Edexcel International A Level Maths: Pure 3

Topic Questions

6.1 Numerical Methods

1a
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2 marks

The diagram below shows part of the graphspace y equals straight f left parenthesis x right parenthesis space where straight f left parenthesis x right parenthesis equals 2 x squared minus 2 x cubed plus 3.

q1a-10-1-solving-equations-easy-a-level-maths-pure

(i)
Find straight f open parentheses 1.5 close parentheses
(ii)
Findspace straight f left parenthesis 1.6 right parenthesis
1b
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3 marks

Write down an interval, in the form a less than alpha space less than b, such that straight f open parentheses alpha close parentheses equals 0, explain clearly your choice of values for a and b.

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2
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3 marks

A solution to the equation straight f open parentheses x close parentheses equals 0 is x equals 3.1, correct to two significant figures.

(i)
Write down the lower bound, l, and the upper bound, u, of 3.1.

(ii)
Assuming straight f left parenthesis x right parenthesis is continuous in the interval l less than x less than u, what can you say about the values of straight f left parenthesis u right parenthesis and straight f open parentheses l space close parentheses?

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3a
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2 marks

Show that the equation space x cubed minus 5 x equals 2 spacecan be rewritten as

               x equals 1 fifth left parenthesis x cubed minus 2 right parenthesis.

3b
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3 marks

Starting with x subscript 0 equals 1, use the iterative formula

            x subscript n plus 1 end subscript equals 1 fifth open parentheses space x subscript n superscript 3 minus 2 close parentheses

to find values for x subscript 1 comma space x subscript 2 and x subscript 3, giving each to four decimal places where appropriate.

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4
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3 marks

The graph ofspace y equals straight f space open parentheses theta close parentheses where straight f left parenthesis theta right parenthesis space equals sec space theta is shown below. θ is measured in radians and -pi less or equal than theta less or equal than pi.

q7-10-1-solving-equations-easy-a-level-maths-pure

Given that sec space theta equals fraction numerator 1 over denominator cos space theta end fraction .

(i)
Find straight f left parenthesis 1.5 right parenthesis space and straight f left parenthesis 1.6 right parenthesis.
(ii)
Explain how, in this case, the change of sign rule fails to locate a root of straight f left parenthesis theta right parenthesis space in the interval (1.5 , 1.6).

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5
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3 marks

A student is trying to find a solution to the equation straight f open parentheses x close parentheses equals 0 spaceusing an iterative formula.

The student rearranges straight f open parentheses x close parentheses equals 0 space into the form x equals straight g left parenthesis x right parenthesis.

The diagram below shows a sketch of the graphs of y equals straight g left parenthesis x right parenthesis and y equals x.

q8-10-1-solving-equations-easy-a-level-maths-pure

The student is trying to find the root alpha, starting with an initial estimate x subscript 0.
Show on the diagram, how the iterative formula will converge and find the root alpha.
Mark the x-axis with the positions of x subscript 1 and x subscript 2.

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6a
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3 marks

A bypass is to be built around a village.
On the graph below the road through the village is modelled by the line  y equals x.The bypass is modelled by the equation  y equals space square root of fraction numerator 40 x over denominator x plus 1 end fraction end root.

cie-p3-q6-maths

The bypass runs from the origin to the point P left parenthesis p comma p right parenthesis.

Use the iterative formula x subscript n plus 1 end subscript equals square root of fraction numerator 40 x subscript n over denominator x subscript n plus 1 end fraction end root  with x subscript 0 equals 5 to find the value of p, correct to three significant figures.

6b
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3 marks
(i)
Calculate space straight f left parenthesis 5.835 right parenthesis and straight f left parenthesis 5.845 right parenthesis where space straight f left parenthesis x right parenthesis equals x minus space square root of fraction numerator 40 x over denominator x plus 1 end fraction end root  .

(ii)
Hence use the sign change rule to show your answer to part (a) is correct to three significant figures.

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7
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4 marks

The game of Tanball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

A particular player decides to roll the ball at an angle of  pi over 4 radians.
This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

cie-p3-q7-maths

The boundary of the winning zone is given by part of the curve with equation

 y equals 1 minus tan open parentheses square root of 3 open parentheses x plus 1 close parentheses end root close parentheses.

(i)
Using the iterative formula x subscript n plus 1 end subscript equals 1 minus tan left parenthesis square root of 3 left parenthesis x subscript n plus 1 right parenthesis end root right parenthesis comma with initial starting value x subscript 0 equals 1.5, find the estimates x subscript 1 comma x subscript 2 and x subscript 3, writing each to five decimal places.

