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

Solve the equation

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

In the triangle ABC,  and  .

Show that to 1 d.p.

The function f(x) is defined as

 Show that f(x) can be written in the form

Explain why the inverse of f(x) does not exist and suggest an adaption to its domain so the inverse does exist.

Two sequences are being used to model the value of a car,  years after it was new.  At new, the car’s value is £25 000.

Model 1 is an arithmetic sequence.

Model 2 is a geometric sequence.

Both models predict the same value for the car, £7 500, when it is exactly 8 years old.

Find the common difference for Model 1 and the common ratio for Model 2, giving answers to three significant figures where appropriate.

Find the value of the car according to Model 2 in the year Model 1 predicts its value to be £5000.

State one benefit of Model 2 over Model 1 for estimating the value of older cars.

The alternating voltage, , in an electrical circuit,  seconds after it is switched on, is modelled by the function

Show that

can be written as

sin

where and .

The diagram below shows a sketch of the curves with equations

and

Find the x-coordinates of the intersections of the two graphs.

Show that the area of the shaded region labelled R is given by

Use calculus to find the area of the shaded region labelled R.

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

Using the variables V, for the volume of water in the container, and t, for time, write down a differential equation involving the term , for the volume of water in the container.

Differentiate with respect to x, simplifying your answer:

The function  is to be transformed by a sequence of functions, in the order detailed below:

1. A reflection in the -axis.
2. A vertical stretch of scale factor 4.
3. A translation by .

Write down an expression for the combined transformation in terms of .

Use calculus and the substitution  to find the exact value of

The diagram below shows circles C₁ and C₂ which intersect at the two points A and B. Circle C₁ has equation  , and points A and B lie along the line with equation  . Circle also passes through the point (-13, 2). 

Find an equation of circle C₂.

Show that cos sin can be written in the form  cos, where  and  is an acute angle measured in radians.

Hence, or otherwise, solve the equation cos sin ,for .
Give your answers to three significant figures.

A sequence is defined for by the recurrence relation

where  and  are real numbers.

Show that the sequence is periodic, and determine its order.

Given that  and   

determine the possible values of  and .

A large weather balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m³) and time (seconds) write down a differential equation to describe the relationship between volume and time as the weather balloon is inflated.

Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m³, find the particular solution to your differential equation.

Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m^3.  If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?

The curve C is defined by

Points A and B have coordinates and respectively.

The tangents to C at points A and B intersect at the point P.
The tangent to C at point A intersects the x-axis at point Q.
The tangent to C at point B intersects the x-axis at point R.

Find the area of triangle PQR.

Below is a proof by contradiction that there is no largest multiple of 7.

Line 1:	Assume there is a number, S, say, that is the largest multiple of 7.
Line 2:
Line 3:	Consider the number .
Line 4:
Line 5:
Line 6:	So  is a multiple of 7.
Line 7:	This is a contradiction to the assumption that S is the largest multiple of 7.
Line 8:	Therefore, there is no largest multiple of 7.

The proof contains two omissions in its argument.

Identify both omissions and correct them.