Exam-Style Problems

End-of-chapter review exercise 19 — problem 1 · Maths4U Original · 6 marks

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Notes & worked examples

Info

Maths4U Original · M4U-EXT-9231-FP2-T480-P01

Let \(u=\ln 8\). Use exact exponential definitions.

Find sinh u. Explain your reasoning. [2]
Find cosh u. Explain your reasoning. [2]
Find tanh u. Explain your reasoning. [2]

Solution

Hint

The argument is real unless stated otherwise. Substitute \(\displaystyle e^{u}=a\) and \(\displaystyle e^{− u}= \frac{1}{a}\).

Worked solution

(a)

\(\displaystyle \frac{8- \frac{1}{8}}{2}= \frac{63}{16}\)

Answer: \(\displaystyle \frac{63}{16}\).

Answer: \(\displaystyle \frac{63}{16}\)

(b)

\(\displaystyle \frac{8+ \frac{1}{8}}{2}= \frac{65}{16}\)

Answer: \(\displaystyle \frac{65}{16}\).

Answer: \(\displaystyle \frac{65}{16}\)

(c)

\(\displaystyle \frac{\sinh u}{\cosh u}= \frac{63}{65}\)

Answer: \(\displaystyle \frac{63}{65}\).

Answer: \(\displaystyle \frac{63}{65}\)

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.

(a) · 2 marks

1 mark: Substitute \(\displaystyle e^{u}=a\) and \(\displaystyle e^{− u}= \frac{1}{a}\).
1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: The fundamental identity has a minus sign.

(b) · 2 marks

1 mark: Add the exponential and its reciprocal before dividing by two.
1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: Sech is a reciprocal, not an inverse function.

(c) · 2 marks

1 mark: Divide the two hyperbolic values.
1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: The fundamental identity has a minus sign.

Total: 6 marks.

End-of-chapter review exercise 19 — problem 2 · Maths4U Original · 6 marks

17028

Notes & worked examples

Info

Maths4U Original · M4U-EXT-9231-FP2-T480-P02

Use the real inverse functions, with \(\displaystyle a=9,\ p=\frac{4}{5}\).

Find arsinh(a). Explain your reasoning. [2]
Find the nonnegative value arcosh(a). Explain your reasoning. [2]
Find artanh(p). Explain your reasoning. [2]

Solution

Hint

Arcosh has real domain [1,∞), with range [0,∞). Use the real logarithmic inverse-sinh expression.

Worked solution

(a)

\(\ln(a+\sqrt{a^2+1})=2.89344398588587\)

Answer: \(2.89344398588587\).

Answer: \(\approx 2.89344398588587\)

(b)

\(\ln(a+\sqrt{a^2-1})=2.88727095035762\)

Answer: \(2.88727095035762\).

Answer: \(\approx 2.88727095035762\)

(c)

\(\displaystyle \frac{1}{2}\ln [\frac{1+ p}{1- p}]= 1.09861228866811\)

Answer: \(1.09861228866811\).

Answer: \(\approx 1.09861228866811\)

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.

(a) · 2 marks

1 mark: Use the real logarithmic inverse-sinh expression.
1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: Sinh⁻¹x is not \(\displaystyle \frac{1}{\sinh x}\) when inverse notation is intended.

(b) · 2 marks

1 mark: Choose the nonnegative inverse-cosh branch.
1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: Do not drop the negative solution of \(\displaystyle \cosh u=c\) without a branch condition.

(c) · 2 marks

1 mark: Use the logarithmic form with |p|<1.
1 mark: Obtain the stated result with the required sign, units and domain.

Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.

Common error: Sinh⁻¹x is not \(\displaystyle \frac{1}{\sinh x}\) when inverse notation is intended.

Total: 6 marks.