Numerical Methods (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

4 hours44 questions
1a
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2 marks

The figure below shows the curve with equation y equals straight f open parentheses x close parentheses where

straight f left parenthesis x right parenthesis equals 2 x squared minus 2 x cubed plus 3

  • The equation straight f open parentheses x close parentheses equals 0 has only one solution, x equals alpha

  • You may assume that straight f open parentheses x close parentheses is continuous for all values of x

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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1 mark

Use part (a) to write down an interval containing the root alpha, in the form

a less than alpha less than b

where a and b are constants to be found.

2
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3 marks

A function straight f open parentheses x close parentheses is continuous for all values of x.

The equation straight f open parentheses x close parentheses equals 0has only one solution, x equals 3.1, correct to 2 significant figures.

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

(ii) Write down a statement about the signs of straight f left parenthesis u right parenthesis and straight f open parentheses l space close parentheses.

3a
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1 mark

Show that the equation

x cubed minus 5 x equals 2

can be written 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 the values of x subscript 1 comma space x subscript 2and x subscript 3.

Give your answers correct to 4 decimal places, where necessary.

4a
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2 marks

The functionspace straight f left parenthesis x right parenthesis spaceis given by

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

Show there is a root, alpha, of the equation straight f open parentheses x close parentheses equals 0 in the interval 0.5 less than x less than 0.6.

4b
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2 marks

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

(ii) Show that the Newton-Raphson method is given by the iteration formula

x subscript n plus 1 end subscript equals x subscript n minus fraction numerator x subscript n minus straight e to the power of negative x subscript n end exponent over denominator 1 plus straight 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 the values of x subscript 1 comma space x subscript 2 and x subscript 3, giving your answers correct to 5 decimal places.

(ii) Assuming that the answers are converging to alpha, use the unrounded values of x subscript 2 and x subscript 3 to estimate alpha to the highest degree of accuracy possible.

5
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6 marks

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

Graph showing a curve extending rightwards with a shaded area between x=1 and x=2 under the curve. Axes marked as x and y.

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

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

(i) Given that 4 trapezia of equal width are used, calculate the width of one trapezium, h.

(ii) Complete the table of values below, giving each value correct to 3 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, giving your answer correct to 2 significant figures.

6
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3 marks

The figure below show the curve with equation y equals straight f open parentheses theta close parentheses where

  • straight f left parenthesis theta right parenthesis space equals fraction numerator 1 over denominator cos space theta end fraction

  • theta is measured in radians

  • negative pi less or equal than theta less or equal than pi

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

(i) Find straight f left parenthesis 1.5 right parenthesis and straight f left parenthesis 1.6 right parenthesis.

A student claims that the answers to part (i) show that a root of straight f open parentheses theta close parentheses equals 0 lies in the interval open square brackets 1.5 comma space 1.6 close square brackets.

(ii) Explain why the student is incorrect.

7
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3 marks

The solution to the equation straight f open parentheses x close parentheses equals 0 is x equals alpha.

The equation straight f open parentheses x close parentheses equals 0 can be rearranged to x equals straight g open parentheses x close parentheses.

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

Starting with an initial estimate of x subscript 0, show on the diagram how the iteration formula

x subscript n plus 1 end subscript equals straight g open parentheses x subscript n close parentheses

converges to alpha.

Indicate, on the x-axis, the positions of x subscript 1 and x subscript 2.

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 used to estimate the area of the shaded region shown above, given by

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) If all the y-values in the table below are used, write down the number of x-values, the number of trapezia and the width of each trapezium.

x

4

5

6

7

8

9

10

y 

1.59

2.08

2.52

2.92

3.30

3.70

4.00

(ii) Use the trapezium rule to find an estimate of the shaded area.

(iii) State, with a reason, whether your answer to part (ii) is an overestimate or an underestimate.

1a
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3 marks

The table below shows corresponding values of x and y for y equals square root of fraction numerator x over denominator 1 plus x end fraction end root

The values of y are given to 4 significant figures.

x

0.5

1

1.5

2

2.5

y

0.5774

0.7071

0.7746

0.8165

0.8452

Use the trapezium rule, with all the values of y in the table, to find an estimate for

integral subscript 0.5 end subscript superscript 2.5 end superscript square root of fraction numerator x over denominator 1 plus x end fraction end root space straight d x

giving your answer to 3 significant figures.

