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

Given that is small, show that 

sin  tan  cos 

Hence, or otherwise, find approximate solution(s) to the equation

sin tan cos 

Show that the function is decreasing for all

In the binomial expansion of     where , the coefficient of the  term is equal to the coefficient of the  term.

Show that .

Given further that  find the values of and .

Given that 

Sketch the graph of  , showing clearly the coordinates of the points where the curve crosses or touches the coordinate axes.

The functions  and  are defined by the equations

The graph of  touches the -axis at the point .  Find the value of , giving your answer as an exact value.

Solve the equation

cos⁴  cos 

State your answers as multiples of .

Given that

tan

find the values of such that  .

A manufacturer claims their flask will keep a hot drink warm for up to 7 hours.

In this sense, warm is considered to be  or higher.

Assuming a hot drink is made at and its temperature inside the flask is after exactly 7 hours, find:

a linear model for the temperature of the drink inside the flask of the form , and

an exponential model for the temperature of the drink inside the flask of the form

where  is the temperature of the drink in the flask after  hours and and  are constants.

Compare the rate of change of the temperature of the drink inside the flask of both models after 3 hours.

A user of the flask suggests that hot drinks are only kept warm for 5 hours.
Suggest a reason why the user’s experience may not be up to the claims of the manufacturer.

The diagram below shows part of the graph of .

Find the total area of the two shaded regions.

The diagram below shows part of the function where.

Correct to three significant figures, and.

Explain why using the sign change rule with these values would not necessarily be helpful in finding the root close to .

Using suitable values of x, show that there is a root close to .

Without using a calculator, show that

Prove that for all values of x,  where .

A ball is dropped and bounces such that the height of each bounce is 80% of the previous bounce.

The first bounce of the ball reaches a height of 1.60 m.
Find the height the ball bounces on its 8^th bounce 

The ball is considered as no longer bouncing once its bounce height fails to reach 1% of its first bounce height.
Find the number of bounces the ball makes.

Find the total distance travelled by the ball from when it first hits the ground to when it stops bouncing.

Give one reason why this model may be unrealistic.

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

sin cos

Express

sin cos

in the form

sin 

where and  are constants to be found.   and is α acute.

In the UK, domestic electricity runs at a frequency, , of 50 Hertz (Hz).

The constant , is given by .

The diagram below shows a sketch of the curves with equations

    and      

Show that the two curves intersect at the points (3, 4), (4, 3) and (5, 0).

By using the substitution  , show that

where c is the constant of integration.

Using calculus, and your results from parts (a) and (b), show that the total shaded area enclosed by the two curves in the diagram is equal to

The curve C is described by the equation

Show that the tangents of the two points on C wheremeet at the point .

Note:  For this question ensure you are working in degrees.

A wave tank is used to simulate the sea at high tide.
At a certain point along the tank the height of water is measured relative to the calm water level which has a height of .
The height of water in the tank is modelled by the function

where cm is the height of water and  seconds is the time after the peak of the first wave passes the measuring point.

Sketch a graph of against for .