(ii)
Continue using the iterative feature on your calculator to find the value of x correct to three significant figures.

(iii)
Write down the y-coordinate of the point where this player’s ball should cross the winning boundary, give your answer to three significant figures.

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8a
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2 marks

According to legend, unicorn tears can heal an injury almost instantly.

If a unicorn tear is applied to a burn of initialize size B space mm squared on human skin it will heal according to the model

         b open parentheses t close parentheses space equals space B minus t cubed space plus square root of t space space space space space space space space space space space space space space space space t greater or equal than 0

where b is the area of the burn, in square millimeters, at time t seconds after the unicorn tear has been applied.

Show that the equation straight b open parentheses t close parentheses equals 0 can be written as

            t equals cube root of B space plus square root of t end root.

8b
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4 marks

Use the iterative formula t subscript n plus 1 end subscript space equals cube root of 40 space plus square root of t subscript n end root end root comma with initial value t subscript 0 equals 3, to find how many seconds it takes a burn of size 40 space mm squared to heal once a unicorn tear is applied .

Give your final answer to three significant figures.

8c
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3 marks

An alternative iterative formula is t subscript n plus 1 space end subscript equals open parentheses t subscript n superscript space 3 end superscript minus B close parentheses squared,

(i)
Using t subscript 0 equals 3, find t subscript 1 comma end subscript space t subscript 2 space and space t subscript 3, for the same initial burn size as part (b), giving each to three significant figures.
(ii)
Explain how you can deduce whether this sequence of estimates is converging or diverging.

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1a
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2 marks

The diagram below shows part of the function y space equals space straight f left parenthesis x right parenthesis wherespace straight f left parenthesis x right parenthesis equals 3 x squared sin squared space x space minus 2.

q1a-10-1-solving-equations-hard-a-level-maths-pure

Correct to three significant figures, straight f open parentheses 0.9 close parentheses equals negative 0.509 andspace straight f left parenthesis 3.4 right parenthesis equals 0.265.
Explain why using the sign change rule with these values would not necessarily be helpful in finding the root close to x equals 0.98.

1b
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2 marks

Using suitable values of x, show that there is a root close to x equals 0.98.

1c
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2 marks

Show that the root close to space x equals 0.98 spaceis 0.982, correct to three significant figures.

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2a
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2 marks

The diagram below shows a sketch of the graphs y equals x, and  y equals cube root of 3 x squared plus 2 x minus 1 end root.

pmoTqjiD_q1a-10-1-solving-equations-hard-a-level-maths-pure

An iterative formula is used to find roots to the equation x cubed minus 3 x squared minus 2 x plus 1 equals 0

On the diagram above show that the iterative formula

         x subscript n plus 1 end subscript equals cube root of 3 x subscript n squared plus 2 x subscript n minus 1 space end root

would converge to the root close to space x equals 3.5 spacewhen using a starting value of x subscript 0 equals 0.5.

2b
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2 marks
(i)
Use x subscript 0 equals 0.5 in the iterative formula from part (a) to find three further approximations to the root close tospace x equals 3.5.
 Give each approximation correct to three significant figures.
(ii)
Comment on your approximations and what they suggest about convergence to the root close to x equals 3.5.
2c
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3 marks

Confirm that the root close to space x equals 3.5 spaceis 3.49 correct to three significant figures.

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3
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3 marks

The diagrams below show the graphs of four different functions.

q5a-10-1-solving-equations-hard-a-level-maths-pure

q5-2-10-1-solving-equations-hard-a-level-maths-pure

Match each graph above with the correct statement below. 

  1. The sign change rule with values of x equals 2 and x equals 4 would indicate a root but has failed due to the discontinuity (asymptote) at x equals 3.
  2. The sign change rule with values of x equals 1 and space x equals 5 spacewould indicate no root but has failed because there are two roots in the interval (1 , 5).
  3. The sign change rule with values ofspace x equals 3 space and x equals 5 would indicate no root but fail as there are two roots in the interval (3 , 5).
  4. The sign change rule with values of x equals 3 and x equals 5 would indicate no root but has failed to find the root as the graph has a turning point at x equals alpha.

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4a
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2 marks

The diagram below shows the graphs ofspace y equals x space and y equals g left parenthesis x right parenthesis.

