Solving Equations (A Level only) (OCR A Level Maths: Pure)

Exam Questions

4 hours36 questions
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 x subscript 2and x subscript 3, giving each to four decimal places where appropriate.

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4a
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2 marks

The functionspace straight f left parenthesis x right parenthesis spaceis defined as

straight f open parentheses x close parentheses equals x minus e to the power of negative x end exponent    x element of straight real numbers.

Use the sign change rule to show there is a root, alpha, ofspace straight f left parenthesis x right parenthesis space in the interval 0.5 less than alpha less than 0.6.

4b
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4 marks
(i)
Find straight f apostrophe left parenthesis x right parenthesis.

(ii)
Show that, in this instance, the Newton-Raphson method would be given by the iteration

x subscript n plus 1 end subscript equals x subscript n minus fraction numerator x subscript n minus e to the power of negative x subscript n end exponent over denominator 1 plus e to the power of negative x subscript n end exponent end fraction

4c
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4 marks
(i)
Use the Newton-Raphson method with x subscript 0 equals 0.55 to find values of x subscript 1 comma x subscript 2 and x subscript 3, giving each to five decimal places.

(ii)
Use your answers to part (i) to estimate to the highest degree of accuracy possible.

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5
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6 marks

The diagram below shows part of the graph with equation y equals 5 minus 3 e to the power of negative x end exponent.

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

integral subscript 1 superscript 2 5 minus 3 e to the power of negative x end exponent space space straight d x

(i)
Given that 4 strips are to be used, calculate the width of each strip, h.

(ii)
Complete the table of values below, giving each entry correct to three significant figures.

x 

1

1.25

1.5

1.75

2

y 

3.90

   

4.48

 

(iii)
Use the trapezium rule with the values from the table in part (ii) to find an estimate of the shaded area.

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6
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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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7
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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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8
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8 marks

The diagram below shows part of the graph with equation y equals open parentheses x minus 2 close parentheses to the power of begin inline style 2 over 3 end style end exponent.

QBPQiCZA_q8-10-1-solving-equations-easy-a-level-maths-pure

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

integral subscript 4 superscript 10 open parentheses x minus 2 close parentheses to the power of 2 over 3 end exponent space straight d x

(i)

All of the values in the table below will be used in the trapezium rule.
Write down the number of ordinates that will be used, the number of strips and the width of each strip.

x 

4

5

6

7

8

9

10

y 

1.59

2.08

2.52

2.92

3.30

3.70

4.00

(ii)
Apply the trapezium rule, using the values above, to find an estimate of the shaded area.
(iii)

State, with a reason, whether your answer to part (ii) is an over-estimate or an under-estimate.

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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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3a
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2 marks

The function straight f left parenthesis x right parenthesis is defined as

 space straight f open parentheses x close parentheses equals x squared minus ln space left parenthesis x plus 2 right parenthesis space   space x greater than 0

Use the sign change rule to show there is a root to the equationspace straight f open parentheses x close parentheses equals 0 space in the interval 1 less than x less than 1.2.

3b
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2 marks

Find straight f apostrophe open parentheses x close parentheses.

3c
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4 marks

Use the Newton-Raphson method with x subscript 0 equals 1 to find the root in the interval 1 less than x less than 1.2 correct to three decimal places.

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4a
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1 mark

The diagram below shows part of the graph with equation y equals 2 to the power of ln space x end exponent.

q4-10-1-solving-equations-medium-a-level-maths-pure

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral.

integral subscript 5 superscript 10 2 to the power of ln space x end exponent space d x

Given that 4 strips are to be used, calculate h, the width of each strip.

4b
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2 marks

Complete the table of values below, giving each entry correct to three significant figures.

x 

5

6.25

7.5

8.75

10

y 

3.05

 

4.04

   
4c
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3 marks

Find an estimate of the shaded area using the values from the table in part (b).

4d
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1 mark

State whether your answer to part (c) is an overestimate or an underestimate.

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5
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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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6a
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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.

6b
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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.

6c
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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.

6d
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2 marks

Confirm that α = 4.146 correct to three decimal places.

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7a
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1 mark

The diagram below shows the graph of straight f open parentheses x close parentheses equals 2 x minus open parentheses ln space x close parentheses cubed minus 3,   x greater than 0, where α and β are roots of the functionspace straight f left parenthesis x right parenthesis.

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

The Newton-Raphson method is to be used to estimate the values of α and β.

Draw a line on the diagram to indicate a starting value (x subscript 0) that would lead the Newton-Raphson method to fail in finding either root.
(It is not required that you state the value of x subscript 0.)

