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OCR A Level Further Maths

Topic questions Pure mathematics Differentiating hyperbolic functions

Differentiating hyperbolic functions

6 questions · 49 marks · about 1 hour

Differentiating hyperbolic functions questions

TopicDifferentiating hyperbolic functions

OCR H245 Differentiating hyperbolic functions 6 questions

Pure Core 1 June 2024, Question 7 9 marks

(a)

By using the definitions of cosh u and sinh u in terms of eu and e-u, show that

sinh 2 u ≡ 2 sinh u cosh u.

[2]
Reveal mark scheme Mark scheme for question 7(a)
(b)
Figure for question 7
[4]
Reveal mark scheme Mark scheme for question 7(b)
(c)

You are now given that C has exactly one point of inflection.

Use your answer to part (b) to determine the exact coordinates of this point of inflection.

Give your answer in a logarithmic form where appropriate.

[3]
Reveal mark scheme Mark scheme for question 7(c)

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)
Figure for question 7
[1]
Reveal mark scheme Mark scheme for question 7(a)(i)
(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]
Reveal mark scheme Mark scheme for question 7(a)(ii)
(b)(i)
Figure for question 7
[1]
Reveal mark scheme Mark scheme for question 7(b)(i)
(b)(ii)
Figure for question 7
[1]
Reveal mark scheme Mark scheme for question 7(b)(ii)
(b)(iii)
Figure for question 7
[4]
Reveal mark scheme Mark scheme for question 7(b)(iii)
(c)(i)
Figure for question 7
[1]
Reveal mark scheme Mark scheme for question 7(c)(i)
(c)(ii)
Figure for question 7
[1]
Reveal mark scheme Mark scheme for question 7(c)(ii)

Pure Core 1 November 2021, Question 8 8 marks

You are given that f (x) = 4 sinh x + 3 cosh x.

(a)

Show that the curve y = f (x) has no turning points.

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

Determine the exact solution of the equation f (x) = 5.

[5]
Reveal mark scheme Mark scheme for question 8(b)

Pure Core 1 November 2020, Question 8 10 marks

(a)

Using exponentials, show that cosh 2u ≡ 2 sinh2u + 1.

[2]
Reveal mark scheme Mark scheme for question 8(a)
(b)

By differentiating both sides of the identity in part (a) with respect to u, show that

sinh 2u ≡ 2 sinh u cosh u.

[1]
Reveal mark scheme Mark scheme for question 8(b)
(c)
Question 8, shown as an image
[5]
Reveal mark scheme Mark scheme for question 8(c)
(d)
Figure for question 8
[2]
Reveal mark scheme Mark scheme for question 8(d)

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