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

Write the quadratic function  in the form  where a, b and c are integers to be found.

Write down the minimum point on the graph of .

Sketch the graph of , clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

A sequence is defined for by the recurrence relation

where  is a constant.

Given that the sequence is periodic with order 2, and given as well that ,

Find the value of  .

For the value of  found in part (a),

Calculate  

The function is defined as

Use the sign change rule to show there is a root to the equation in the interval .

Find .

Use the Newton-Raphson method with to find the root in the interval correct to three decimal places.

Stephen opens a savings account with £600.

Compound interest is paid annually at a rate of 1.2%.

At the start of each new year Stephen pays another £600 into his account.

Show that at the end of two years Stephen has £1221.69 in the account

Show that at the end of year , the amount of money, in pounds, Stephen will have in his account is given by

Hence show that the total amount, in pounds, in Stephen’s account after  years is

In the triangle ABC,  and  .

Show that to 1 d.p.

The line intersects the circle at exactly two points.  Find the range of possible values of c.

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years

A model for the mass of carbon-14,  g, in an object originally containing 100 g,
at time years is

where is a constant.

Find the value of , giving your answer to three significant figures.

The object is considered as having no radioactivity once the mass of carbon-14 it contains falls below 0.5 g. Find out how old the object would have to be considered non-radioactive.

A different object currently contains 25g of carbon-14.
In 500 years’ time how much carbon-14 will remain in the object?

Express    as partial fractions.

Given that  is small such that and higher powers of  can be ignored show that

Find the percentage error between your calculator answer and the approximation in part (a) when , giving your answer to one decimal place.

For which values of  is the approximation in part (a) valid?

Show that the equation  tan² sec   can be written as

sec² sec

Hence, or otherwise, solve the equation

tan²  sec 

Give your answers to three significant figures.

A stretch of road along Equality Street has a speed limit of 70 mph and traffic is monitored by six average speed cameras.

Ricky is driving his car along Equality Street and passes the first camera at time zero.  Ricky’s speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below. 

Use all the results above to estimate the distance between the first and last camera.

A driver will receive a speeding ticket if their average speed between the first and last camera exceeds the speed limit.
Should Ricky receive a speeding ticket or not?  Justify your answer.

The diagram below shows a sketch of the curves with equations

       and       

Find the x-coordinates of the points of intersection of the two graphs.

Use calculus to find the total shaded area enclosed by the two graphs.

A soft ball is thrown upwards from the top of a building.
The height,  m of the ball above the ground after  seconds is modelled by the function

What does the constant H indicate in the function?

At what time is the ball at the same height as when it was thrown?

Find in terms of  how long it takes for the ball to first hit the ground.

How much longer does a ball launched from a 25 m tall building stay in the air for compared to a ball launched from a 15 m tall building?

The graph of the curve C shown below is defined by the parametric equations

Find an expression for    in terms of  .

Differentiate  with respect to x.

Prove by contradiction that if is odd, then x must be odd.