4.07a-e Hyperbolic functions, their derivatives and inverses Quiz: OCR Further Maths, Unit 1

The 20 questions

1. Which expression represents the definition of sinh x in terms of exponentials?

   • (e^x - e^(-x))/2
   • (e^x - e^(-x))/x
   • (e^x + e^(-x))/2
   • (e^(-x) - e^x)/2

2. Which of the following is the correct fundamental identity relating cosh x and sinh x?

   • sinh^2 x - cosh^2 x = 1
   • cosh^2 x + sinh^2 x = 1
   • cosh^2 x - sinh^2 x = 0
   • cosh^2 x - sinh^2 x = 1

3. What is the derivative of cosh x with respect to x?

   • -sinh x
   • -cosh x
   • cosh x
   • sinh x

4. What is the derivative of sinh x with respect to x?

   • cosh x
   • sinh x
   • -sinh x
   • -cosh x

5. What is the derivative of tanh x with respect to x?

   • coth^2 x
   • cosech^2 x
   • sech^2 x
   • -sech^2 x

6. What is the domain of the inverse hyperbolic function arcosh x?

   • All real x
   • x >= 1
   • x >= 0
   • x > 0

7. Which logarithmic expression is equal to arsinh x?

   • ln(x - sqrt(x^2 + 1))
   • ln(x + sqrt(x^2 - 1))
   • 0.5 ln(x + 1)
   • ln(x + sqrt(x^2 + 1))

8. What is the derivative of arsinh x with respect to x?

   • 1 / sqrt(1 - x^2)
   • 1 / sqrt(x^2 - 1)
   • 1 / sqrt(x^2 + 1)
   • 1 / (x^2 + 1)

9. What is the exact numerical value of cosh 0?

   • 0.5哺乳
   • 0
   • 1
   • -1

10. What is the exact numerical value of sinh 0?

    • 1
    • 0
    • -1
    • 0.5哺乳

11. According to Osborn's rule, what is the hyperbolic equivalent of 1 + tan^2 x = sec^2 x?

    • 1 - tanh^2 x = sech^2 x
    • 1 - tanh^2 x = -sech^2 x
    • 1 + tanh^2 x = sech^2 x
    • 1 + tanh^2 x = -sech^2 x

12. What is the real solution to the equation cosh x = 1?

    • x = -1
    • x = 0
    • x = 1
    • x = ln 2

13. What is the derivative of arcosh x for x > 1?

    • 1 / sqrt(x^2 - 1)
    • 1 / sqrt(1 - x^2)
    • 1 / sqrt(x^2 + 1)
    • 1 / (x^2 - 1)

14. What is the derivative of artanh x with respect to x for |x| < 1?

    • 1 / sqrt(1 - x^2)
    • 1 / (1 + x^2)
    • 1 / (x^2 - 1)
    • 1 / (1 - x^2)

15. What is the exact numerical value of tanh 0?

    • 0
    • 1
    • undefined
    • 0.5

16. What is the exact value of arcosh 2 expressed as a natural logarithm?

    • ln(2 + sqrt(3))
    • ln(2 + sqrt(5))
    • ln(3 + sqrt(2))
    • ln(2 - sqrt(3))

17. What is the integral of 1 / sqrt(x^2 + 9) with respect to x?

    • 3 arsinh(x) + C
    • arsinh(x / 3) + C
    • artanh(x / 3) + C
    • arcosh(x / 3) + C

18. Which logarithmic expression is equal to artanh x for |x| < 1?

    • 0.5 ln(x^2 - 1)
    • 0.5 ln((1 - x) / (1 + x))
    • ln((1 + x) / (1 - x))
    • 0.5 ln((1 + x) / (1 - x))

19. What is the integral of 1 / sqrt(x^2 - 16) with respect to x for x > 4?

    • arcosh(x / 4) + C
    • artanh(x / 4) + C
    • arsinh(x / 4) + C
    • 4 arcosh(x) + C

20. What is the exact value of x if sinh x = 3/4?

    • ln 3
    • ln(3/2)
    • ln 2
    • ln(7/4)

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