E4-E6 Partial fractions, inverse trig functions and trig substitution Quiz: AQA Further Maths, Unit 2

The 20 questions

1. What is the derivative of arcsin(x) with respect to x, for -1 < x < 1?

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

2. What is the derivative of arctan(x) with respect to x?

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

3. For a > 0, which standard integral result is correct for the integrand 1/(x^2 + a^2)?

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

4. Which standard result gives the integral of 1/sqrt(a^2 - x^2) for |x| < a?

   • arcsin(x/a) + C
   • arccos(x/a) + C
   • (1/a) arcsin(x/a) + C
   • ln|x + sqrt(x^2 - a^2)| + C

5. In partial fractions, what is the correct decomposition form of (3x + 5)/((x + 1)^2 (x + 3))?

   • A/(x + 1) + B/(x + 1)^2 + C/(x + 3)^2
   • A/(x + 1) + B/(x + 3)
   • A/(x + 1)^2 + C/(x + 3)
   • A/(x + 1) + B/(x + 1)^2 + C/(x + 3)

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

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

7. When the substitution x = a sin(theta) is applied to an integral with sqrt(a^2 - x^2), what does dx become?

   • cos(theta) d(theta)
   • a sin(theta) d(theta)
   • a cos(theta) d(theta)
   • a sec^2(theta) d(theta)

8. Evaluate the definite integral of 1/sqrt(4 - x^2) from x = 0 to x = 1.

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

9. Evaluate the definite integral of 1/(x^2 + 4) from x = 0 to x = 2.

   • pi/4
   • pi/8
   • pi/2
   • pi/16

10. Partial fractions: if (5x - 1)/((x - 2)(x + 1)) = A/(x - 2) + B/(x + 1), what are A and B?

    • A = 3, B = 2
    • A = -3, B = 2
    • A = 2, B = 3
    • A = 3, B = -2

11. Find the integral of (2x + 3)/((x + 1)(x + 2)) with respect to x.

    • ln|(x + 1)/(x + 2)| + C
    • ln|(x + 1)(x + 2)| + C
    • 1/(x + 1) + 1/(x + 2) + C
    • 2 ln|(x + 1)(x + 2)| + C

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

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

13. What is the derivative of arcsin(2x) with respect to x, for |x| < 1/2?

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

14. Using x = 3 tan(theta), the integral of 1/(x^2 + 9) with respect to x simplifies to which integral in theta?

    • integral of 3 d(theta)
    • integral of (1/9) sec^2(theta) d(theta)
    • integral of (1/3) sec(theta) d(theta)
    • integral of (1/3) d(theta)

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

    • pi/3
    • pi/12
    • pi/6
    • pi/2

16. Using x = a sin(theta), the integral of sqrt(a^2 - x^2) with respect to x becomes an integral in theta with integrand equal to which expression?

    • a^2 cos(theta)
    • a^2 sin^2(theta)
    • a cos(theta)
    • a^2 cos^2(theta)

17. For the partial fraction decomposition of (2x + 1)/(x^2 (x + 1)) written as A/x + B/x^2 + C/(x + 1), what is the value of B?

    • 2
    • 0
    • 1
    • -1

18. What is the second derivative of arctan(x) evaluated at x = 0?

    • 1
    • 0
    • -1
    • 2

19. Evaluate the definite integral of 1/(4x^2 + 4x + 5) from x = 0 to x = 1.

    • (1/4) arctan(7/4)
    • (1/4) arctan(4/7)
    • (1/2) arctan(4/7)
    • (1/8) arctan(4/7)

20. Which trigonometric substitution is most suitable for the integral of 1/(x^2 sqrt(x^2 - 1)) with respect to x?

    • x = sec(theta), giving sqrt(x^2 - 1) = tan(theta)
    • x = sin(theta), giving sqrt(x^2 - 1) = cos(theta)
    • x = cos(theta), giving sqrt(x^2 - 1) = sin(theta)
    • x = tan(theta), giving sqrt(x^2 - 1) = sec(theta)

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