OCR MEI Further Pure Core 2022 June — Question 3 6 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2022
SessionJune
Marks6
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Mark schemeDownload PDF ↗
TopicHyperbolic functions
TypeSolve using substitution u = cosh x or u = sinh x
DifficultyStandard +0.3 This is a straightforward Further Maths hyperbolic equation requiring substitution of cosh²x = 1 + sinh²x to form a quadratic in sinh x, then using the logarithmic definition to find exact solutions. While it requires knowledge of hyperbolic identities and careful algebraic manipulation, it follows a standard template with no novel insight needed—slightly easier than average even for Further Maths.
Spec4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials
3 In this question you must show detailed reasoning.
Solve the equation \(3 \cosh x = 2 \sinh ^ { 2 } x\), giving your solutions in exact logarithmic form.
Question 3:
AnswerMarks
3DR
3cosh𝑥 = 2sinh2𝑥 = 2(cosh2𝑥−1)
⇒ 2cosh2𝑥−3cosh𝑥−2 = 0
⇒ (2cosh𝑥+1)(cosh𝑥−2) = 0
1
⇒ cosh𝑥 = − or 2
2
⇒ 𝑥 = ln(2+√3)
AnswerMarks
or −ln(2+√3)M1
A1
M1
A1
A1
A1
AnswerMarks
[6]3.1a
1.1
1.1
1.1
1.1
AnswerMarks
1.1sinh2𝑥 = cosh2𝑥−1 used
Solve their quadratic
Don't need to see –½ if factorisation or quadratic formula shown
or ln(2−√3)
Question 3:
3 | DR
3cosh𝑥 = 2sinh2𝑥 = 2(cosh2𝑥−1)
⇒ 2cosh2𝑥−3cosh𝑥−2 = 0
⇒ (2cosh𝑥+1)(cosh𝑥−2) = 0
1
⇒ cosh𝑥 = − or 2
2
⇒ 𝑥 = ln(2+√3)
or −ln(2+√3) | M1
A1
M1
A1
A1
A1
[6] | 3.1a
1.1
1.1
1.1
1.1
1.1 | sinh2𝑥 = cosh2𝑥−1 used
Solve their quadratic
Don't need to see –½ if factorisation or quadratic formula shown
or ln(2−√3)
3 In this question you must show detailed reasoning.\\
Solve the equation $3 \cosh x = 2 \sinh ^ { 2 } x$, giving your solutions in exact logarithmic form.

\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q3 [6]}}
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