Practice Paper Pure 3 (Edexcel International A Level Maths: Mechanics 1)

Practice Paper Questions

1
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5 marks

Solve the equation

cos 2 theta equalscos theta minus 1 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

State your answers as multiples of straight pi.

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2a
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2 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 amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, t hours after the drug was administered by injection. A  and k are constants.

The graph below shows values of In D plotted against t with a line of best fit drawn.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-medium

(i)        Use the graph and line of best fit to estimate ln space D at time t equals 0.

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

2b
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1 mark

Use your answers to part (a) to write down an equation for the line of best fit in the form ln space D equals m t plus ln space c,  where m and c are constants.

2c
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1 mark

Show that D equals A e to the power of negative k t end exponent can be rearranged to give ln space D equals negative k t plus ln space A

2d
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2 marks

Hence find estimates for the constants A spaceand k.

2e
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2 marks

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

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3
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6 marks

The diagram below shows the graph of y equals straight f open parentheses x close parentheses, where straight f open parentheses x close parentheses is the function defined by

space straight f open parentheses x close parentheses equals fraction numerator sin space 3 x over denominator e to the power of 2 x minus 3 end exponent end fraction space comma space   space space space space space space space space space 0 less or equal than x less or equal than fraction numerator 2 pi over denominator 3 end fraction

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

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

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4a
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3 marks

On the same axes, sketch the graphs of y equals vertical line straight f left parenthesis x right parenthesis vertical line  and y equals vertical line g left parenthesis x right parenthesis vertical line   where

straight f left parenthesis x right parenthesis space equals space 3 x space minus space 1 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 2 x space plus space 2 space space space space space space space space space space x element of straight real numbers

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

4b
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3 marks

Solve the equation vertical line straight f left parenthesis x right parenthesis vertical line space equals space vertical line g left parenthesis x right parenthesis vertical line.

4c
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1 mark

Which of the solutions to vertical line straight f left parenthesis x right parenthesis vertical line space equals space vertical line g left parenthesis x right parenthesis vertical line is also a solution to straight f left parenthesis x right parenthesis space equals space g left parenthesis x right parenthesis?

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5a
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3 marks

The minimum point on the graph of y equals straight f left parenthesis x right parenthesis has coordinates left parenthesis 4 space comma negative 8 right parenthesis as shown on the diagram below.

q10a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-hard

Sketch the graph of y equals vertical line straight f left parenthesis 2 x right parenthesis vertical line minus 3 and state the coordinates of the maximum point.

5b
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2 marks

Find the exact distance between the minimum point on the graph of y equals straight f left parenthesis x right parenthesis and the maximum point on the graph of y equals vertical line straight f left parenthesis 2 x right parenthesis vertical line minus 3.

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6a
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3 marks

Show that, for theta not equal to k pi (where k is an integer),

fraction numerator 2 minus 2 space cos squared theta over denominator sin space 2 theta end fraction space space equals tan space theta

6b
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4 marks

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

integral subscript straight pi over 6 end subscript superscript straight pi over 3 end superscript space fraction numerator 2 minus 2 cos squared theta space over denominator sin space 2 theta end fraction space straight d theta equals 1 half ln space 3 space

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7a
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1 mark

The village of Crinkley Bottom lies on a straight road, as modelled by the line y equals x 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 y equals x squared sin space x .

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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 P open parentheses p comma space p close parentheses.

On the diagram show how using the iterative formula  x subscript n plus 1 end subscript equals x subscript n superscript 2 sin space x spacewith 0 less than x subscript 0 less than p will lead to convergence at the southern roundabout 

7b
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3 marks

Use the alternative iterative method

x subscript n plus 1 end subscript equals square root of fraction numerator x over denominator sin space x end fraction end root

with x subscript 0 equals 1, to find the position of the roundabout at P to four significant figures.

7c
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2 marks

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

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8a
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3 marks

Show that 6 cos theta minus 8 sin theta can be written in the form  R cosopen parentheses theta plus straight alpha close parentheses, where R greater than 0 and alpha is an acute angle measured in radians.

8b
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3 marks

Hence, or otherwise, solve the equation 3 cos theta minus 4 sin theta minus 2 equals 0,for 0 less or equal than space x less or equal than 2 straight pi
Give your answers to three significant figures.

8c
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2 marks

Write down the minimum value of 6 cos theta minus 8 sin theta and the smallest positive value of theta for which it occurs.  Give your value of theta  to three significant figures.

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9a
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3 marks

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

open parentheses 2 sin space 3 x space minus cos space 3 x close parentheses space e to the power of 6 minus x space end exponent

9b
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3 marks

open parentheses x squared minus x close parentheses squared ln space 5 x

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10
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5 marks

A is the point on the graph of  space y equals arctan space x spacesuch that the tangent to the graph at A passes through the point open parentheses 0 comma space space 1 half close parentheses.  Show that the x-coordinate of A satisfies the equation

x minus tan space open parentheses fraction numerator open parentheses 1 plus x close parentheses squared over denominator 2 open parentheses 1 plus x squared close parentheses space end fraction close parentheses equals 0

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11
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5 marks

It is given that

fraction numerator left parenthesis x plus a right parenthesis cubed over denominator x space plus space b end fraction equals left parenthesis x squared plus 7 x plus 13 right parenthesis plus fraction numerator 1 over denominator x space plus space b end fraction

where a and b are integers.

Find the values of a space and space b.

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12
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7 marks

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

               integral subscript 0 superscript 1 space fraction numerator 2 a over denominator a x plus b end fraction space straight d x space equals integral subscript b superscript 2 a end superscript space 1 over x space straight d x

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