Practice Paper Pure 3 (Edexcel International A Level Maths: Pure 4)

Practice Paper Questions

1a
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2 marks

The function f(x) is defined as

straight f colon space x space rightwards arrow from bar space fraction numerator x squared plus 1 over denominator x squared end fraction space space space space space space space space space space x element of straight real numbers comma space x space not equal to space 0

 Show that f(x) can be written in the form

 straight f colon space x space rightwards arrow from bar space 1 space plus space 1 over x squared

                             

1b
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2 marks

Explain why the inverse of f(x) does not exist and suggest an adaption to its domain so the inverse does exist.

1c
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4 marks

The domain of f(x) is changed to x > 0.
Find an expression for f−1(x)  and state its domain and range.

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2
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8 marks

Simplify fully

(i)
fraction numerator x squared plus 5 x over denominator x cubed plus 3 x squared minus 10 x end fraction
 

(ii)
fraction numerator x squared minus 4 over denominator x to the power of 4 minus 16 end fraction
 
(iii)
fraction numerator left parenthesis x plus 2 right parenthesis squared plus left parenthesis x plus 2 right parenthesis left parenthesis x plus 4 right parenthesis over denominator x squared plus 5 x plus 6 end fraction 

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3a
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3 marks

On the same axes sketch the graphs of y equals p open parentheses x close parentheses and y equals p to the power of negative 1 end exponent open parentheses x close parentheses comma where p open parentheses x close parentheses equals open vertical bar 2 x close vertical bar comma space x less or equal than 0.

3b
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3 marks

Find an expression for p to the power of negative 1 end exponent open parentheses x close parentheses and state its domain.

3c
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3 marks

Show that p to the power of negative 1 end exponent open parentheses x close parentheses equals 1 half p open parentheses negative 1 half x close parentheses.

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4a
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1 mark

Scientists introduced a small number of rare breed deer to a large wildlife sanctuary.

The population of deer, within the sanctuary, is modelled by

D equals 20 e to the power of 0.1 t end exponent

D is the number of deer after t years of first being introduced to the sanctuary.

Write down the number of deer the scientists introduced to the sanctuary.

4b
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2 marks

How many years does it take for the deer's population to double?

4c
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1 mark

Give one criticism of the model for population growth.

4d
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2 marks

The scientists suggest that the population of deer are separated after either 25 years or when their population exceeds 400.
Find the earliest time the deer should be separated.

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5a
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5 marks

By letting B equals 2 A, use the identity for tan open parentheses A plus B close parentheses to derive an expression for tan space 3 A in terms of tan A.

5b
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3 marks

Hence, or otherwise, solve the equation

         fraction numerator 6 space tan space x minus 2 space tan cubed space x over denominator 1 minus 3 space tan squared space x end fraction equals 2 space space space space space space space space space 0 less or equal than space x less or equal than straight pi

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6
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5 marks

Use calculus to show that

integral subscript 0 superscript 1 fraction numerator 3 x e to the power of negative 3 x squared end exponent over denominator 3 minus 2 e to the power of negative 3 x squared end exponent end fraction space straight d x equals 1 fourth ln space left parenthesis 3 minus 2 e to the power of negative 3 end exponent space right parenthesis.

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7a
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4 marks

The diagram below shows part of the graph of  y equals straight f open parentheses x close parentheses, wherespace straight f open parentheses x close parentheses spaceis the function defined by

straight f open parentheses x close parentheses equals open parentheses x squared minus 1 close parentheses ln open parentheses x plus 3 close parentheses comma space space space space space x greater than negative 3

q8a-7-3-medium-a-level-maths

Points A, B and C are the three places where the graph intercepts the x-axis.

Find straight f apostrophe open parentheses x close parentheses.

7b
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2 marks

Show that the coordinates of point A are (-2, 0).

7c
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3 marks

Find the equation of the tangent to the curve at point A.

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8a
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2 marks

Show that the equation

x equals 7 e to the power of negative 0.2 t end exponent

can be written as

ln space x equals ln space 7 minus 0.2 straight t

8b
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2 marks

Rewrite the equation ln space y equals 4.1 x plus ln space 8 in the form  y equals A e to the power of k x end exponent.

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9a
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2 marks

Given that x equals sec space 7 y ,

Find  fraction numerator d y over denominator d x end fraction  in terms of y

9b
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4 marks

Hence find fraction numerator d y over denominator d x end fraction in terms of x.

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10a
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3 marks

According to legend, unicorn tears have magical healing powers.

Without unicorn tears, a bruise of initial size A mm2 will heal according to the model

straight f open parentheses t close parentheses equals A e to the power of negative 0.25 t end exponent minus 0.1 t space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where f is the area of the bruise, in square millimetres, at time t days since the bruise first appeared.

Use the iterative formula

t subscript n plus 1 end subscript equals 4 ln space open parentheses 120 over t subscript n close parentheses

withspace t subscript 0 equals 10 space to find how many days it takes a bruise to heal without unicorn tears. Give your answer to three significant figures.

10b
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3 marks

With unicorn tears, a bruise of the same initial size will heal according to the model

u open parentheses T close parentheses equals A T e to the power of negative T end exponent minus 0.1 T space space space space space space space space space space space space space T greater or equal than 0

where time T is measured in seconds.

(i)
Find the initial size of the bruise considered in part (a).
(ii)
Find how many seconds it takes the same size bruise to heal using unicorn tears.

10c
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1 mark

Using your answers from parts (a) and (b), work out approximately how many times quicker a bruise heals when unicorn tears are used.

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11a
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2 marks

Show that

      sin squared open parentheses 3 x plus 2 close parentheses equals 1 half open parentheses 1 minus cos open parentheses 6 x plus 4 close parentheses close parentheses

11b
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3 marks

Hence, or otherwise, find an expression for

      integral sin squared open parentheses 3 x plus 2 close parentheses space d x

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