1b
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1 mark

Using your answer to part (a), deduce an estimate for integral subscript 0.5 end subscript superscript 2.5 end superscript square root of fraction numerator 9 x over denominator 1 plus x end fraction end root space straight d x

1c1 mark

Given that

integral subscript 0.5 end subscript superscript 2.5 end superscript square root of fraction numerator 9 x over denominator 1 plus x end fraction end root space straight d x equals 4.535 to 4 significant figures

comment on the accuracy of your answer to part (b).

2a
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2 marks

The curve with equation y equals 2 ln open parentheses 8 minus x close parentheses meets the line y equals x at a single point, x equals alpha.

Show that 3 less than alpha less than 4.

2b2 marks
Graph with x and y axes, showing intersecting lines y=x and y=2ln(8-x). Axes marked at 4; curves meet at point (4,4).
Figure 2

Figure 2 shows the graph of y equals 2 ln open parentheses 8 minus x close parenthesesand the graph of y equals x.

A student uses the iteration formula

x subscript n plus 1 end subscript equals 2 ln open parentheses 8 minus x subscript n close parentheses comma space space space space space space n element of straight natural numbers

in an attempt to find an approximation for alpha.

Using the graph and starting with x subscript 1 equals 4, determine whether or not this iteration formula can be used to find an approximation for alpha, justifying your answer.

3a4 marks

The curve with equation y space equals space straight f left parenthesis x right parenthesis where

straight f open parentheses x close parentheses equals x squared plus ln open parentheses 2 x squared minus 4 x plus 5 close parentheses

has a single turning point at x space equals space alpha.

Show that alpha is a solution of the equation 2 x cubed minus 4 x squared plus 7 x minus 2 equals 0.

3b
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3 marks

The iterative formula

x subscript n plus 1 end subscript equals 1 over 7 open parentheses 2 plus 4 x subscript n squared minus 2 x subscript n cubed close parentheses

is used to find an approximate value for alpha.

Starting with x subscript 1 equals 0.3, calculate, giving each answer to 4 decimal places,

(i) the value of x subscript 2

(ii) the value of x subscript 4

3c
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2 marks

Using a suitable interval and a suitable function that should be stated, show that alpha is 0.341 to 3 decimal places.

4a
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3 marks

A continuous curve has equation y equals straight f left parenthesis x right parenthesis.

The table shows corresponding values of x and y for this curve, where a and b are constants.

x

3

3.2

3.4

3.6

3.8

4

y

a

16.8

b

20.2

18.7

13.5

The trapezium rule is used, with all the y values in the table, to find an approximate area under the curve between x equals 3 and x equals 4

Given that this area is 17.59, show that a plus 2 b equals 51

4b
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3 marks

Given also that the sum of all the y values in the table is 97.2, find the value of a and the value of b.

5a3 marks

The equation 2 x cubed plus x squared minus 1 equals 0 has exactly one real root.

Show that, for this equation, the Newton-Raphson formula can be written

x subscript n plus 1 end subscript equals fraction numerator 4 x subscript n cubed plus x subscript n squared plus 1 over denominator 6 x subscript n squared plus 2 x subscript n end fraction

5b
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2 marks

Using the formula given in part (a) with x subscript 1 equals 1, find the values of x subscript 2 and x subscript 3

5c1 mark

Explain why, for this question, the Newton-Raphson method cannot be used with x subscript 1 equals 0

6a
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4 marks
Graph of a curve with x and y axes. The curve crosses the x-axis at point α and later has a local maximum marked P in the fourth quadrant. Origin is labelled O.
Figure 2

Figure 2 shows a sketch of part of the curve with equation y equals straight f open parentheses x close parentheses where

straight f open parentheses x close parentheses equals 8 sin open parentheses 1 half x close parentheses minus 3 x plus 9 space space space space space space space space space space space space space x greater than 0

and x is measured in radians.

The point P, shown in Figure 2, is a local maximum point on the curve.

