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AS & A Level · AS/A Level

Further Mathematics

Hyperbolic functions and inverses

Name: ____________________Date: October 10, 2026
  1. 1.

    Using exponential definitions, find sinh(ln 3), cosh(ln 3) and tanh(ln 3) exactly.

    [3 marks] · no calculator
  2. 2.

    Prove cosh^2 x - sinh^2 x = 1 directly from exponential definitions.

    [3 marks] · no calculator
  3. 3.

    Solve cosh x = 5/4 for real x, giving exact values.

    [3 marks] · no calculator
  4. 4.

    Evaluate integral from 0 to ln 2 of sinh(2x) dx exactly.

    [3 marks] · no calculator
  5. 5.

    Derive arsinh x = ln(x + sqrt(x^2 + 1)) from y = arsinh x, and hence differentiate arsinh x for real x.

    [3 marks] · no calculator
  6. 6.

    Evaluate integral from 0 to sqrt(3) of sqrt(1 + x^2) dx using x = sinh u. Give an exact answer.

    [3 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.

Marking points are indicative, not an official mark scheme. Accept equivalent valid methods and supported interpretations that address the task; award each mark once without requiring the model wording.