QpsJxjGx_q1a-10-1-solving-equations-hard-a-level-maths-pure

Show on the diagram, using the value of x subscript 0 indicated, how an iterative process will lead to a sequence of estimates that converge to the x-coordinate of the point P.
Mark the estimates space x subscript 1and x subscript 2 on your diagram.

4b
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3 marks

By finding a suitable iterative formula, use x subscript 0 equals 2 to estimate a root to the equationx minus sin space 0.8 x equals 2.5 correct to two significant figures.

4c
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2 marks

Confirm that your answer to part (b) is correct to two significant figures.

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5a
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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.

q1-10-2-edexcel-modelling-with-numerical-methods-hard

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 left parenthesis p comma p right parenthesis.

On the diagram show how using the iterative formula x subscript n plus 1 end subscript equals x subscript n squared sin space x

with 0 less than x subscript 0 less than p will lead to convergence at the southern roundabout

 

5b
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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 comma to find the position of the roundabout at P to four significant figures.

5c
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2 marks

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

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6a
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3 marks

The game of Logball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

A particular player decides to roll the ball at an angle of 45°, as illustrated in the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

The lower boundary of the winning zone has equation  y equals 3 minus ln left parenthesis x plus 1 right parenthesis squared,  x comma y greater or equal than 0 

 The upper boundary of the winning zone has equation y equals 4 minus ln open parentheses x plus 1 close parentheses squared  ,  x comma y greater or equal than 0

q2-10-2-edexcel-modelling-with-numerical-methods-hard

Using an appropriate iterative formula with initial value x subscript 0 equals 1.8 comma, find the minimum distance this player’s ball needs to travel to stop within the winning zone. Give your answer to two significant figures.

6b
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3 marks

Using another iterative formula with initial value x subscript 0 equals 2.5 comma, find the maximum distance this player’s ball can travel yet remain within the winning zone.
Give your answer to two significant figures.

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7a
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3 marks

According to legend, unicorn tears have magical healing powers.

When a unicorn tear is applied to a bruise of size A space mm squared it will heal according to the model

         straight f open parentheses t close parentheses equals A e to the power of negative 0.15 t end exponent minus space 0.2 t space space space space space 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 straight f is the area of the bruise, in square millimeters, at time t seconds after the unicorn tear is applied.

Show that, for a bruise of initial size 20 space mm squared, the equation straight f open parentheses t close parenthesesequals 0 can be rearranged into the form

            t equals 20 over 3 In open parentheses 100 over t close parentheses.

7b
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3 marks

Using the equation from part (a) as an iterative formula and initial value t subscript 0 equals 12 comma find how many seconds it takes a bruise of siae 20 space mm squared to heal once a unicorn tear is applied. Give your answer to three significant figures. 

7c
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2 marks

It is rumoured that a unicorn tear can heal bruises one-hundred-thousand times faster than they would heal naturally. Approximately how many days would it take a bruise of initial size 20 space mm squared to heal without a unicorn tear?

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1a
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3 marks

The diagram below shows part of the graph space y equals straight f left parenthesis x right parenthesis spacewhere space straight f left parenthesis x right parenthesis equals 2 x space cos space left parenthesis 3 x right parenthesis minus 1.

q1a-10-1-solving-equations-medium-a-level-maths-pure

(i)
Find straight f open parentheses 1.6 close parentheses and straight f open parentheses 1.7 close parentheses, giving your answers to three significant figures.
(ii)
Briefly explain the significance of your results from part (i).
1b
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3 marks

One of the solutions to the equation straight f open parentheses x close parentheses equals 0 is x equals 2.55, correct to three significant figures.

(i)
Write down the upper and lower bound of 2.55.
(ii)
Hence, use the sign change rule to confirm that this is a solution
(to three significant figures) to the equation straight f open parentheses x close parentheses equals 0.

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2a
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2 marks

Show that the equation  x cubed plus 3 equals 5 x can be rewritten as

            x equals cube root of 5 x minus 3 end root

2b
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3 marks

Starting with x subscript 0 equals 1.8, use the iterative formula

         x subscript n plus 1 end subscript equals cube root of 5 x subscript n minus 3 end root

to find a root of the equation x cubed plus 3 equals 5 x, correct to two decimal places.

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3
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2 marks

Part of the graph of y equals tan space straight theta is shown below, where straight theta is measured in radians.

q5-10-1-solving-equations-medium-a-level-maths-pureExplain why the change of sign rule would fail if attempting to locate a root of the function space straight f open parentheses straight theta close parentheses equals tan space straight theta using the values of θ = 1.55 and θ = 1.65.

