at the point where , giving your answer in the form where and c are integers to be found.

A geometric series is given by 

Write down the common ratio, , of the series.

Given that the series is convergent, and that , calculate the value of .

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 

Using the points marked and , find an equation for the line of best fit, giving your answer in the form , where and  are constants to be found.

Hence find estimates for the constants  and 

The patient is allowed a second injection of the drug once the amount of drug in the bloodstream falls below 1% of the initial dose.
Find, to the nearest minute, how long until the patient is allowed a second injection of the drug.

A sketch of the graph with equation where  is shown below.

Points and  are the -axis intercepts and point  is the maximum point on the graph.

On the diagram above, sketch the graph of labelling the image of the points  and  with and .

Show that the area of is twice the area of triangle .

Use the substitution  to show that

where c is the constant of integration.

The curve has parametric equations

Show that at the point (0 , 6),and find the value of    at this point.

The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P

A large block of ice used by sculptors is in the shape of a cuboid with dimensions x m by 2x m by 5x m.  The block melts uniformly with its surface area decreasing at a constant rate of k m² s^-1. You may assume that as the block melts, the shape remains mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

In the case when, the block of ice remains solid enough to be sculpted as long as the rate of melting, by volume, does not exceed .

Find the value of x for the largest block of ice that can be used for ice sculpting under such conditions, giving your answer as a fraction in its lowest terms.

The number of daylight hours, , in the UK,  days after the spring equinox (the day in spring when the number of daylight hours is 12) is modelled using the function

sin 

where are  constants.

Write down the value of .

Given that the maximum number of daylight hours is 16.5, write down the value of .

For how many days of the year does the number of daylight hours remain below 10? Give your answer as a whole number of days.

If the spring equinox falls on the 21^st March, find the dates throughout the year when there are 16 hours of daylight.

The model needs to be adjusted every four years. Suggest a reason why.

A particle of mass 0.2 kg is acted upon by a force of . 

Given that the magnitude of the acceleration experienced by the particle under the influence of this force is 65 m s-², find the possible values of p.

An additional second force with a magnitude of qj N now acts on the particle, where  .   Under the influence of the resultant of the two forces, the particle now experiences an acceleration of magnitude .

Find the possible values of q.

Prove that the square of an odd number is always odd.