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

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  with a line of best fit drawn.

(i)        Use the graph and line of best fit to estimate at time .

(ii)       Work out the gradient of the line of best fit.

Use your answers to part (a) to write down an equation for the line of best fit in the form ,  where and  are constants.

Show that  can be rearranged to give 

Hence find estimates for the constants and .

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

The diagram below shows the graph of , where is the function defined by

The points A and B are maximum and minimum points, respectively.

Find the difference between the -values of the coordinates of  and , giving your answer correct to 3 decimal places.

On the same axes, sketch the graphs of   and    where

Label the points at which the graphs intersect the coordinate axes.

Solve the equation .

Which of the solutions to is also a solution to ?

The minimum point on the graph of  has coordinates  as shown on the diagram below.

Sketch the graph of  and state the coordinates of the maximum point.

Find the exact distance between the minimum point on the graph of  and the maximum point on the graph of .

Show that, for (where k is an integer),

Use calculus and your result from part (a) to show that

The village of Crinkley Bottom lies on a straight road, as modelled by the line on the graph below.  Rush hour traffic causes much air pollution in the village so to improve the air quality around Crinkley Bottom a bypass is to be built.
The path of the bypass is modelled by part of the equation .

The bypass is to be built with a roundabout south of the village at the origin and a northern roundabout which re-joins the road through Crinkley Bottom at the point .

On the diagram show how using the iterative formula  with will lead to convergence at the southern roundabout 

Use the alternative iterative method

with , to find the position of the roundabout at P to four significant figures.

Verify that your answer to part (b) is correct to four significant figures.

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.

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

Differentiate with respect to x, simplifying your answers as far as possible:

A is the point on the graph of  such that the tangent to the graph at passes through the point .  Show that the x-coordinate of A satisfies the equation

It is given that

where  and  are integers.

Find the values of .

Show that there are no positive values of and that satisfy the equation