4.02e-n Modulus-argument form, polynomial roots and Argand diagrams Quiz: OCR Further Maths, Unit 1

The 20 questions

1. What is the modulus-argument form of a complex number z?

   • r(sin θ + i cos θ)
   • r(i cos θ + sin θ)
   • r(cos θ - sin θ)
   • r(cos θ + i sin θ)

2. What is the standard range for the principal argument θ of a complex number?

   • 0 < θ ≤ 2π
   • -π < θ ≤ π
   • 0 ≤ θ < 2π
   • -π ≤ θ < π

3. What is the modulus of the complex number z = 3 + 4i?

   • 5
   • 7
   • 1
   • 25

4. What is the principal argument of the complex number z = 1 + i?

   • π/2
   • π/4
   • π/3
   • π/6

5. What is the principal argument of the complex number z = -1 + i√3?

   • 2π/3
   • π/3
   • -π/3
   • -2π/3

6. If a polynomial with real coefficients has a complex root z, what must also be a root?

   • Its negation -z
   • Its inverse 1/z
   • Its square z²
   • Its conjugate z*

7. For a cubic equation ax³ + bx² + cx + d = 0, what is the sum of its roots?

   • -d/a
   • c/a
   • -b/a
   • b/a

8. What is the product of the roots of the cubic equation x³ - 6x² + 11x - 6 = 0?

   • 11
   • -6
   • -11
   • 6

9. What geometric locus is represented by the equation |z - (2 + 3i)| = 5 on an Argand diagram?

   • Perpendicular bisector
   • Circle
   • Ellipse
   • Half-line

10. What geometric locus is represented by |z - a| = |z - b| on an Argand diagram?

    • Half-line
    • Circle
    • Perpendicular bisector
    • Tangent line

11. What geometric shape is represented by the locus equation arg(z - z₁) = θ on an Argand diagram?

    • Full straight line
    • Parabola
    • Circle
    • Half-line

12. What is 2(cos(π/3) + i sin(π/3)) written in Cartesian form x + iy?

    • √3 + i
    • √3 - i
    • 1 + i√3
    • 1 - i√3

13. If |z₁| = 3 and |z₂| = 4, what is the value of |z₁z₂|?

    • 7
    • 12
    • 1
    • 25

14. If arg(z₁) = π/2 and arg(z₂) = π/6, what is arg(z₁/z₂)?

    • 3π/4
    • π/12
    • π/3
    • 2π/3

15. What is the sum of the roots of x⁴ - 4x³ + 5x² - 2x + 8 = 0?

    • -2
    • -4
    • 4
    • 5

16. If 2 + i is a root of x² + px + q = 0 with real coefficients, what is p?

    • 5
    • 4
    • -5
    • -4

17. If 2 + i is a root of x² + px + q = 0 with real coefficients, what is q?

    • 4
    • -5
    • 5
    • -4

18. What is the centre of the circle defined by |z - 3 + 4i| = 2?

    • 3 - 4i
    • -3 + 4i
    • 3 + 4i
    • -3 - 4i

19. Which Argand diagram transformation corresponds to multiplying a complex number by i?

    • stretch scale factor i
    • 90 degrees clockwise rotation
    • 90 degrees anticlockwise rotation
    • reflection in real axis

20. What is the gradient of the perpendicular bisector locus defined by |z - 3| = |z - 5i|?

    • 3/5
    • 5/3
    • -5/3
    • -3/5

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