4.08e-h Mean values, partial fractions and inverse trigonometric integration Quiz: OCR Further Maths, Unit 1

The 20 questions

1. Which divisor is used to calculate the mean value of f(x) over the interval [a, b]?

   • a - b
   • (b - a)^2
   • b - a
   • b + a

2. What is the mean value of f(x) = 2x over the interval [0, 4]?

   • 16
   • 4
   • 8
   • 2

3. What is the mean value of f(x) = x^2 over the interval [0, 3]?

   • 3
   • 4.5
   • 27
   • 9

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

   • arcsin(x) + C
   • ln|1 - x^2| + C
   • arccos(x) + C
   • arctan(x) + C

5. What is the indefinite integral of 1 / (1 + x^2) with respect to x?

   • ln|1 + x^2| + C
   • arctan(x) + C
   • arcsin(x) + C
   • arccosh(x) + C

6. What is the indefinite integral of 1 / sqrt(9 - x^2) with respect to x?

   • (1/3)arcsin(x/3) + C
   • arcsin(3x) + C
   • arctan(x/3) + C
   • arcsin(x/3) + C

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

   • 3arctan(x/3) + C
   • arctan(x/3) + C
   • (1/9)arctan(x/3) + C
   • (1/3)arctan(x/3) + C

8. What is the exact value of the integral of 1 / (1 + x^2) from 0 to 1?

   • pi / 3
   • pi / 2
   • pi / 4
   • pi / 6

9. What is the exact value of the integral of 1 / sqrt(1 - x^2) from 0 to 0.5?

   • pi / 6
   • pi / 4
   • pi / 3
   • pi / 2

10. Which form is correct for the partial fractions of 1 / ((x - 1)(x + 2))?

    • A/(x-1) + B/(x+2)
    • (Ax+B)/(x-1)(x+2)
    • A/(x-1)^2 + B/(x+2)
    • A/(x-1) * B/(x+2)

11. What is the correct partial fraction form for 1 / (x(x + 1)^2)?

    • A/x + (Bx+C)/(x+1)^2
    • A/x + B/(x+1)
    • A/x + B/(x+1)^2
    • A/x + B/(x+1) + C/(x+1)^2

12. What is the indefinite integral of 1 / (x(x + 1)) with respect to x?

    • 1 / ln|x(x+1)| + C
    • ln|x+1| - ln|x| + C
    • ln|x(x+1)| + C
    • ln|x/(x+1)| + C

13. What is the mean value of f(x) = sin(x) over the interval [0, pi]?

    • 2 / pi
    • 0
    • 2
    • pi / 2

14. In the partial fraction decomposition 1 / ((x - 1)(x + 1)) = A / (x - 1) + B / (x + 1), what is A?

    • 2
    • -1/2
    • 1
    • 1/2

15. What is the derivative of arcsin(x/a) with respect to x?

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

16. What is the indefinite integral of 1 / sqrt(4 - 9x^2) with respect to x?

    • (1/2)arcsin(3x/2) + C
    • (1/3)arcsin(2x/3) + C
    • arcsin(3x/2) + C
    • (1/3)arcsin(3x/2) + C

17. What is the mean value of f(x) = 1 / (1 + x^2) over [0, 1]?

    • 1 / 2
    • pi / 4
    • pi / 8
    • pi / 2

18. What is the indefinite integral of 1 / (x^2 + 4) with respect to x?

    • (1/4)arctan(x/2) + C
    • arctan(x/2) + C
    • 2arctan(x/2) + C
    • (1/2)arctan(x/2) + C

19. What is the indefinite integral of (2x + 3) / (x^2 + 1) with respect to x?

    • 2ln(x^2+1) + arctan(x) + C
    • ln(x^2+1) + 3arctan(x) + C
    • ln(x^2+1) + arctan(x) + C
    • 3ln(x^2+1) + 2arctan(x) + C

20. What is the correct partial fraction form for 1 / ((x - 2)(x^2 + 1))?

    • A/(x-2) + (Bx+C)/(x^2+1)
    • (Ax+B)/(x-2) + C/(x^2+1)
    • A/(x-2) + B/(x^2+1)
    • A/(x-2) + B/(x+1)

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