Hyperbolic functions questions

Questions to attempt

6 questions on Hyperbolic functions, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-8.1 · Understand the definitions of hyperbolic functions sinh x, cosh x and tanh x, including their domains and ranges, and be able to sketch their graphs.

   Q1

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   1.

   Using the exponential definitions, find the exact values of $\sinh(\ln3)$, $\cosh(\ln3)$ and $\tanh(\ln3)$.

   (4)

   (Total for Question 1 is 4 marks)

   Reveal answer and method

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2. CP-8.1 · Understand the definitions of hyperbolic functions sinh x, cosh x and tanh x, including their domains and ranges, and be able to sketch their graphs.

   Q2

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   2.

   State the domain, range and $y$-intercept of $y=\cosh x$.

   (3)

   (Total for Question 2 is 3 marks)

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3. CP-8.2 · Differentiate and integrate hyperbolic functions.

   Q3

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   3.

   Find $\displaystyle\int(5\cosh(3x)-2\sinh x)\,dx$.

   (3)

   (Total for Question 3 is 3 marks)

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4. CP-8.3 · Understand and be able to use the definitions of the inverse hyperbolic functions and their domains and ranges.

   Q4

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   4.

   Solve $\operatorname{arsinh}(2x-1)=\ln2$ exactly.

   (3)

   (Total for Question 4 is 3 marks)

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5. CP-8.4 · Derive and use the logarithmic forms of the inverse hyperbolic functions.

   Q5

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   5.

   Solve $\operatorname{arsinh}x=\ln5-\ln2$, giving $x$ exactly.

   (4)

   (Total for Question 5 is 4 marks)

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6. CP-8.5 · Integrate functions of the form (x^2 + a^2)^(-1/2) and (x^2 - a^2)^(-1/2) and be able to choose substitutions to integrate associated functions.

   Q6

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   6.

   Using $x=3\sinh u$, evaluate $\displaystyle\int_0^4\sqrt{x^2+9}\,dx$ exactly.

   (6)

   (Total for Question 6 is 6 marks)

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CP-8 Hyperbolic functions revision notes