remember remember Chapter Complex s 19 1. The magnitude (or modulus or absolute value) of z = x + yi is the length of the line segment from (0, 0) to z and is denoted by z, x + yi or mod z.. z = x + y and zz = z. y. arg z = θ where tan θ = --. x. z i n, n N produces an anticlockwise rotation of 90n degrees. 5. z = r cos θ + r sin θi = r cis θ in polar form.. Arg z is the angle θ in the range < θ. D Complex s in polar form In the following exercise give arg z or Arg z correct to three decimal places where it is not easily expressed as a multiple of. 1 a Represent z = + i on an Argand diagram. b Calculate the exact distance of z from the origin. Find the modulus of each of the following. 17 a z = 5 + 1i b z = 5 i c z = + 7i d z = i e z = + i f z = ( + i) Complex 1 Mathcad 1 19 If z = + i, w = i and u = + 5i then: i represent each of the following on an Argand diagram ii calculate the magnitude in each case. a z w b u + z c w u d w + z e z + w u f z a Show the points z 1 = + 0i, z = + 5i, z = 7 + 5i and z = 9 + 0i on the complex plane. b Calculate the area of the shape formed when the four points are connected by straight line segments in the order z 1 to z to z to z and back to z 1. 5 a Show the points z = 1 + i, u = and w = + 1i on the complex plane. b Calculate the area of the triangle produced by joining the three points with straight line segments. 0 Find the argument of z for each of the following in the interval [0, ]. (Give exact answers where possible.) a z = + i b z = + i c z = 5 5i d z = + i e z = i f z = 10i g z = i h z = 7 i z = i j z = 55
10 Maths Quest 1 Specialist Mathematics 1 7 Convert each of the following into Arguments. 11 15 5 a ----- b -------- c -------- d ----- 19 0 1 1 e -------- f -------- g -------- h -------- 7 5 1 Find the modulus and Argument of each of the following complex s. a i b 5 + 5i c 1 i d + i e 7 10i f i g ( + i) 9 Express each of the following in polar form z = r cis θ where θ = Arg z. a z = 1 + i b z = + i c z = 5 5i d z = 5 15i 1 1 1 e z = -- ------i f z = -- + --i 10 Express each of the following complex s in Cartesian form. a cis ----- b cis -- c 5 cis ----- 5 d cis -- e 7 cis 7 ----- f cis -- g cis 11 1 If z = 50i and w = 5 + 5i the value of z + w is: A B 15 C 17 D 5 E 9 The perimeter of the triangle formed by the line segments connecting the points i, 1 i and + i is: A 1 B 0 C 10 D 17 E 5 1 The Argument of i is: A -- B -- 5 C ----- D -- E -- 1 In polar form, 5i is: A cis -- 5 B cis 5 C cis ----- D 5 cis 5 E 5 cis -- WorkSHEET 15 The Cartesian form of cis ----- is: 1 1 1 A -- + ------i B ----- + ------i C --------- + --i D ----- + ------i E.1 7 1 --------- --i
Chapter Complex s 19 E Basic operations on complex s in polar form 5 1 Express each of the following in the form r cis θ where θ (, ]. a cis -- cis-- b 5 cis ----- cis -- c cis 5 ----- cis -- d cis ----- 5 cis e 7 cis 7 ----- cis----- 5 1 1 Express the resultant complex s in question 1 in Cartesian form. GC Power of program TI GCprogram Casio Power of Express the following products in polar form. 7 a ( + i)( + i) b ( i)( i) c ( + i)( 1 i) 9 0 1 Express each of the following in the form r cis θ where θ (, ]. 5 a 1 cis ----- cis b cis -- ----- 9 cis -- c 0 cis -- 5 cis -- d cis----- cis-------- 11 5 7 1 7 e 5 cis ----- 1 10 cis ----- 5 5 If z = cis----- and w = cis -- then express each of the following in: i polar form ii Cartesian form. a z b w c z d w 5 If z = 1 i and w = + i, write the following in Cartesian form. a z b w c z d w 5 e ----- f z w 7 Determine ( + i) ( 1 i) in Cartesian form. ( i) Write ---------------------------- in the form x + yi. ( i) z w 9 a 5 cis -- cis -- is equal to: A i B 10i C D i E b If = + + ( )i then z is: z ( ) A 1 + i B i C 1 i D + i E 1 i c If z = 1 i and w = + i then ----- is equal to: A + i B C D i E w z
10 Maths Quest 1 Specialist Mathematics 10 If z = cis----- and w = cis, find the modulus and the argument of. -- ----- 11 If z = + i and w = i, determine (z + w) 9. z w 1 Find z + w, if z = i and w = i. 1 If z 1 = 5 cis -----, z and, find the modulus and the 5 = cis----- z = 10 cis 1 ----- z argument of 1 z -------------------. z 1 By finding z if z = cis θ, show that cos θ = cos θ cos θ sin θ + sin θ and that sin θ = cos θ sin θ cos θ sin θ. 15 Using z = r cis θ, verify that zz = z. 1 If z n = (1 + i) n, determine the smallest value of n N so that z n is equal to: ( ) n ( ) n ( ) n i ( ) n i a b c d. Factorisation of polynomials in C A polynomial in z is an expression of the form P(z) = a n z n + a n 1 z n 1 + a n z n +... + a 1 z + a 0, where n N is the degree (highest power) of P(z) and a n (with a n 0) are the coefficients. If a n R, that is, all the coefficients are real, then P(z) is said to be a polynomial over R. Similarly, if at least one of the a n is complex, P(z) is said to be a polynomial over C. For example, P(z) = z 5z + is a polynomial of degree over R and P(z) = iz + z i is a polynomial of degree over C. The fundamental theorem of algebra Firstly recall that R C and the factor theorem, which states: If (x a) is a factor of the polynomial P(x), then P(a) = 0. In 1799 the German mathematician Carl Friedrich Gauss proved that every polynomial over C has a solution that is. That is, if P n (z) is a polynomial of degree n over C, then there exists a z 0 C such that P n (z 0 ) = 0. This important result can be used to show that a polynomial of degree n, with n N, has n solutions. The proof relies on a repeated application of the fundamental theorem of algebra and the factor theorem. Firstly, the fundamental theorem of algebra guarantees that there is a z 0 C such that P n (z 0 ) = 0. The factor theorem states that if P n (z 0 ) = 0 for some z 0 then (z z 0 ) is a factor of P n (z) so that P n (z) = (z z 0 )P n 1 (z), where P n 1 (z) is a polynomial of degree n 1. Now by applying the fundamental theorem of algebra to P n 1 (z) there is a z 1 C such that P n 1 (z 1 ) = 0 and the factor theorem ensures that P n 1 (z) = (z z 1 )P n (z).
