H5-H7 Hyperbolic identities and proofs Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Which is an antiderivative of 1/sqrt(x^2 + 4) with respect to x?

   • arcosh(x/2)
   • (1/2) arsinh(x/2)
   • arsinh(x/2)
   • arsinh(x)

2. Which identity is correct for hyperbolic functions?

   • cosh^2(x) + sinh^2(x) = 1
   • cosh^2(x) - sinh^2(x) = 1
   • cosech^2(x) = 1 - coth^2(x)
   • sech^2(x) = tanh^2(x) - 1

3. Which identity follows from cosh^2(x) - sinh^2(x) = 1?

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

4. Which expression equals cosh(2x)?

   • 2 sinh(x) cosh(x)
   • cosh^2(x) - sinh^2(x)
   • cosh^2(x) + sinh^2(x)
   • 2 cosh^2(x)

5. Which expression equals sinh(2x)?

   • cosh(x) sinh(x)
   • sinh^2(x) + cosh^2(x)
   • 2 sinh(x)
   • 2 sinh(x) cosh(x)

6. Which identity is correct?

   • cosech^2(x) = coth^2(x) + 1
   • cosech^2(x) = 1 - coth^2(x)
   • cosech^2(x) = coth(x) - 1
   • cosech^2(x) = coth^2(x) - 1

7. Given cosh(x) = 5/4 with x > 0, what is sinh(x)?

   • 1/4
   • -3/4
   • 9/16
   • 3/4

8. Given sinh(x) = 3/4, what is tanh(x)?

   • 3/4
   • 5/3
   • 4/5
   • 3/5

9. Which is an antiderivative of 1/sqrt(x^2 - 1) for x > 1?

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

10. Which expression is an antiderivative of 1/(9 - x^2) for |x| < 3?

    • (1/9) artanh(x/3)
    • (1/3) arsinh(x/3)
    • (1/3) artanh(x/3)
    • artanh(x/3)

11. Evaluate the integral of 1/sqrt(4 + x^2) from 0 to 2.

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

12. Which is an antiderivative of 1/sqrt(x^2 - 4) for x > 2?

    • arcosh(x/2)
    • (1/2) arcosh(x/2)
    • arcosh(x)
    • arsinh(x/2)

13. Find the integral of 1/(1 - 4x^2) for |x| < 1/2.

    • (1/2) artanh(2x) + C
    • (1/4) artanh(2x) + C
    • artanh(2x) + C
    • (1/2) arsinh(2x) + C

14. Which identity gives cosh(x + y)?

    • cosh(x) + cosh(y)
    • cosh(x) cosh(y) + sinh(x) sinh(y)
    • sinh(x) cosh(y) + cosh(x) sinh(y)
    • cosh(x) cosh(y) - sinh(x) sinh(y)

15. Which identity gives sinh(x + y)?

    • sinh(x) sinh(y) + cosh(x) cosh(y)
    • sinh(x) + sinh(y)
    • sinh(x) cosh(y) - cosh(x) sinh(y)
    • sinh(x) cosh(y) + cosh(x) sinh(y)

16. Using sech^2(x) = 1 - tanh^2(x), what is the derivative of tanh(x) written in terms of tanh(x)?

    • tanh^2(x) - 1
    • 1 - tanh^2(x)
    • 1 + tanh^2(x)
    • 2 tanh(x)

17. Find the integral of 1/sqrt(x^2 + 2x + 5) with respect to x.

    • arcosh((x + 1)/2) + C
    • arsinh((x + 1)/2) + C
    • (1/2) arsinh((x + 1)/2) + C
    • arsinh(x + 1) + C

18. Why is the integral of tanh(x) equal to ln(cosh(x)) + C?

    • The derivative of ln(cosh(x)) is sinh(x)/cosh(x) = tanh(x)
    • The derivative of tanh(x) is sech^2(x)
    • cosh(x) equals e^x
    • tanh(x) is an odd function

19. Evaluate the integral of cosh(x) from 0 to ln(2).

    • 5/4
    • 3/2
    • 3/4
    • 1/2

20. Find sech^2(ln 3).

    • 16/25
    • 4/5
    • 25/9
    • 9/25

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