7b
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3 marks

Show that

fraction numerator d y over denominator d x end fraction equals 2 minus fraction numerator 3 open parentheses ln space x close parentheses squared over denominator x end fraction

7c
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4 marks

Use the Newton-Raphson method withspace x subscript 0 equals 1 space to find β correct to five significant figures.

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8a
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2 marks

The diagram below shows the graph with equation y equals square root of e to the power of negative x end exponent plus 2 x end root.

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

The area shaded is to be estimated using the trapezium rule where h equals 1.

 

(i)
Write down the number of strips to be used.

(ii)
Write down the number of ordinates to be used.

8b
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4 marks

Apply the trapezium rule as described above to estimate the shaded area, giving your answer to three significant figures.

8c
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1 mark

Describe a way in which the estimate calculated in part (b) could be improved.

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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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3a
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2 marks

The function space straight f left parenthesis x right parenthesis spaceis defined as

 space straight f open parentheses x close parentheses equals sin space 3 x minus ln space 2 x     x greater than 0, where x is in radians.

Find straight f apostrophe left parenthesis x right parenthesis.

3b
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4 marks

Use the Newton-Raphson method with x subscript 0 equals 0.8 to find a root, α, of the equation straight f open parentheses x close parentheses equals 0, correct to four decimal places.

3c
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1 mark

The graph of y equals straight f left parenthesis x right parenthesis has a local maximum point at x equals straight beta. Briefly explain why the Newton-Raphson method would fail if the exact value of β was used for x subscript 0.

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4
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5 marks

The diagram below shows part of the graph with equation y equals 3 x minus e to the power of x squared end exponent.

q3a-10-1-solving-equations-hard-a-level-maths-pure

Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

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5
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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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6a
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2 marks

The diagram below shows the graphs of space 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.

6b
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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.

6c
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2 marks

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

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7a
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1 mark

The diagram below shows part of the graph of space y equals straight f open parentheses x close parentheses spacewhere straight f open parentheses x close parentheses equals 0.3 e to the power of sin space x end exponent space minus 0.5.

gfVjCPAq_q1a-10-1-solving-equations-hard-a-level-maths-pure

Write down the x-coordiante of the point marked M on the graph.

7b
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2 marks

The first two positive roots of the function straight f open parentheses x close parentheses, α and β, are marked on the graph above. The Newton-Raphson method is to be used to find a sequence of estimates for the root β.

Indicate on the graph above a value of x subscript 0 in the interval (α , β) that would lead to the Newton-Raphson method converging to the root (i) α  and (ii) β.

7c
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5 marks

Using the Newton-Raphson method with x subscript 0 equals 2, find four more estimates for the root β. Verify that your final estimate gives the value of β correct to five significant figures.

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8a
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2 marks

Use two separate diagrams to show how the trapezium rule can lead to an underestimate or an overestimate when used to estimate the area under a curve.

8b
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4 marks

Use the trapezium rule with space h equals 0.25 spaceto find an estimate for the area bounded by the curve with equation y equals 1 plus 0.3 x squared sin space x , the lines with equations x equals 1 and x equals 2 and the x-axis.
Give your answer to three significant figures.

8c
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1 mark

The integral 

integral subscript 1 superscript 2 left parenthesis 1 plus 0.3 x squared sin space x right parenthesis space space space d x

can be evaluated exactly by applying the method of integration by parts (twice).
Suggest a reason why it may be preferrable to use a numerical method, such as trapezium rule, to estimate the integral rather than use integration by parts to find its exact value.

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9a
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3 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis where the function straight f left parenthesis x right parenthesis is defined by

straight f open parentheses x close parentheses equals 10 minus 5 x squared minus fraction numerator 1 over denominator 2 x plus 4 end fraction    x greater than negative 2

NJMlean__q1a-10-1-solving-equations-hard-a-level-maths-pure

The function straight f left parenthesis x right parenthesis has a root close to x equals 1.4.

Using the iterative formula

x subscript n plus 1 end subscript equals square root of 2 minus fraction numerator 1 over denominator 10 x plus 20 end fraction end root

with x subscript 0 equals 1.4, find an estimate of the root near x equals 1.4 to six decimal places

9b
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3 marks

Given thatspace straight f apostrophe open parentheses x close parentheses equals fraction numerator 1 over denominator 2 left parenthesis x plus 2 right parenthesis squared end fraction minus 10 x, use the Newton-Raphson method with x subscript 0 equals 1.4 to find an estimate of the root near x equals 1.4 to six decimal places.

9c
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1 mark

Justify which of the methods in this case was more efficient at finding the root close to x equals 1.4 to six decimal places.

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10
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3 marks

Use the two diagrams below to show how rectangles can be used to give an upper and lower bound when estimating the area under a curve using the trapezium rule.

q10-1-10-1-solving-equations-hard-a-level-maths-pure

q10-2-10-1-solving-equations-hard-a-level-maths-pure

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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, 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.

