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

Practice Paper Questions

1a
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5 marks

Show that the derivative function of the curve given by 

ln space y minus 2 x y cubed equals 8

is given by

fraction numerator d y over denominator d x end fraction equals fraction numerator 2 y to the power of 4 over denominator 1 minus 6 x y cubed end fraction.

1b
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3 marks

Find the equation of the normal to the curve given in part (a) at the point where y equals 1, giving your answer in the form a x plus b y plus c equals 0 comma where a comma space b and c are integers to be found.

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2a
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4 marks

Given that y greater than 2, find the general solution to the differential equation

      fraction numerator d y over denominator d x end fraction equals x squared left parenthesis y minus 2 right parenthesis

2b
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3 marks

Find the general solution to the differential equation

      fraction numerator d y over denominator d x end fraction equals sin to the power of 2 space end exponent 2 y

giving your answer in the form x equals straight f left parenthesis y right parenthesis.

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3
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4 marks

Write  fraction numerator x squared plus 3 x plus 10 over denominator x squared plus 8 x plus 15 end fraction  in the form  A plus fraction numerator B over denominator x plus 3 end fraction plus fraction numerator C over denominator x plus 5 end fraction , where A comma B spaceand C are integers to be found.

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4a
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2 marks

The functions straight f left parenthesis x right parenthesis and  straight g left parenthesis x right parenthesis are given as follows

               straight f left parenthesis x right parenthesis equals open parentheses 4 plus 3 x close parentheses to the power of 1 half end exponent space space space space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals open parentheses 9 minus 2 x close parentheses to the power of negative 1 half end exponent

Expand straight f left parenthesis x right parenthesis, in ascending powers of x up to and including the term in x squared.

4b
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2 marks

Expand straight g open parentheses x close parentheses, in ascending powers of x up to and including the term in x squared.

4c
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2 marks

Find the expansion of square root of fraction numerator 4 plus 3 x over denominator 9 minus 2 x end fraction end root  in ascending powers of x, up to and including the term in x squared.

4d
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2 marks

Find the values of x for which your expansion in part (c) is valid.

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5a
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2 marks

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

x equals 3 sin space 3 theta     y equals 6 cos space 2 theta   space minus space straight pi over 2 space less or equal than theta less or equal than space straight pi over 2

q4a-9-2-medium-a-level-maths

(i)
Write down the value of  fraction numerator straight d y over denominator straight d theta end fraction  at the point (0 , 6).

(ii)
Write down the value of  fraction numerator straight d x over denominator straight d theta end fraction at the points (-3 , 3) and (3 , 3).
5b
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3 marks

Find an expression for  fraction numerator straight d y over denominator straight d x end fraction  in terms of theta.

5c
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4 marks
(i)
Find the values of x, y and  fraction numerator straight d y over denominator straight d x end fraction  at the point where  theta equals space pi over 12.

(ii)
Hence show the equation of the tangent to C at the point where space theta equals space pi over 12 space spaceis
      2 square root of 2 x plus 3 y minus open parentheses 9 square root of 3 plus 6 close parentheses equals 0

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6
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3 marks

A curve C has parametric equations

       space x equals 2 t minus 1 space     space y equals 4 t squared plus 3

Find a Cartesian equation for the curve C in the form y equals f open parentheses x close parentheses.

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7
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5 marks

Use calculus and the substitution space x equals cos space theta to find the exact value of

      integral subscript fraction numerator 1 over denominator square root of 2 end fraction end subscript superscript fraction numerator square root of 3 over denominator 2 end fraction end superscript fraction numerator 1 over denominator square root of 1 minus x squared end root space end fraction straight d x

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8
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6 marks

Find the coordinates of the point on the line bold r equals 2 bold i minus 12 bold j plus 3 bold k plus s open parentheses bold i minus 6 bold j plus 4 bold k close parentheses space that is closest to the point  P open parentheses 2 comma space 3 comma space minus 1 close parentheses comma  and hence determine the minimum distance from point P to the line.

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9
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6 marks

Use integration by parts to find, in terms of e, the exact value of

      integral subscript 0 superscript 1 left parenthesis 5 x minus 4 right parenthesis e to the power of 3 x end exponent space straight d x

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10
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5 marks

The diagram below shows the graph of the curve with equation y equals 4 minus x squared.

q7-6-2-medium-cie-a-level-maths

(i)
Find the x-coordinates of the points where  the graph of y space equals space 4 space minus space x squaredintercepts the x-axis.
(ii)
The shaded region, R, is to be rotated 360 degree around the  x-axis.
Find the volume of the shape generated.

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11a
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2 marks

A large weather balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m3) and time (seconds) write down a differential equation to describe the relationship between volume and time as the weather balloon is inflated.

11b
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6 marks

Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m3, find the particular solution to your differential equation.

11c
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2 marks

Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m3.  If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?

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12
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4 marks

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

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