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Complex Numbers

Complex Numbers. The more straight forward questions on Complex Numbers have been found in Question 2 of the HSC since 1991.

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Complex Numbers

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  1. Complex Numbers The more straight forward questions on Complex Numbers have been found in Question 2 of the HSC since 1991. Longer (and usually more difficult) problems will appear in the latter half of the paper and are often combined with concepts from other topics such as Polynomials, Induction and Binomial Expansions. (i) Basic Operations

  2. Example The complex number 2 + i is a root of the quadratic equation (i) Find the other root (ii) Determine the value of m

  3. (ii) Mod-Arg Form z = x + iy where

  4. (iii) Conjugate Properties (iv) De Moivre’s Theorem

  5. Examples 1. Express the complex number in modulus argument form. Hence express in the form a + ib where a and b are real. 2. Express in the terms of

  6. (v) Roots of Complex Numbers To find the square root of a complex number we can; a) Let the solution be a + ib and solve simultaneous equations in a and b b) Convert into mod arg form and use De Moivre’s Theorem

  7. c) Use the “alternative” formula Example Find the square root of 8 + 6i, giving your answer in the form x + iy

  8. For cube roots (and higher) we use De Moivre’s Theorem (unless of course the polynomial in the equation is easily factorised) Equations of the form have n roots which are equally spaced on the circle of radius r, where r is the modulus of any root. Example

  9. The equation has n roots which lie on the unit circle, centre (0,0), and if is a complex root of this equation, then the roots are; and we can show that; Examples

  10. 3. If is a complex cube root of unity, use the fact that to; c) Form the cubic equation with roots

  11. Some important results

  12. Examples 1. Resolve into real linear and quadratic factors. Hence prove sum of roots = 0

  13. 3. Solve

  14. (vi) Geometrical Representation of Complex Numbers Addition If a point A represents and point B represents then point C representing is such that the points OACB form a parallelogram. Subtraction If a point D represents and point E represents then the points ODEB form a parallelogram. Note:

  15. x y Multiplication If we multiply by the vector OA will rotate by an angle of in an anti-clockwise direction. If we multiply by it will also multiply the length of OA by a factor of r Note: will rotate OA anticlockwise 90 degrees. REMEMBER: A vector is HEAD minus TAIL

  16. Examples

  17. (vii) Locus y 5 1 -3 Locus problems can be done by an intuitive method or by letting z = x + iy and using methods in Coordinate Geometry Examples 1. Sketch on an Argand Diagram, the region satisfying x

  18. y 1 -1 -3 2. Sketch on an Argand Diagram, the region satisfying perpendicular bisector of the line joining i and –3i 3. Describe the locus described by

  19. 4. Describe the locus described by locus is ellipse

  20. y x 2 4 r (2,y)

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