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4a
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1 mark

The diagram below shows the graphs of y equals x subscript blank and space y equals ln space open parentheses x minus 1 close parentheses plus 3 space.

q6-10-1-solving-equations-medium-a-level-maths-pure

The iterative formula

         x subscript n plus 1 end subscript equals ln open parentheses space x subscript n minus 1 close parentheses plus 3

is to be used to find an estimate for a root, alpha, of the functionspace straight f left parenthesis x right parenthesis.

Write down an expression for straight f left parenthesis x right parenthesis.

4b
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2 marks

Using an initial estimate, x subscript 0 space equals space 2, show, by adding to the diagram above, which of the two points (S or T) the sequence of estimates x subscript 1 comma x subscript 2 comma x subscript 3 comma horizontal ellipsis will converge to.
Hence deduce whether alpha  is the x-coordinate of point S or point T.

4c
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2 marks

Find the estimates x subscript 1 comma x subscript 2 comma space x subscript 3 and x subscript 4, giving each to three decimal places.

4d
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2 marks

Confirm that alpha space equals space 4.146 space correct to three decimal places.

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5a
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1 mark

The village of Greendale lies on a straight road, as modelled by the line y equals x spaceon the graph below.  To ease rush hour congestion, a bypass is to be built around Greendale.
The path of the bypass is modelled by the equation space y equals 6 square root of 1 minus fraction numerator 1 over denominator x plus 1 end fraction end root.

cie-p3-q6-maths

The bypass runs from the origin to the point P left parenthesis p comma p right parenthesis.

On the diagram show how using the iterative formula x subscript n plus 1 end subscript equals 6 square root of 1 minus fraction numerator 1 over denominator x subscript n plus 1 end fraction end root with 0 less than x subscript 0 less than p will lead to convergence at the point  P.

5b
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3 marks

Use the iterative method  x subscript n plus 1 end subscript equals 6 square root of 1 minus fraction numerator 1 over denominator x subscript n plus 1 end fraction end root with x subscript 0 equals 5 to find the value of p, correct to two decimal places.

 

5c
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2 marks

Use the sign change rule with the function straight f left parenthesis x right parenthesis equals x minus 6 square root of 1 minus fraction numerator 1 over denominator x plus 1 end fraction end root to show your answer to part (b) is correct to two decimal places.

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6a
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3 marks

The game of Curveball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

 A particular player decides to roll the ball at an angle of 45°.
This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

cie-p3-q7-maths-2

The boundary of the winning zone is given by part of the curve with equation

y equals 1 half sin squared open parentheses e to the power of x close parentheses.

Use the iterative formula x subscript n plus 1 end subscript equals 1 half sin squared left parenthesis e to the power of x right parenthesiswith initial starting value x subscript 0 equals 0.5, to show that the x-coordinate of the point where this player’s ball should cross the winning zone boundary is 0.497 to three significant figures.

6b
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2 marks

Use your answer to part (a) to find the minimum distance the ball should travel for this player to win Curveball.

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7a
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2 marks

According to legend, a unicorn can heal an injury almost instantly by touching it with its horn.

When a unicorn touches a cut in human skin of length L space mm, it will heal according to the model

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

where straight f is the length of the cut in millimeters, at time t seconds after the unicorn has touched the injury with its horn.

(i)
Write down the value straight f would be when the cut is completely healed.
(ii)
Show that, for a cut in human skin of length 5 space mm the equation straight f open parentheses t close parentheses equals 0 can be rearranged into the form
            t equals negative In open parentheses t over 5 close parentheses
7b
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4 marks

Use the iterative formula  t subscript n plus 1 end subscript equals negative In open parentheses t subscript n over 5 close parentheses, with initial value t subscript 0 equals 1 comma to find how many seconds it takes a cut of size 5 space mm to heal once a unicorn has touched it with its horn.

(i)
Write down the values of the estimates t subscript 1 comma space t subscript 2 space and space t subscript 3 to four decimal places.
(ii)
Give your final answer to two significant figures.
(iii)
State the number of iterations required for convergence to two significant figures.
7c
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1 mark

Briefly explain why the model should also restrict the range of straight f open parentheses t close parentheses to be greater than or equal to zero?