1 Maths Quest 1 Specialist Mathematics 9 1 Find the values of a and b (a, b R) if: a (z + 1) is a factor of z iz + aiz + b b (z i) is a factor of az z + biz + 1i c (z + i) is a factor of z + aiz + iz + (1 + i)b. 1 Explain why at least one of the zeros of a polynomial of degree n (where n is an odd natural ) is a real. 1 Write down a polynomial of degree, whose coefficients are all real, that has i and as two of its zeros. 15 Find the values of a, (a R) for which ai is a solution to: a P(z) = z + z + z + 10 b P(z) = z + iz 11z i. 1 Factorise z + i over C. 17 a Show that P(1) = 0 for P(z) = z (1 + i)z + (i 1)z + (7 + i)z i. b Find the polynomial Q(z) if P(z) = (z 1)Q(z). c Determine the values of a C, b R if Q(z) is of the form Q(z) = (z a) + b. 1 Factorise z + z + z + 10z + 15 over C given that z + 5 i is a factor. 19 Factorise P(z) = 9z + (9i 1)z + (5 1i)z + 5i over C if P( i) = 0. 0 Determine the value of a R if i is to be a zero of a z a 11 + = --------------. z tip! Graphics Calculator tip! Roots of complex s Casio tip removed. 1. To select complex mode, press MODE and select Radian mode; scroll down and select a+bi and press ENTER.. To find the cube roots of z = i, start by finding one of the roots as follows. Press MATH, select :, enter ( i) and press ENTER. So one cube root is z 1 = 1 i.. Since cube roots occur at angles of -----, the second cube root can be found by multiplying z 1 by cis -----. Scrolling shows that this root is 0. + 1.i.. The third cube root is found by multiplying z 1 by cis -----. Scrolling shows that this root is 1. 0.i. Note that the cube root was recalled using nd [ENTRY] twice; cis ----- was also recalled using nd [ENTRY] twice and then edited to make it cis -----.
Chapter Complex s 15 G Solving equations in C 0 1 1 Solve the following quadratic equations over C. a x + x + 5 = 0 b x x + 5 = 0 c x 1x + 19 = 0 d x 1x + 1 = 0 e x x + = 0 Solve the following equations over C. a z z z + 10 = 0 b z z + z = 0 c z 7z + 10z = 0 d z 1z + 5z = 0 e z 0z + z 0 = 0 For f(z) = z, g(z) = z z + 1 and h(z) = z 5z + 5z show that f(z) g(z) = h(z) and hence determine the values of z such that h(z) = 0. Solve these equations over C. a x + 5x + 1 = 0 b z z = 0 c 9z + 5z = 0 d x + 1x + 9 = 0 5 The solutions to the equation (z ) + = 0 are: A z = + i, z = i B z = i, z = + i C z = + i, z = i D z = i, z = + i E z = 9 + 1i, z = 9 1i 5 Find the square roots of each of the following in Cartesian form. a 1 + i b 11 + 0i c 1 + i 7 Find i in Cartesian form. If one of the square roots of a(1 + i) is a cis --, the other square root is: A a cis ----- 7 B a cis ----- 7 7 C a cis ----- D a cis ----- 9 E a cis ----- 9 9 Use De Moivre s theorem to solve the following equations, in polar form. a z = i b z = + i c z = + i d z = i e z = 1 i f z + i = 0 1 -- 10 Find ( 15i) and determine the value of the sum of the roots. 11 a Find the cube root of. b Show the results on an Argand diagram. 1 Solve the following equations in Cartesian form. a z = 1 b z = 5 c z = d z = 7 1 Find all z satisfying a z 5 = 1 b z + 1 = 0. Express your answers in polar form. GC Roots of program TI GCprogram Casio Roots of WorkSHEET.