Using calculus and the sketch in Figure 2, find the x coordinate of P, giving your answer to 3 significant figures.

6b1 mark

The curve crosses the x-axis at x equals alpha, as shown in Figure 2.

Given that, to 3 decimal places, straight f left parenthesis 4 right parenthesis equals 4.274 and straight f left parenthesis 5 right parenthesis equals negative 1.212, explain why α must lie in the interval open square brackets 4 comma space 5 close square brackets

6c
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2 marks

Taking x subscript 0 equals 5 as a first approximation to alpha, apply the Newton-Raphson method once to straight f left parenthesis x right parenthesis to obtain a second approximation to alpha.

Show your method and give your answer to 3 significant figures.

7a
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2 marks

The diagram below shows part of the graph y equals straight f left parenthesis x right parenthesis where

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

Show that a solution to the equation straight f open parentheses x close parentheses equals 0 exists in the interval 1.6 less than x less than 1.7

7b
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3 marks

Using a suitable interval. show that x equals 2.55 is a solution to the equation straight f open parentheses x close parentheses equals 0, correct to 3 significant figures.

8a
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2 marks

Show that the equation x cubed plus 3 equals 5 x can be written in the form

x equals cube root of a x minus b end root

where a and b are integers to be found.

8b
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1 mark

Taking x subscript 0 equals 1.8 as the first approximation, use the iteration formula

x subscript n plus 1 end subscript equals cube root of a x subscript n minus b end root

with your values of a and b in part (a) to find, by repeated iteration, a solution to the equation

x cubed plus 3 equals 5 x

9a
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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 space space space space space space space space space space space x greater than 0

Show that there is a solution to the equation straight f open parentheses x close parentheses equals 0 in the interval 1 less than x less than 1.2

9b
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2 marks

Find an expression for straight f apostrophe open parentheses x close parentheses

9c
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2 marks

Taking x subscript 0 equals 1 as the first approximation, apply the Newton-Raphson method repeatedly to find a solution to the equation straight f open parentheses x close parentheses equals 0 in the interval open square brackets 1 comma space 1.2 close square brackets.

Give your answer correct to 3 decimal places.

10a
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1 mark

The diagram below shows part of the curve with the 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 used to estimate the area of the shaded region, given by

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

Four equally spaced trapezia are used, each of width h.

Find h.

10b
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2 marks

Complete the table of values below, correct to 3 significant figures.

x

5

6.25

7.5

8.75

10

y

3.05

 

4.04

 

 

10c
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3 marks

Using the trapezium rule with all the values of y in the table, find an estimate for

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

10d
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1 mark

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

11
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2 marks

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

q5-10-1-solving-equations-medium-a-level-maths-pure

A student uses a change of sign argument to show that the interval open square brackets 1.55 comma space 1.56 close square brackets contains a solution to the equation tan space theta equals 0

Explain whether, or not, this is a valid method.

12a
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1 mark

The figure below shows the line y equals x and the curve y equals ln space open parentheses x minus 1 close parentheses plus 3

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

The iteration formula

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

with x subscript 0 equals 2 is used to find an estimate for a root, alpha, of the equationspace straight f left parenthesis x right parenthesis equals 0.

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

12b
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2 marks

Draw a staircase diagram on the graph in part (a) to determine whether the iteration formula starting with x subscript 0 equals 2 finds an approximation for the x-coordinate of point S or the x-coordinate of point T.

12c
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2 marks

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

12d
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2 marks

Using a suitable interval, show that alpha equals 4.146 to 3 decimal places.

13a
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1 mark

The graph of y equals straight f open parentheses x close parentheses where

straight f open parentheses x close parentheses equals 2 x minus open parentheses ln space x close parentheses cubed minus 3 space space space space space space space space space space x greater than 0

is shown below, where alpha and beta are solutions to the equation straight f left parenthesis x right parenthesis equals 0

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

The Newton-Raphson method is to be used to estimate the values of alpha and beta.

Indicate on the diagram the starting value, x subscript 0, that would lead to the Newton-Raphson method failing to find either solution, alpha or beta.

[You do not need to calculate the value of x subscript 0.]

