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

Practice Paper Questions

1a
Sme Calculator
2 marks

The function,space straight f left parenthesis x right parenthesis space is defined by straight f open parentheses x close parentheses equals 1 over e to the power of x minus x plus 1 space space space space space space space space space space space space space space x element of straight real numbers

Show that the equation straight f open parentheses x close parentheses equals 0 can be written in the form

space space x equals e to the power of negative x end exponent plus 1

1b
Sme Calculator
2 marks

On the same diagram sketch the graphs of y equals x and y equals e to the power of negative x end exponent plus 1.

1c
Sme Calculator
2 marks

The equation straight f open parentheses x close parentheses equals 0 has a root, α, close to x equals 1.
The iterative formula  xn+1=e-xn+1 with x0=2 is to be used to find correct to three significant figures.

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root α.

1d
Sme Calculator
3 marks
(i)
Find the values ofspace x subscript 1 comma space x subscript 2 spaceand x subscript 3, giving each correct to three significant figures.
(ii)
How many iterations are required before x subscript n and x subscript n minus 1 end subscript agree to two decimal places?

Did this page help you?

2
Sme Calculator
4 marks

Solve the equation

3 sec4 theta plus 16 equals 16 plus 16 spacetan2 theta comma space space space space space space space space space space space minus straight pi less or equal than theta less or equal than straight pi

giving your answers to three significant figures where appropriate.

Did this page help you?

3
Sme Calculator
5 marks

Solve the equation

8 spacecos4 theta minus 5 cos 2 theta minus 2 equals 0 space space space space space space space space space space 0 less or equal than theta less or equal than straight pi

State your answers as multiples of straight pi.

Did this page help you?

4a
Sme Calculator
4 marks

Given that the graph of  begin mathsize 20px style y equals straight f left parenthesis x right parenthesis end style passes through the point left parenthesis 0 comma 4 right parenthesis and that

               begin mathsize 20px style straight f apostrophe open parentheses x close parentheses equals fraction numerator 12 x squared plus 7 over denominator 2 open parentheses 4 x cubed plus 7 x plus 4 close parentheses end fraction end style

Find straight f left parenthesis x right parenthesis

4b
Sme Calculator
1 mark

Explain why x equals negative 1 half must be excluded from the domain of straight f open parentheses x close parentheses.

Did this page help you?

5
Sme Calculator
3 marks

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

fraction numerator x squared minus 5 x plus 1 over denominator x plus 1 end fraction fraction numerator x plus 2 over denominator left parenthesis x plus 1 right parenthesis squared end fraction fraction numerator space x squared minus 5 x plus 1 over denominator left parenthesis x plus 1 right parenthesis cubed end fraction

Identify which fraction is improper and write it in the form  A x plus B plus fraction numerator C over denominator x plus 1 end fraction, where A comma space B and C are integers to be found.

Did this page help you?

6a
Sme Calculator
3 marks

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

 straight f left parenthesis x right parenthesis space equals space left parenthesis x space minus space 1 right parenthesis squared space minus space 4 space space space space space space space space space space space x element of straight real numbers comma x greater or equal than 1

g left parenthesis x right parenthesis equals space 1 space plus space square root of left parenthesis x plus 4 right parenthesis space end root space space space space space space space space space space x element of straight real numbers comma x greater or equal than negative 4

Find 

(i)  fg(x)

(ii)  gf(x)

6b
Sme Calculator
2 marks

Write down straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

6c
Sme Calculator
2 marks

The graphs of f(x) and straight f to the power of negative 1 end exponent left parenthesis x right parenthesis are drawn on the same axes.
Describe the transformation that would map one graph onto the other.

6d
Sme Calculator
2 marks

Find the coordinates of the point where the graphs of  y = f(x) and  y space equals space straight f to the power of negative 1 end exponent left parenthesis x right parenthesis meet. 

Did this page help you?

7a
Sme Calculator
3 marks

An exponential model of the form

D equals A e to the power of negative k t end exponent

is used to model the concentration of a pain-relieving drug (D mg/ml) in a patient’s bloodstream t hours after the drug was administered by injection.  A{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} and k{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} are constants.

The graph below shows values of ln D{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} plotted against t{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

q11a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-veryhard

Find estimates for the constants A {"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}and k{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}.

7b
Sme Calculator
2 marks

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.

Did this page help you?

8a
Sme Calculator
3 marks

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

 straight f left parenthesis x right parenthesis space equals space vertical line x to the power of 3 space end exponent minus space 8 vertical line space space space space space space space space space space x element of straight real numbers

g left parenthesis x right parenthesis space equals space vertical line x space space space space space space space space space space space space space space space space space space space space space space space x element of straight real numbers space space space

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

8b
Sme Calculator
3 marks

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.

Did this page help you?

9a
Sme Calculator
4 marks

Find  fraction numerator d y over denominator d x end fraction  for each of the following:

y equals cos open parentheses space x squared minus 3 x plus 7 close parentheses plus sin space open parentheses e to the power of x close parentheses space

9b
Sme Calculator
3 marks

y equals ln space open parentheses 2 x cubed close parentheses space

Did this page help you?

10a
Sme Calculator
2 marks

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, y g, in an object originally containing 100 g,
at time t years is

y equals 100 e to the power of negative k t end exponent

where k is a constant.

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

10b
Sme Calculator
2 marks

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.

10c
Sme Calculator
2 marks

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

Did this page help you?

11
Sme Calculator
5 marks

Solve the equation

6 cross times 3 to the power of x minus 1 end exponent equals 6 to the power of 2 x end exponent,

giving your answer in the form fraction numerator ln space a over denominator ln space b end fraction, where a and b are integers to be found.

Did this page help you?

12a
Sme Calculator
4 marks

Solve the equation 5 sin theta plus 2 cos theta equals 3,for  negative straight pi less or equal than theta less or equal than straight pi.
Give your answers to three significant figures.

12b
Sme Calculator
2 marks

Write down the maximum value of 5 sin theta plus 2 cos theta and the second positive value of theta  for which it occurs.  Give your value of θ  to three significant figures.

Did this page help you?

13
Sme Calculator
5 marks

Given that x satisfies the equation  arc sin x equals k, where  negative straight pi over 2 less than k less than 0,

(i)

state the range of possible values of x,

(ii)
express both cos k and tan k in terms of x.

Did this page help you?