D1-D2 Roots and coefficients of polynomials Quiz: AQA Further Maths, Unit 2

The 20 questions

1. For x^2 + bx + c = 0, what is alpha + beta?

   • -b
   • b
   • c
   • -c

2. For the cubic ax^3 + bx^2 + cx + d = 0, what is alpha beta gamma?

   • c/a
   • -c/a
   • d/a
   • -d/a

3. If alpha and beta are roots of x^2 - 5x + 6 = 0, what is alpha^2 + beta^2?

   • 25
   • 19
   • 13
   • 12

4. If alpha and beta are roots of x^2 - 5x + 6 = 0, what is alpha^3 + beta^3?

   • 30
   • 27
   • 35
   • 65

5. For x^3 - 2x^2 + 3x - 4 = 0 with roots alpha, beta, gamma, what is alpha beta + beta gamma + gamma alpha?

   • 3
   • -2
   • 4
   • -3

6. Which equation has roots 2 alpha and 2 beta, where alpha and beta are roots of x^2 - 5x + 6 = 0?

   • x^2 - 10x - 24 = 0
   • x^2 - 10x + 24 = 0
   • x^2 - 10x + 6 = 0
   • x^2 - 5x + 24 = 0

7. Which equation has roots 1/alpha and 1/beta, where alpha and beta are roots of x^2 - 5x + 6 = 0?

   • x^2 - 5x + 6 = 0
   • 6x^2 - 5x + 1 = 0
   • 6x^2 + 5x + 1 = 0
   • 6x^2 - 5x - 1 = 0

8. For x^3 - 3x + 1 = 0 with roots alpha, beta, gamma, what is alpha^2 + beta^2 + gamma^2?

   • 3
   • 9
   • 6
   • -3

9. For a quartic x^4 + ax^3 + bx^2 + cx + d = 0, what is the sum of the roots?

   • -b
   • a
   • -a
   • b

10. The cubic x^3 - 6x^2 + 11x - 6 = 0 has roots 1, 2 and 3. Which cubic has the reciprocals of these roots as its roots?

    • 6x^3 - 11x^2 + 6x - 1 = 0
    • x^3 - 6x^2 + 11x - 6 = 0
    • 6x^3 - 11x^2 + 6x + 1 = 0
    • 6x^3 + 11x^2 + 6x - 1 = 0

11. If alpha and beta are roots of x^2 - 4x + 1 = 0, what is (alpha - beta)^2?

    • 12
    • 4
    • 15
    • -12

12. A real cubic has roots 1, 1 + i and 1 - i. What is the coefficient b in x^3 + ax^2 + bx + c?

    • 2
    • -4
    • 0
    • 4

13. If alpha and beta are roots of x^2 + 2x + 5 = 0, what is alpha^3 + beta^3?

    • 8
    • 22
    • 12
    • -22

14. Which quadratic has roots alpha + 2 and beta + 2, where alpha and beta are roots of x^2 - 3x + 1 = 0?

    • x^2 - 7x + 11 = 0
    • x^2 - 3x + 1 = 0
    • x^2 - 3x + 11 = 0
    • x^2 - 7x + 1 = 0

15. If alpha and beta are roots of x^2 - kx + 4 = 0 and alpha^2 + beta^2 = 9, what is k?

    • k = plus or minus 3
    • k = plus or minus sqrt(8)
    • k = plus or minus 4
    • k = plus or minus sqrt(17)

16. For x^3 - 2x + 3 = 0 with roots alpha, beta, gamma, what is alpha^2 beta^2 + beta^2 gamma^2 + gamma^2 alpha^2?

    • -4
    • 6
    • 2
    • 4

17. Which quadratic has roots 1 + sqrt(2) and 1 - sqrt(2)?

    • x^2 + 2x - 1 = 0
    • x^2 - x - 1 = 0
    • x^2 - 2x - 1 = 0
    • x^2 - 2x + 1 = 0

18. For x^3 + 2x^2 - x + 5 = 0 with roots alpha, beta, gamma, what is 1/alpha + 1/beta + 1/gamma?

    • -1/5
    • 1/5
    • 5
    • -5

19. For 2x^2 - 3x + 5 = 0 with roots alpha and beta, what is alpha/beta + beta/alpha?

    • -5/11
    • -11/10
    • -3/5
    • 11/10

20. For x^3 - 3x^2 + 2x - 5 = 0 with roots alpha, beta, gamma, what is (alpha + beta)(beta + gamma)(gamma + alpha)?

    • 1
    • -1
    • 6
    • 5

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