13b
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3 marks

Find an expression for straight f apostrophe open parentheses x close parentheses.

13c
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3 marks

Apply the Newton-Raphson method with x subscript 0 equals 1 to find beta.

Give your answer to 5 significant figures.

14a
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4 marks

The curve with equation y equals square root of straight e to the power of negative x end exponent plus 2 x end root is shown below.

The shaded area is represented by

integral subscript 2 superscript 8 square root of straight e to the power of negative x end exponent plus 2 x end root d x

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

Use the trapezium rule with 6 equally spaced trapezia to find an estimate for the area of the shaded region.

Give your answer to 3 significant figures.

14b
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1 mark

Explain how the accuracy of the estimate in part (a) can be improved.

15a
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2 marks

Sketch two separate diagrams to show how the trapezium rule can lead to either an underestimate or an overestimate.

15b
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4 marks

Use the trapezium rule with step size h equals 0.25 to find an estimate for the area bounded by the curve with equation y equals 1 plus 0.3 x squared sin space x and the lines x equals 1, x equals 2 and the x-axis.

Give your answer to 3 significant figures.

16a
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4 marks

The trapezium rule is to be used to approximate

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

The table below shows values of x and straight f open parentheses x close parentheses.

x 

4

4.5

5

5.5

6

6.5

7

7.5

8

 straight f open parentheses x close parentheses

3.16

3.39

3.61

3.81

4

4.18

4.36

4.53

4.69

Using the values in the table, find an estimate for the integral using

(i) 2 trapezia,
(ii) 4 trapezia,
(iii) 8 trapezia.

16b
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1 mark

Explain which estimate from part (a) is likely to be the most accurate approximation of

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

1a
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3 marks

The table below shows corresponding values of x and y for y equals log subscript 3 2 x

The values of y are given to 2 decimal places as appropriate.

x

3

4.5

6

7.5

9

y

1.63

2

2.26

2.46

2.63

Using the trapezium rule with all the values of y in the table, find an estimate for

integral subscript 3 superscript 9 log subscript 3 2 x space straight d x

1b
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3 marks

Using your answer to part (a) and making your method clear, estimate

(i) space integral subscript 3 superscript 9 log subscript 3 open parentheses 2 x close parentheses to the power of 10 space straight d x

(ii) space integral subscript 3 superscript 9 log subscript 3 18 x space straight d x

2a4 marks
Graph in the first quadrant of convex (i.e. "concave up") curve C,  with a minimum turning point marked at point P . Axes are labelled x and y, with origin O at their intersection.
Figure 1

Figure 1 shows a sketch of the curve C with equation

y equals fraction numerator 4 x squared plus x over denominator 2 square root of x end fraction minus 4 ln x space space space space space space space space space space space x greater than 0

Show that

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 12 x squared plus x minus 16 square root of x over denominator 4 x square root of x end fraction

2b3 marks

The point P, shown in Figure 1, is the minimum turning point on C.

Show that the x coordinate of P is a solution of

x equals open parentheses 4 over 3 minus fraction numerator square root of x over denominator 12 end fraction close parentheses to the power of 2 over 3 end exponent

2c
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3 marks

Use the iteration formula

x subscript n plus 1 end subscript equals open parentheses 4 over 3 minus fraction numerator square root of x subscript n end root over denominator 12 end fraction close parentheses to the power of 2 over 3 end exponent space space space space space space space space space space space spacewith x subscript 1 equals 2

to find

(i) the value of x subscript 2 to 5 decimal places,

(ii) the x coordinate of P to 5 decimal places.

3a
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2 marks
Graph showing a curve labelled 'C' in the first quadrant, starting at the origin and curving upwards to the right. Axes are labelled 'x' and 'y'.
Figure 8

Figure 8 shows a sketch of the curve C with equation y equals x to the power of x comma space space space x greater than 0.

The point P open parentheses alpha comma space 2 close parentheses lies on C.

Show that 1.5 less than alpha less than 1.6.

3b
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2 marks

A possible iteration formula that could be used in an attempt to find alpha is

x subscript n plus 1 end subscript equals 2 x subscript n to the power of 1 minus x subscript n end exponent

Using this formula with x subscript 1 equals 1.5, find x subscript 4 to 3 decimal places.

