Complex Numbers Practice 0708 & SP 1. The complex number z is defined by

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1 IB Math Hgh Leel: Complex Nmbers Practce 0708 & SP Complex Nmbers Practce 0708 & SP. The complex nmber z s defned by π π π π z = sn sn. 6 6 Ale - Desert Academy (a) Express z n the form re, where r and hae exact ales. Fnd the cbe roots of z, expressng n the form re, where r and hae exact ales. (Total 6 marks). The polynomal P(z) = z + mz + nz 8 s dsble by (z ++ ), where z and m, n. Fnd the ale of m and of n. (Total 6 marks). Let =+ and =+ where =. (a) () Show that. () By expressng both and n modls-argment form show that π π sn. () π Hence fnd the exact ale of tan n the form a b where a, b. Use mathematcal ndcton to proe that for n n n nπ nπ sn. (c) Let z =. Show that Re z = 0. +, (5) (7) (6) (Total 8 marks) C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of

2 IB Math Hgh Leel: Complex Nmbers Practce 0708 & SP Ale - Desert Academy. (a) Express the complex nmber + n the form ae, where a, b Usng the reslt from (a), show that, where n, has only eght dstnct ales. (c) Hence sole the eqaton z 8 = Fnd, n ts smplest form, the argment of (sn + ( )) where s an acte angle. z 6. Consder w = where z = x + y, y 0 and z + 0. z Gen that Im w = 0, show that z =. 7. (a) Use de More s theorem to fnd the roots of the eqaton z =. Draw these roots on an Argand dagram. z (c) If z s the root n the frst qadrant and z s the root n the second qadrant, fnd n the z form a + b. n π b +. () (5) () (Total 9 marks) (Total 7 marks) (Total 7 marks) (6) () () (Total marks) C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of

3 IB Math Hgh Leel: Complex Nmbers Practce 0708 & SP Ale - Desert Academy 8. Gen that (a + b) = + obtan a par of smltaneos eqatons nolng a and b. Hence fnd the two sqare roots of +. (Total 7 marks) 9. Gen that z = 0 0, sole the eqaton 5z + z * = 6 8, where z* s the conjgate of z. (Total 7 marks) 0. Sole the smltaneos eqatons z + z = z + ( )z = gng z and z n the form x + y, where x and y are real. (Total 9 marks). b 7 9 Fnd b where. b 0 0 (Total 6 marks). Gen that z = (b + ), where b s real and poste, fnd the ale of b when arg z = 60. (Total 6 marks). Consder the complex geometrc seres e θ + e e +... (a) Fnd an expresson for z, the common rato of ths seres. () Show that z <. () (c) Wrte down an expresson for the sm to nfnty of ths seres. () (d) () Express yor answer to part (c) n terms of sn θ and θ. () Hence show that θ + θ + θ +... =.. The roots of the eqaton z + z + = 0 are denoted by α and β? (a) Fnd α and β n the form re θ. (c) Gen that α les n the second qadrant of the Argand dagram, mark α and β on an Argand dagram. Use the prncple of mathematcal ndcton to proe De More s theorem, whch states that nθ + sn nθ = ( θ + sn θ) n for n +. (d) Usng De More s theorem fnd n the form a + b. (e) Usng De More s theorem or otherwse, show that α = β. (f) Fnd the exact ale of αβ* + βα* where α* s the conjgate of α and β* s the conjgate of β. (g) Fnd the set of ales of n for whch α n s real. 5 (0) (Total 6 marks) (6) () (8) () () (5) () (Total marks) C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of

