Lesson B1-B3

B1-B3 Solving polynomial equations and complex conjugates Quiz: AQA Further Maths, Unit 2

20 questions

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Lesson B1-B3, Solving polynomial equations and complex conjugates: 20 multiple choice questions for the AQA Further Maths (7367), Unit 2: Compulsory content, written with Revision Ninja.

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The 20 questions

  1. If z = 2 + 3i, what is its complex conjugate?

    • 2 - 3i
    • 3 + 2i
    • -2 - 3i
    • -2 + 3i

  2. What is the real part of z = 4 - 5i?

    • -5
    • 4
    • 5
    • -5i

  3. A polynomial with real coefficients has root 1 + 2i. Which other number must also be a root?

    • 2 + i
    • -1 - 2i
    • -1 + 2i
    • 1 - 2i

  4. A real cubic has a non-real root. Which description is correct?

    • No real roots at all, since every root of a real cubic is non-real
    • Three non-real roots, which can occur without any conjugate pair in the cubic
    • Three real roots, all distinct, with no non-real roots at all in the cubic
    • Exactly one real root, together with a conjugate pair of non-real roots

  5. What is i^2?

    • i
    • 1
    • -1
    • -i

  6. Solve z^2 - 6z + 13 = 0.

    • z = -3 + 2i or z = -3 - 2i
    • z = 3 + 2i or z = 3 - 2i
    • z = 3 + 4i or z = 3 - 4i
    • z = 6 + 4i or z = 6 - 4i

  7. What is the product of the roots of z^2 - 2z + 5 = 0?

    • -2
    • 5
    • -5
    • 2

  8. Given that x = 2 is a root of x^3 - x^2 - 4x + 4 = 0, which is the factorisation with that root removed?

    • (x - 2)(x^2 - x + 2)
    • (x + 2)(x^2 + x - 2)
    • (x - 2)(x^2 + x - 2)
    • (x - 2)(x^2 + 3x - 2)

  9. A real cubic has roots 2 and 1 + i. What is the third root?

    • 1 - i
    • 2 + i
    • 2 - i
    • -1 + i

  10. Given x = 1 is a root of 2x^3 + x^2 - 7x + 4 = 0, which factorisation is correct?

    • (x + 1)(2x^2 + 3x - 4)
    • (x - 1)(2x^2 - 3x + 4)
    • (x - 1)(2x^2 + x - 4)
    • (x - 1)(2x^2 + 3x - 4)

  11. A real cubic has roots 1 + 2i and -3. What is the coefficient of x^2?

    • 2
    • -1
    • -2
    • 1

  12. The quartic x^4 - 2x^3 + 6x^2 - 8x + 8 has root 1 + i. Which quadratic factor with real coefficients has 1 + i as a root?

    • x^2 - 2x - 2
    • x^2 - 4x + 2
    • x^2 + 2x + 2
    • x^2 - 2x + 2

  13. Which real value of k makes x^2 + kx + 13 = 0 have roots 2 +/- 3i?

    • k = 4
    • k = -4
    • k = -13
    • k = 13

  14. If z = a + bi satisfies z + z* = 6 and z z* = 13 with a > 0 and b > 0, which is z?

    • 3 + 2i
    • 3 - 2i
    • 6 + 13i
    • 2 + 3i

  15. Which real value of x satisfies [[x, 1], [1, x]] applied to [[1], [1]] equal to [[3], [3]]?

    • x = 3
    • x = 1
    • x = 2
    • x = 4

  16. Which set of real roots does x^3 - 3x - 2 = 0 have?

    • x = 1 and x = -2, since these are the other factors of the constant term
    • x = -1 (repeated) and x = 2
    • x = -1 and x = -2, since both are roots of the quadratic factor
    • x = 2 only, since the cubic has only one real root in this case

  17. Over the complex numbers, which statement about x^3 - 2 = 0 is true?

    • It has two real roots and one repeated root
    • It has only one root, the real cube root of 2
    • It has one real root and two non-real conjugate roots
    • It has three real roots

  18. Solve z^2 + 2z + 5 = 0. What is the modulus of each root?

    • 5
    • 2 sqrt(5)
    • sqrt(5)
    • 1

  19. A real cubic has roots 2 + i, 2 - i and one more real root. If the coefficient of x^2 is -6, what is the third root?

    • 6
    • 2
    • -2
    • 4

  20. Which complex numbers z satisfy z^2 = 3 + 4i?

    • z = 2 - i or z = -2 + i
    • z = 3 + 4i or z = -3 - 4i
    • z = 2 + i or z = -2 - i
    • z = 1 + 2i or z = -1 - 2i

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