3c
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2 marks

Describe the long-term behaviour of x subscript n.

4a
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1 mark

The diagram below shows part of the curve y space equals space straight f left parenthesis x right parenthesis where

straight f left parenthesis x right parenthesis equals 3 x squared sin squared space x space minus 2 space space space space space space space space minus fraction numerator 3 pi over denominator 2 end fraction less than x less than fraction numerator 3 pi over denominator 2 end fraction

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

You are given that straight f open parentheses 0.9 close parentheses equals negative 0.509 andspace straight f left parenthesis 3.4 right parenthesis equals 0.265, to 3 significant figures,

A student wishes to find an estimate of the root of straight f open parentheses x close parentheses equals 0 that is close to x equals 0.98

Explain why a change of sign in the interval open square brackets 0.9 comma space 3.4 close square brackets is not necessarily helpful to the student.

4b
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2 marks

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

4c
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2 marks

Show that the root close to x equals 0.98 is 0.982, correct to 3 significant figures.

5a
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2 marks

The graph below shows a sketch of the line y equals x and the curve  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 iteration formula is used to find the three roots of the equation

x cubed minus 3 x squared minus 2 x plus 1 equals 0

Draw a staircase diagram on the graph above to show that the iteration 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

with a starting value of x subscript 0 equals 0.5 converges to the largest positive root of the equation.

5b
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3 marks

(i) Use the iteration formula from part (a) with x subscript 0 equals 0.5 to find x subscript 1 comma space x subscript 2 and x subscript 3, to 3 significant figures.

(ii) If the root is close to x equals 3.5, describe the speed of convergence to the root of the values x subscript 1 comma space x subscript 2 and x subscript 3.

5c
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2 marks

Show that the root close to x equals 3.5 is 3.49, correct to 3 significant figures.

6a
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5 marks

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

 space straight f open parentheses x close parentheses equals sin space 3 x minus ln space 2 x space space space space space space space space space x greater than 0

where x is in radians.

Apply the Newton-Raphson method with a starting approximation of x subscript 0 equals 0.8 to find a solution to the equation

sin space 3 x equals ln space 2 x

giving your answer correct to 4 decimal places.

6b
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1 mark

The graph of y equals straight f left parenthesis x right parenthesis has a local maximum point with coordinates open parentheses beta comma space straight f open parentheses beta close parentheses close parentheses.

Describe what happens when applying the Newton-Raphson method with a starting approximation of x subscript 0 equals beta.

7
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5 marks

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

The area between the curve and the x-axis from x equals 0.5 to x equals 1 is shaded.

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

Use the trapezium rule with 5 equally spaced trapezia to find an estimate for the area of the shaded region.

Give your answer to 3 significant figures.

8
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3 marks

Graphs of y equals straight f open parentheses x close parentheses for four different functions are shown below.

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. straight f open parentheses x close parentheses is not continuous and it is possible to have an interval showing no sign change that contains exactly one root

  2. straight f open parentheses x close parentheses is not continuous and it is possible to have an interval showing no sign change that contains more than one root

  3. straight f open parentheses x close parentheses is continuous and it is possible to have an interval showing no sign change that contains exact one root

  4. straight f open parentheses x close parentheses is continuous and it is possible to have an interval showing no sign change that contains more than one root

9a
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2 marks

The diagram below shows the line y equals x and the curve y equals straight g left parenthesis x right parenthesis.

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

Draw a cobweb diagram on the graph above, using the starting approximation x subscript 0 indicated.

You must show

  • the first two estimates, x subscript 1 and x subscript 2

  • convergence to a root of the equation x minus straight g open parentheses x close parentheses equals 0

9b
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3 marks

By forming a suitable iteration formula with x subscript 0 equals 2 and using repeated iteration, find a root of the equation

x minus sin space 0.8 x equals 2.5

correct to 2 significant figures.

9c
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2 marks

Using a suitable interval and a suitable function that should be stated, show that your answer to part (b) is correct to 2 significant figures.

