5.3-5.4 Mean values and integration by partial fractions Quiz: Pearson Edexcel Further Maths, Unit 5

The 20 questions

1. What formula gives the mean value of a function f(x) over [a, b]?

   • (1 / 2) * integral f(x) dx
   • (f(a) + f(b)) / 2
   • (1 / (b - a)) * integral f(x) dx
   • integral f(x) dx

2. What is the main purpose of using partial fractions when integrating?

   • Split rational functions
   • Eliminate square roots
   • Differentiate polynomials
   • Find turning points

3. For a denominator with two distinct linear factors (x - a)(x - b), what is the standard partial fraction form?

   • A/(x - a) + B/(x - b)^2
   • A/(x - a)^2 + B/(x - b)^2
   • (Ax + B)/((x - a)(x - b))
   • A/(x - a) + B/(x - b)

4. For an irreducible quadratic factor ax^2 + c in the denominator, what numerator is used in its partial fraction term?

   • A only, a constant
   • Ax + B
   • A/x + B
   • Ax^2 + Bx + C

5. What is the mean value of x^2 over the interval [0, 3]?

   • 6
   • 1
   • 3
   • 9

6. What is the mean value of 2x over the interval [1, 3]?

   • 2
   • 4
   • 8
   • 6

7. What is the mean value of sin x over the interval [0, pi]?

   • 2/pi
   • 1
   • 1/pi
   • pi/2

8. What is the mean value of e^x over the interval [0, 1]?

   • e - 1
   • e
   • 1/(e - 1)
   • 1

9. What is the mean value of 1/x over the interval [1, e]?

   • 1/e
   • 1/(e - 1)
   • 1
   • e - 1

10. What is the partial fraction decomposition of 1/((x - 1)(x + 2))?

    • (1/3)(1/(x - 1) - 1/(x + 2))
    • (1/3)(1/(x - 1) + 1/(x + 2))
    • 1/(x - 1) - 1/(x + 2)
    • (1/3)(1/(x + 1) - 1/(x - 2))

11. What is the partial fraction decomposition of (3x + 5)/((x + 1)(x + 2))?

    • 2/(x + 1) + 1/(x + 2)
    • 2/(x + 1) - 1/(x + 2)
    • 1/(x + 1) + 2/(x + 2)
    • 3/(x + 1) + 5/(x + 2)

12. What is the value of the integral from 0 to 1 of 1/((x + 1)(x + 2)) dx?

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

13. What is the value of the integral from 2 to 3 of 1/((x - 1)(x + 2)) dx?

    • (1/3) ln(5/8)
    • ln(8/5)
    • (1/3) ln(8/5)
    • (1/3) ln 2

14. What is the value of the integral from 0 to 1 of (3x + 5)/((x + 1)(x + 2)) dx?

    • ln 6
    • ln 4
    • ln 3
    • ln(3/2)

15. Which expression is an antiderivative of 1/(x(x^2 + 1))?

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

16. For (x + 3)/((x - 2)(x + 1)), which partial fraction decomposition is correct?

    • 5/(3(x - 2)) - 2/(3(x + 1))
    • 2/(3(x - 2)) - 5/(3(x + 1))
    • 5/(3(x - 2)) + 2/(3(x + 1))
    • 1/(x - 2) + 1/(x + 1)

17. Which partial fraction decomposition is correct for 2/((x + 1)(x^2 + 1))?

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

18. Find c in [1, 3] such that f(c) equals the mean value of f(x) = x^2 on [1, 3].

    • c = sqrt(13)
    • c = 13/6
    • c = sqrt(13/3)
    • c = 2

19. What is the mean value of 1/x^2 over the interval [1, 2]?

    • 3/2
    • 1
    • 1/4
    • 1/2

20. What is the value of the integral from 3 to 5 of 1/(x^2 - 1) dx?

    • (1/2) ln(3/4)
    • (1/2) ln 2
    • ln(4/3)
    • (1/2) ln(4/3)

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