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OCR A Level Further Maths

Topic questions Pure mathematics Forming differential equations

Forming differential equations

5 questions · 61 marks · about 1 hour

Forming differential equations questions

TopicForming differential equations

OCR H245 Forming differential equations 5 questions

Pure Core 1 June 2024, Question 10 10 marks

Figure for question 10
(a)
Figure for question 10
[2]
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(b)

Using an integrating factor, determine the first time at which B is stationary according to the

model.

[8]
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Pure Core 1 June 2023, Question 7 11 marks

Figure for question 7
(a)

The engineer applies the equation “ F = ma ” to the model of the motion of P and derives the following differential equation.

Question 7, shown as an image
(a)(i)
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(a)(ii)

When t = 0 the displacement of P is 6 m, and it is travelling towards O with a speed of 5 ms-1.

Without attempting to solve the differential equation, find the acceleration of P

when t = 0.

[2]
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(b)(i)
Figure for question 7
[1]
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(b)(ii)
Figure for question 7
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(b)(iii)
Figure for question 7
[4]
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(c)(i)
Figure for question 7
[1]
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(c)(ii)
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[1]
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Pure Core 2 June 2023, Question 8 11 marks

A surge in the current, I units, through an electrical component at a time, t seconds, is to be modelled. The surge starts when t = 0 and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, I is modelled by the following differential equation.

Question 8, shown as an image
(a)

By using an integrating factor show that, according to the model, while the surge is occurring,

I is given by I = (2t - t2)(sin-1(t - 1) + 5).

[6]
Reveal mark scheme Mark scheme for question 8(a)
(b)

The surge lasts until there is again no current through the component.

Determine the length of time that the surge lasts according to the model.

[2]
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(c)

Determine, according to the model, the rate of increase of the current at the start of the surge.

Give your answer in an exact form.

[3]
Reveal mark scheme Mark scheme for question 8(c)

Pure Core 2 November 2021, Question 8 16 marks

Figure for question 8
(a)
Question 8, shown as an image
[3]
Reveal mark scheme Mark scheme for question 8(a)
(b)

Find an expression for v in terms of t for t ⩾ 0.

[5]
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(c)

By considering the behaviour of v as t becomes large explain why, according to the model, P ’s

speed must reach a maximum value for some t > 0.

[2]
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(d)

Determine the maximum speed considered in part (c).

[2]
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(e)

Determine the greatest value of t for which the model is valid.

[4]
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Pure Core 1 November 2020, Question 10 13 marks

Figure for question 10
(a)

Show that the motion of the particle can be modelled by the following differential equation.

Question 10, shown as an image
[2]
Reveal mark scheme Mark scheme for question 10(a)
(b)

Solve the differential equation in part (a), giving the particular solution for v in terms of t,

λ and μ.

[7]
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(c)
Figure for question 10
[2]
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(d)
Figure for question 10
[1]
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(e)

Without solving the differential equation in part (d), write down what will happen to the

velocity in the long term according to this refined model.

[1]
Reveal mark scheme Mark scheme for question 10(e)

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