10a
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1 mark

The diagram below shows part of the graph of y equals straight f open parentheses x close parentheses where

straight f open parentheses x close parentheses equals 0.3 straight e to the power of sin space x end exponent minus 0.5

Two roots of the equation straight f open parentheses x close parentheses equals 0 are shown, alpha and beta.

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

Write down the x-coordinate of the maximum point, M, on the graph.

10b
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6 marks

The Newton-Raphson method is applied to find an estimate for the root beta.

The starting approximation, x subscript 0, is the smallest positive integer value greater than the x-coordinate of the maximum point M.

Find the first four estimates and use a suitable interval to find beta correct to 5 significant figures.

11a
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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 space space space space space space space 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 positive root close to x equals 1.4

An iteration formula is given by

x subscript n plus 1 end subscript equals square root of k minus fraction numerator 1 over denominator 10 x plus 20 end fraction end root to the power of blank

where k is an integer to be found.

Use repeated iteration to find an estimate of the positive root, to 6 decimal places, using a starting value of x subscript 0 equals 1.4

11b
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5 marks

Apply the Newton-Raphson method with x subscript 0 equals 1.4 to find an estimate of the positive root, to 6 decimal places.

11c
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1 mark

Compare the rates at which the estimates converge between the different methods in part (a) and part (b).

12a
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1 mark

The function, straight f left parenthesis x right parenthesis, is defined by

straight f open parentheses x close parentheses equals 1 over straight 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 straight e to the power of a x end exponent plus b

where a and b are integers to be found.

12b
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2 marks

On the same diagram, sketch the graphs of y equals x and y equals straight e to the power of a x end exponent plus b, using your values of a and b from part (a).

12c
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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.

Draw a cobweb diagram on the graph in part (b) to show how the iteration formula

x subscript n plus 1 end subscript equals e to the power of a x subscript n end exponent plus b

with x subscript 0 equals 2 converges to alpha.

12d
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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 answer correct to 3 significant figures.

(ii) How many iterations, n, are required before x subscript n and x subscript n minus 1 end subscript agree with each other to 2 decimal places?

12e
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1 mark

The root alpha lies in the interval p less than x less than q.

Find the values of p and q that give the largest interval such that alpha can be found to 2 decimal places.

13a
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3 marks

Apply 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 4 significant figures.

13b
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2 marks

Show that there is a solution to the equation in the interval [0.605 , 0.615].

Without further calculation, state the value of this solution to the highest degree of accuracy possible.

14
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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 where x greater than 0.

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

Use the trapezium rule in step sizes of 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 3 significant figures.

1a
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3 marks
Graph showing a curve with shaded area R under it, between x=2 and x=4, with axes labelled x and y.
Figure 2

Figure 2 shows a sketch of part of the curve with equation

y equals open parentheses ln x close parentheses squared space space space x greater than 0

The finite region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x equals 2, the x-axis and the line with equation x equals 4.

The table below shows corresponding values of x and y, with the values of y given to 4 decimal places.

x

2

2.5

3

3.5

4

y

0.4805

0.8396

1.2069

1.5694

1.9218

Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of R, giving your answer to 3 significant figures.

1b5 marks

Use algebraic integration to find the exact area of R, giving your answer in the form

a open parentheses ln 2 close parentheses squared plus b ln 2 plus c

where a, b and c are integers to be found.

2a5 marks

A curve has equation y equals straight f open parentheses x close parentheses, where

straight f open parentheses x close parentheses equals fraction numerator 7 x straight e to the power of x over denominator square root of straight e to the power of 3 x end exponent minus 2 end root end fraction space space space space space space space space space space space space space space space space x greater than ln cube root of 2

Show that

straight f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 7 straight e to the power of x open parentheses straight e to the power of 3 x end exponent open parentheses 2 minus x close parentheses plus A x plus B close parentheses over denominator 2 open parentheses straight e to the power of 3 x end exponent minus 2 close parentheses to the power of 3 over 2 end exponent end fraction

where A and B are constants to be found.