4 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme Complex Nmbers Practce 0708 & SP MarkScheme. (a) z AA = 8 r = 8 θ or 60 z 8e z z e e e, e e e z Notes: Do not allow any form other than re. Both answers mst be gen for fnal A.. METHOD Usng factor theorem (M) Sbstttng z = nto P(z) M (6 + n) + (m n) = 0 A Eqatng both real and magnary parts to zero M Hence m = and n = 6 AA N METHOD Usng Conjgate root theorem M Mltply (z + )(z + + ) = z + z + M Let P(z) = (z + z + )(z a) (M) a = 8 a = A Hence m = and n = 6 AA N. (a) () Usng * where * = (M) A = AA Note: Award A for a correct nmerator and A for a correct denomnator. AG () and arg sn AA and arg sn AA A A A A Ale - Desert Academy N N0 [6] [6] C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of

5 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme sn MA sn AG () METHOD Usng arg to form arctan (M)(A) tan A (Let P(n) be ( + = M = A N0 METHOD sn (M) tan and sn C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM (A) = M = A N0 Note: Please check that has been consdered n ether lne or lne. n n n n ) sn ) For n = : sn, so P() s tre A Assme P(k) s tre, M k k k k sn (A) Consder P(k + ) k k A M k k k = sn sn A = k k k sn A Ale - Desert Academy N0 Page of

6 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme Ale - Desert Academy C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of P(k) tre mples P(k + ) tre, P() tre so P(n) tre n +. R N0 (c) METHOD + = ( + ) + ( + ) (M)(A) = ( ) + ( ) (A) M Re (A) = A = 0 AG N0 Note: If the canddate explans that to show that Re z = 0, t s only necessary to consder then award as aboe. METHOD (M)(A) (A) M Re (A) = A = 0 AG N0 Note: If the canddate explans that to show that Re z = 0, t s only necessary to consder then award as aboe. METHOD sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn

7 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme Ale - Desert Academy C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of (M)(A) = A = M Re A = A = 0 AG N0 METHOD (M)(A) = A = M Re A = A = 0 AG N0 [8]. (a) + = ( a = + = ) sn sn sn sn sn sn sn sn b a e

8 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme (c) + = e AA N EITHER n n e e Let n = 0,,,,, 5, 6, 7 Hence the eght dstnct ales are, e MA M A Ale - Desert Academy There are only eght dstnct answers snce the next answer wold be e whch s and hence the argments to all frther answers wold be the same as the frst eght pls a mltple of. R OR n n e e Snce ( θ θ e ) e n 0 Hence n can only take the ales 0,,,,, 5, 6 and 7. From part f we rase each of these roots to power 8 then the answer s. Hence these are the eght roots to ths eqaton. z =, e MA M A R (M) 5. (sn + ( )) = sn ( ) + sn ( ) MA Let be the reqred argment. sn θ θ tan = M sn θ θ sn θ θ = (M) θ θ θ sn θ θ = A θ θ = tan A = A 6. METHOD tan θ θ Sbstttng z = x + y to obtan w x n n 5 x y y xy w x y 7 x y 5 7 A (A) A [9] [7] C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page 5 of

9 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme Use of (x y + xy) to make the denomnator real. M x yx y xy = A x y x y Im w yx y x y x y x y y x y = x y x y A Im w = 0 x y = 0 e z = as y 0 RAG N0 METHOD w (z + ) = z (A) w(x y + + xy) = x + y A Eqatng real and magnary parts w (x y + ) = x and wx =, y 0 MA x y Sbstttng w to ge x x x x A y x x or eqalent (A) x + y =, e z = as y 0 RAG 7. (a) z = ( ) Let = r( θ + sn θ) r π θ = z = = (A) π nπ π nπ = sn M 6 6 = π π sn 6 6 Note: Award M aboe for ths lne f the canddate has forgotten to add π and no other solton gen. = = π π sn π nπ sn 7π 7π sn 6 6 5π 6 5π sn 6 π nπ A A M Ale - Desert Academy [7] C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page 6 of