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

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3a
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3 marks

The function f(x) is defined as

  straight f open parentheses x close parentheses equals 5 cos space x sin space 2 x space space minus 3 space space space space space space space space space space space space space space space space x element of straight real numbers

Show thatspace straight f apostrophe open parentheses x close parentheses equals 10 cos space x space left parenthesis 1 minus 3 sin squared x space right parenthesis.

3b
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3 marks

Use the Newton-Raphson method with x subscript 0 equals 0.3 to find a root of the equation space straight f open parentheses x close parentheses equals 0 spacecorrect to five significant figures.

3c
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1 mark

Write down the exact value of a root to the equation straight f open parentheses x close parentheses equals negative 3.

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4a
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4 marks

The trapezium rule is to be used to find an estimate for the integral

 

integral subscript 4 superscript 8 space straight f left parenthesis x right parenthesis space space straight d x

The table below shows values for x and f(x), rounded to three significant figures where appropriate.

x 

4

4.5

5

5.5

6

6.5

7

7.5

8

 f(x)

3.16

3.39

3.61

3.81

4

4.18

4.36

4.53

4.69

Using the values in the table find 

(i)      an estimate for the integral using 2 strips,
(ii)     an estimate for the integral using 4 strips,
(iii)    an estimate for the integral using 8 strips.

4b
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2 marks

Justify which of the estimates from part (a) will be the most accurate estimate for the integral.

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5
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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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6
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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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7a
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2 marks

Draw a diagram to show how the Newton-Raphson method produces a series of estimates that converge to a root, α.  On your diagram you should indicate the values α, x0, x1 and x2.

7b
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3 marks

Use the Newton-Raphson method with x subscript 0 equals 1.5 to find a solution to equation

x to the power of 5 minus 2 x to the power of 4 plus 3 x cubed minus 4 x squared plus 1 equals 0

correct to four significant figures.

7c
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2 marks

Verify that there is another solution in the interval (0.605 , 0.615) and state the value of the root to the highest degree of accuracy possible.

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8a
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4 marks

The diagram below shows the graph of y equals 4 minus 2 x to the power of ln space x end exponent space comma space space space x greater than 0.

q7a-10-1-solving-equations-very-hard-a-level-maths-pure-screenshots

Use the trapezium rule with h = 0.2 to find an estimate of the integral

integral subscript 1 superscript 2 open parentheses 4 minus 2 x to the power of ln space x end exponent close parentheses space d x

to three significant figures.

8b
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1 mark

Using the integration feature on your calculator, find the value of

integral subscript 1 superscript 2 open parentheses 4 minus 2 x to the power of ln space x end exponent close parentheses d x

Give your answer to three significant figures.

8c
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2 marks

Assuming your calculator provides the exact answer to the integral, find the percentage error of your estimate from part (a).

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9a
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3 marks

The diagram below shows the graph of space y equals straight f left parenthesis x right parenthesis spacewhere the functionspace straight f left parenthesis x right parenthesis space is defined by 

  straight f open parentheses x close parentheses equals 5 x plus 2 over x squared minus 12 space space space space space space space space space space space space space space space space space space space space space space x greater than 0

q9a-10-1-solving-equations-very-hard-a-level-maths-pure-screenshots

The function f(x) has a root close to x equals 0.4.

Estimates for this root could be found using iteration or the Newton-Raphson method.

(i)
Suggest a suitable starting value left parenthesis x subscript 0 right parenthesis for both methods.
(ii)
Rearrange  f(x) into the form x equals straight g left parenthesis x right parenthesis
(iii)
Find an expression for f '(x)

9b
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3 marks

Using your answers to part (a) use an iterative method to find the root of straight f open parentheses x close parentheses close to x equals 0.4 to four decimal places.

9c
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3 marks

Using your answers to part (a) use the Newton-Raphson method to find the root of straight f left parenthesis x right parenthesis close to x equals 0.4 to four decimal places.

9d
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1 mark

Comment on the efficiency of the two methods in finding the root close tospace x equals 0.4 space to four decimal places.

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10a
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3 marks

The diagram below shows a sketch of the graph of y equals x open parentheses x minus 6 close parentheses squared space space space space space space space space space space space space space space space space x greater or equal than 0

q10a-10-1-solving-equations-very-hard-a-level-maths-pure-screenshots

The graph has a local maximum point at (2 , 32) as indicated on the diagram.

Use the trapezium rule with 5 ordinate values to estimate the area shaded.

10b
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3 marks

Using the appropriate working values from part (a), find an upper and lower bound for the area shaded.

10c
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1 mark

Suggest a reason why using the trapezium rule in this case is not appropriate.

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