| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Prove hyperbolic identity from exponentials |
| Difficulty | Standard +0.3 This is a structured proof question with clear signposting through multiple parts. Part (a) is a routine algebraic manipulation of standard hyperbolic identities (2 marks). Part (b)(i) requires only stating the range of tanh x for positive x (trivial, 1 mark). Part (b)(ii) uses the previous results to prove an inequality, requiring students to recognize that 2/(1+tanh²x) > 1 when 0 < tanh x < 1, which is straightforward given the scaffolding. While it involves proof and hyperbolic functions (Further Maths content), the heavy scaffolding and routine algebraic nature make it slightly easier than average overall. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
| Answer | Marks |
|---|---|
| 12(a) | Uses hyperbolic identities |
| Answer | Marks | Guidance |
|---|---|---|
| quotient | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| result | 2.1 | R1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 12(b)(i) | Correctly states range ACF | 1.2 |
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 12(b)(ii) | Uses range of f to show that | |
| 1+tanh2 x<2 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1+tanh2 x | 2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| scores 0/3 | 2.1 | R1 |
| Subtotal | 3 | |
| Question total | 6 | |
| Q | Marking instructions | AO |
Question 12:
--- 12(a) ---
12(a) | Uses hyperbolic identities
correctly to expresstanh2xas a
quotient | 1.1a | M1 | sinh2x 2sinhxcoshx
tanh2x= =
cosh2x cosh2 x+sinh2 x
2sinhxcoshx
cosh2 x 2tanhx
= =
cosh2 x+sinh2 x 1+tanh2 x
cosh2 x
Completes a reasoned
argument to obtain the required
result | 2.1 | R1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 12(b)(i) ---
12(b)(i) | Correctly states range ACF | 1.2 | B1 | 0<f(x)<1
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 12(b)(ii) ---
12(b)(ii) | Uses range of f to show that
1+tanh2 x<2 | 3.1a | M1 | For x>0,
0<tanhx<1⇒1<1+tanh2 x<2
2
∴ >1
1+tanh2 x
2tanhx
and >tanhx
1+tanh2 x
∴tanh2x > tanhx
2
Deduces that >1
1+tanh2 x | 2.2a | M1
Completes a reasoned
argument to obtain the required
result
Working backwards from the
result to the range of tanhx
scores 0/3 | 2.1 | R1
Subtotal | 3
Question total | 6
Q | Marking instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item Starting from the identities for $\sinh 2x$ and $\cosh 2x$, prove the identity
$$\tanh 2x = \frac{2 \tanh x}{1 + \tanh^2 x}$$
[2 marks]
\item \begin{enumerate}[label=(\roman*)]
\item The function f is defined by
$$f(x) = \tanh x \quad (x > 0)$$
State the range of f
[1 mark]
\item Use part (a) and part (b)(i) to prove that $\tanh 2x > \tanh x$ if $x > 0$
[3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 2023 Q12 [6]}}