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1a
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2 marks

The diagram below shows part of the graph with equation straight f left parenthesis x right parenthesis equals x tan open parentheses space pi minus x close parentheses minus 3 .

q1a-10-1-solving-equations-very-hard-a-level-maths-pure-screenshots

A student searches for a root of the equation straight f open parentheses x close parentheses equals 0.
They find that straight f open parentheses 1.5 close parentheses equals negative 24.2 and that f open parentheses 1.6 close parentheses equals 51.8.
The student concludes that there is a root in the interval 1.5 less than x less than 1.6.
Explain why the student’s conclusion is incorrect.

1b
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1 mark

Verify that x equals 0 is a solution to the equation straight f open parentheses x close parentheses plus 3 equals 0.

1c
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1 mark

Explain why the sign change rule would fail if searching for the rootspace x equals 0 space of the equation straight f open parentheses x close parentheses plus 3 equals 0.

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2a
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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

2b
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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.

2c
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2 marks

The equation straight f open parentheses x close parentheses equals 0 has a root, alpha, close to x equals 1.
The iterative formula  x subscript n plus 1 end subscript equals e to the power of negative x subscript n end exponent plus 1 with x subscript 0 equals 2 is to be used to find alpha 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 alpha.

2d
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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?
2e
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1 mark

The root alpha  lies in the interval p less than x less than q.
Write down the values of p and q such that alpha can be deduced accurate to two decimal places from the interval.

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3
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3 marks

Sketch three separate graphs with values of space x equals p spaceand x equals q, to show how the sign change rule would fail to find a root α in the interval (p , q) for the following reasons:.

(i)

Sign change rule indicates a root but there isn’t one due to a discontinuity in the graph.

(ii)
Sign change rule indicates no root but there is a root at a turning point.
(iii)
Sign change rule indicates no root but there are in fact two roots in the interval p , q.

On each diagram, clearly labelled p, q and the root α.

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4
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2 marks

Sketch two separate diagrams to show how an iterative formula of the form

      x subscript n plus 1 end subscript equals straight g left parenthesis x subscript n right parenthesis

 can diverge in two different ways when being used to find an estimate for a root to the equation straight f left parenthesis x right parenthesis equals 0.

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5a
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1 mark

The village of Camberwick Green lies on a straight road, as modelled by the line y equals x
 on the graph below.  Rush hour traffic through the village causes both congestion and air pollution.  To ease congestion and improve the air quality around Camberwick Green a bypass , modelled by part of the equation y equals 1 plus 2 space ln left parenthesis x squared right parenthesis   is to be built.

4mI3nnPc_q1-10-2-edexcel-modelling-with-numerical-methods-hard

The bypass is to be built with a roundabout south of the village at point S and a northern roundabout which re-joins the road through Camberwick Green at the point P left parenthesis p comma p right parenthesis.

Write down the coordinates of the southern roundabout at point S.

5b
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3 marks
(i)
Show on the diagram how an iterative method with a starting value (x subscript 0) in the interval left parenthesis 1 space comma p right parenthesis space will converge to the northern roundabout at point P.

(ii)
Use a suitable iterative formula with an appropriate initial value (x subscript 0) to find the value of p to five significant figures.
5c
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2 marks

Find the length of road through Camberwick Green that will benefit from the construction of the bypass. Decide on an appropriate unit of measurement.

 

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6
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6 marks

The game of Funcball is played on a flat table. A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

The winning zone is modelled by the function

                    straight f left parenthesis x right parenthesis equals ln left parenthesis 3 x plus 4 right parenthesis minus 0.25 x squared

The lower boundary of the winning zone has equation y equals straight f left parenthesis x right parenthesis comma space space x comma y greater or equal than 0

The upper boundary of the winning zone has equation y equals straight f left parenthesis x right parenthesis plus 1 comma space space x comma y greater or equal than 0,

A particular player decides to roll the ball at an angle of 45°, as illustrated by the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

yUH2wGe6_cie-p3-q7-maths

Using iterative formulas with initial values x subscript 0 equals 1.5 and x subscript 0 equals 2.1 as appropriate, find the exact distances between which the ball must stop for this player to win Funcball.
Give your answers to two significant figures.

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7a
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3 marks

According to legend, unicorn tears have magical healing powers.

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

         straight f open parentheses t close parentheses space equals space A e to the power of negative 0.25 t end exponent space minus space 0.1 t 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 millimeters, at time t days since the bruise first appeared.

Use the iterative formula

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

with t subscript 0 equals 10 to find how many days it takes a bruise to heal without unicorn tears.

Give your answer to three significant figures.

7b
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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 space 0.1 T 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 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.
7c
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1 mark

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

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