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Edexcel A Level Further Maths

Topic questions Pure mathematics Improper integrals

Improper integrals

9 questions · 63 marks · about 1.5 hours

Improper integrals questions

TopicImproper integrals

Edexcel 9FM0 Improper integrals 9 questions

Paper 1: Core Pure Mathematics 1 June 2025, Question 9 9 marks

(i)

The curves with equations

Question 9, shown as an image

intersect at just one point P

(i)(a)

Use algebra to show that the x coordinate of P satisfies the equation

Question 9, shown as an image
[3]
Reveal mark scheme Mark scheme for question 9(i)(a)
(i)(b)

Show that ex maths symbol 3 is a solution of this equation.

[1]
Reveal mark scheme Mark scheme for question 9(i)(b)
(i)(c)

Hence state the exact coordinates of P.

[1]
Reveal mark scheme Mark scheme for question 9(i)(c)
(ii)

Show that

Figure for question 9
[4]
Reveal mark scheme Mark scheme for question 9(ii)

Paper 1: Core Pure Mathematics 1 June 2022, Question 9 6 marks

(i)(a)

Explain why maths symbol0maths symbolcosh x dx is an improper integral.

[1]
Reveal mark scheme Mark scheme for question 9(i)(a)
(i)(b)

Show that maths symbol0maths symbolcosh x dx is divergent.

[3]
Reveal mark scheme Mark scheme for question 9(i)(b)
(ii)

4 sinhx = pcoshxwhere p is a real constant

Given that this equation has real solutions, determine the range of possible values for p

[2]
Reveal mark scheme Mark scheme for question 9(ii)

Paper 1: Core Pure Mathematics 1 June 2022, Question 9 6 marks

(i)(a)
Figure for question 9
[1]
Reveal mark scheme Mark scheme for question 9(i)(a)
(i)(b)
Figure for question 9
[3]
Reveal mark scheme Mark scheme for question 9(i)(b)
(ii)

4 sinhx = pcoshxwhere p is a real constant

Given that this equation has real solutions, determine the range of possible values for p

[2]
Reveal mark scheme Mark scheme for question 9(ii)

Paper 1: Core Pure Mathematics 1 November 2021, Question 5 7 marks

(i)

Evaluate the improper integral

[3]
Reveal mark scheme Mark scheme for question 5(i)
(ii)

The air temperature, θ maths symbolC, on a particular day in London is modelled by the equation

Question 5, shown as an image

where t is the number of hours after midnight.

(ii)(a)

Use calculus to show that the mean air temperature on this day is 8maths symbolC, according to the model.

[3]
Reveal mark scheme Mark scheme for question 5(ii)(a)
(ii)(b)

Given that the actual mean air temperature recorded on this day was higher than 8maths symbolC,

explain how the model could be refined.

[1]
Reveal mark scheme Mark scheme for question 5(ii)(b)

Paper 1: Core Pure Mathematics 1 November 2021, Question 5 7 marks

(i)

Evaluate the improper integral

[3]
Reveal mark scheme Mark scheme for question 5(i)
(ii)

The air temperature, θ maths symbolC, on a particular day in London is modelled by the equation

Question 5, shown as an image

where t is the number of hours after midnight.

(ii)(a)

Use calculus to show that the mean air temperature on this day is 8maths symbolC, according to the model.

[3]
Reveal mark scheme Mark scheme for question 5(ii)(a)
(ii)(b)

Given that the actual mean air temperature recorded on this day was higher than 8maths symbolC,

explain how the model could be refined.

[1]
Reveal mark scheme Mark scheme for question 5(ii)(b)

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