H3-H4 Inverse hyperbolic functions Quiz: AQA Further Maths, Unit 2

The 20 questions

1. What is the domain of arcosh(x)?

   • -1 <= x <= 1
   • All real numbers
   • x >= 0
   • x >= 1

2. What is the domain of arsinh(x)?

   • All real numbers
   • -1 < x < 1
   • x >= 1
   • x > 0

3. What is the domain of artanh(x)?

   • x > 0
   • x >= 1
   • All real numbers
   • -1 < x < 1

4. What is the range of arcosh(x)?

   • 0 < y < 1
   • y >= 1
   • All real numbers
   • y >= 0

5. Which expression is the logarithmic form of arsinh(x)?

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

6. Which expression is the logarithmic form of artanh(x)?

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

7. What is the derivative of arsinh(x) with respect to x?

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

8. What is the derivative of arcosh(x) for x > 1?

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

9. What is the derivative of artanh(x) for -1 < x < 1?

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

10. What is arcosh(1)?

    • 1
    • 0
    • ln(2)
    • -1

11. What is the value of arsinh(1)?

    • sqrt(2)
    • ln(1 + sqrt(2))
    • ln(sqrt(2) - 1)
    • ln(2)

12. What is the value of artanh(1/2)?

    • (1/2) ln(2/3)
    • ln(3)
    • (1/2) ln(3)
    • ln(3/2)

13. What is the value of arcosh(2)?

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

14. Which function is an antiderivative of 1/sqrt(x^2 + 1)?

    • artanh(x)
    • arsinh(x)
    • arcosh(x)
    • arsinh(x^2)

15. If y = arcosh(x), which expression gives x in terms of y?

    • cosh(y)
    • tanh(y)
    • sinh(y)
    • cosh(2y)

16. Find x such that arsinh(x) = ln(3).

    • 4/3
    • 1/3
    • 3/2
    • 8/3

17. Find x such that arcosh(x) = ln(3).

    • 5/3
    • 10/3
    • 3
    • 4/3

18. Evaluate the integral of 1/sqrt(x^2 + 1) from 0 to sqrt(3).

    • ln(2 + sqrt(3))
    • sqrt(3) + 1
    • ln(sqrt(3) - 2)
    • ln(1 + sqrt(3))

19. Which property of arsinh(x) holds for all real x?

    • arsinh(-x) = arcosh(x)
    • arsinh(-x) = 1/arsinh(x)
    • arsinh(-x) = -arsinh(x)
    • arsinh(-x) = arsinh(x)

20. Differentiate arsinh(x^2) with respect to x.

    • 2/sqrt(x^4 + 1)
    • x/sqrt(x^4 + 1)
    • 2x/sqrt(x^4 + 1)
    • 2x/sqrt(x^4 - 1)

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