2b2 marks

Hence show that the x coordinates of the turning points of the curve are solutions of the equation

x equals fraction numerator 2 table row blank blank straight e end table to the power of 3 x end exponent minus 4 over denominator table row blank blank straight e end table to the power of 3 x end exponent plus 4 end fraction

2c1 mark

The equation x equals fraction numerator 2 table row blank blank straight e end table to the power of 3 x end exponent minus 4 over denominator table row blank blank straight e end table to the power of 3 x end exponent plus 4 end fraction has two positive roots alpha spaceand beta where beta greater than alpha

A student uses the iteration formula

x subscript n plus 1 end subscript equals fraction numerator 2 table row blank blank straight e end table to the power of 3 x subscript n end exponent minus 4 over denominator table row blank blank straight e end table to the power of 3 x subscript n end exponent plus 4 end fraction

in an attempt to find approximations for alpha and beta

Diagram 1 shows a plot of part of the curve with equation y equals fraction numerator 2 table row blank blank straight e end table to the power of 3 x end exponent minus 4 over denominator table row blank blank straight e end table to the power of 3 x end exponent plus 4 end fraction and part of the line with equation y equals x

Graph showing the straight  line y=x and a curve intersecting it at two points. Vertical dashed lines join the points of intersection to points α and β on the x-axis.
Diagram 1

Using Diagram 1 draw a staircase diagram to show that the iteration formula starting with x subscript 1 equals 1 can be used to find an approximation for beta

2d
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3 marks

Use the iteration formula with x subscript 1 equals 1, to find, to 3 decimal places,

(i) the value of x subscript 2

(ii) the value of beta

2e
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2 marks

Using a suitable interval and a suitable function that should be stated show that alpha equals 0.432 to 3 decimal places.

3a
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1 mark

The diagram below shows part of the graph with equation y equals straight f open parentheses x close parentheses where

straight f left parenthesis x right parenthesis equals x tan open parentheses 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 less than 0 and straight f open parentheses 1.6 close parentheses greater than 0. They then conclude that there is a root in the interval 1.5 less than x less than 1.6.

Explain why the student’s conclusion is not correct.

3b
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2 marks

Explain why a change of sign method would fail when searching for the rootx equals 0 of the equation

straight f open parentheses x close parentheses plus 3 equals 0

4a
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3 marks

The function straight f open parentheses x close parentheses is defined as

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

Show that the Newton-Raphson formula can be written as

x subscript n plus 1 end subscript equals x subscript n minus fraction numerator negative 3 plus 5 cos space x subscript n space sin 2 x subscript n over denominator a cos space x subscript n open parentheses 1 minus b sin squared space x subscript n close parentheses end fraction

where a and b are integers to be found.

4b
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3 marks

Apply the Newton-Raphson method with x subscript 0 equals 0.3 to find, to 5 significant figures, a solution of the equation

10 cos squared space x space sin space x equals 3

4c
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2 marks

A student wants to use the Newton-Raphson method to find a solution to the equation

straight f open parentheses x close parentheses equals negative 3

in the range x greater than 0.

A teacher tells the student that the Newton-Raphson method is not necessary.

Explain why and find a solution to the equation, giving the highest accuracy possible.

5a
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4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis 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 x greater than 0

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

The equation straight f open parentheses x close parentheses equals 0 has a solution close to x equals 0.4

Use repeated iteration with x subscript 0 equals 0.4 to find this solution to 4 decimal places.

You must state your iteration formula clearly.

5b
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3 marks

Apply the Newton-Raphson method to find the same root, using a different starting value of x subscript 0 equals 0.5

Give your answer to 4 decimal places.

5c
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1 mark

Compare the rates at which the estimates converge between the different methods in part (a) and part (b).

6a
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3 marks

The figure below shows a sketch of the curve y equals x open parentheses x minus 6 close parentheses squared.

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

The coordinates of the local maximum point are open parentheses 2 comma space 32 close parentheses.

Use the trapezium rule with 4 equally-spaced trapezia to find an approximation to

integral subscript 1 superscript 5 x open parentheses x minus 6 close parentheses squared space d x

6b
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3 marks

By replacing the trapezia with rectangles that fit above the curve, find an upper bound for the area shaded.

Use rectangles that fit below the curve to find a lower bound.