Practice Paper Pure 3 (Edexcel International A Level Maths: Pure 3)

Solve the equation

sec⁴ tan² 

giving your answers to three significant figures where appropriate.

Solve the equation

cos⁴  cos 

State your answers as multiples of .

Given that the graph of   passes through the point  and that

Find 

Explain why  must be excluded from the domain of .

One of the three algebraic fractions below is improper (‘top-heavy’):

Identify which fraction is improper and write it in the form  , where  and  are integers to be found.

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

Find 

(i)  fg(x)

(ii)  gf(x)

Write down and state its domain and range.

The graphs of f(x) and  are drawn on the same axes.
Describe the transformation that would map one graph onto the other.

Find the coordinates of the point where the graphs of  y = f(x) and  meet. 

An exponential model of the form

is used to model the concentration of a pain-relieving drug (D mg/ml) in a patient’s bloodstream  hours after the drug was administered by injection.  $A$ and $k$ are constants.

The graph below shows values of $\ln D$ plotted against $t$

Find estimates for the constants $A$and $k$.

Find the time, to the nearest minute, at which the rate of decrease of the concentration of the drug in the patient’s bloodstream is 12 mg/ml/hour.

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

Sketch the graph of y = fg(x), stating the coordinates of all points where the graph intercepts the coordinate axes.

There are between 0 and 4 solutions to the equation fg(x) = c , where c is a real
number.  Determine the values of c that produce each number of solutions.

Find    for each of the following:

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, 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?

Solve the equation

,

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

Solve the equation sin cos ,for  .
Give your answers to three significant figures.

Write down the maximum value of sin cos and the second positive value of   for which it occurs.  Give your value of θ  to three significant figures.

Given that  satisfies the equation   , where  ,

state the range of possible values of ,

express both cos and tan  in terms of .