10 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme Ale - Desert Academy = 9π 9π sn A 6 6 Note: Award A for correct answers. Accept any eqalent form. 8 Note: Award A for roots beng shown eqdstant from the orgn and one n each qadrant. A for correct anglar postons. It s not necessary to see wrtten edence of angle, bt mst agree wth the dagram. (c) 5π 5π 8 sn z 6 6 z 7π 7π 8 sn 6 6 MA π π = sn (A) = A N ( a = 0, b = ) 8. a + ab b = + Eqate real and magnary parts a b =, ab = Snce b = a a a a a = 0 Usng factorsaton or the qadratc formla a = ± b = ± C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM A (M) A (M) A (M) = +, AA 9. 5zz* + 0 = (6 8)z* M Let z = a + b = (6 8)(a b) (= 6a 6b 8a 8b) MA Eqate real and magnary parts (M) 6a 8b = 60 and 6b + 8a = 0 a = and b = AA z = A [] Page 7 of [7]

11 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme 0. z + z = z = z z + ( )z = z + ( ) z = MA z z z = 5 z z z z = 5 + A EITHER Let z = x + y (M) x + y x y = 5 + Eqate real and magnary parts M x + y = 5 x + y = y = 8 y = x =.e. z = + AA z = ( ) M z = 7 z = A OR 5 z = M (5 )( ) 5 8 z = MA ( )( ) z = + A z = ( + ) + M z = 7 z = A. METHOD 0 + 0b = ( b)( 7 + 9) (M) 0 + 0b = ( 7 + 9b) + (9 + 7b) AA Eqate real and magnary parts (M) EITHER 7 + 9b = 0 b = (M)A OR 0b = 9 + 7b b = 9 C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Ale - Desert Academy Page 8 of [7] [9]

12 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme b = (M)A METHOD ( b)( b) 7 9 = (M) ( b)( b) 0 b b 7 9 b 0 A Eqate real and magnary parts (M) b 7 Eqaton A b 0 b 9 Eqaton B b 0 From eqaton A 0 0b = 7 7b b = 7 b = ± A From eqaton B 0b = 9 + 9b b 0b + = 0 By factorsaton or sng the qadratc formla b = or A Snce s the common solton to both eqatons b = R. METHOD snce b > 0 (M) arg(b + ) = 0 A b = tan 0 MA b = A N METHOD arg(b + ) = 60 arg(b + b) = 60 M b = tan 60 = MA ( b ) b b = 0 A ( b )( b ) = 0 snce b > 0 (M) b = A N e. (a) z = (M) e z = e A N z A z < AG Ale - Desert Academy [6] [6] C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page 9 of

13 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme a (c) Usng S = (M) r e S = A N e e cs (d) () S = (M) e cs sn (A) ( sn ) Also S = e θ + e e... = cs θ + cs cs... (M) S =... sn sn sn... () Takng real parts, sn... Re A ( sn ) = (sn ) A ( ) ( ) = ( ) (5 ) A = AAG N (a) z = M + re r = A θ = arctan sn ( sn ) = Re sn sn M sn = sn A π C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM A Ale - Desert Academy A [5] Page 0 of

14 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme Ale - Desert Academy = re θ r = θ = arctan π e e π π A A A (c) (d) nθ + sn nθ = ( θ + sn θ) n Let n = Left hand sde = θ + sn θ = θ + sn θ Rght hand sde = ( θ + sn θ) = θ + sn θ Hence tre for n = Assme tre for n = k kθ + sn kθ = ( θ + sn θ) k AA MA M (k + )θ + sn(k + )θ = (θ + sn θ) k ( θ + sn θ) MA = ( kθ + sn kθ)( θ + sn θ) = kθ θ sn kθ sn θ + ( kθ snθ + sn kθ θ) A = (k + )θ + sn(k + )θ A Hence f tre for n = k, tre for n = k + Howeer f t s tre for n = tre for n = etc. R hence proed by ndcton π π 8e e π e A π π = sn (M) = AA (e) a = 8e π A β = 8e π A Snce e π and e π are the same α = β R (f) EITHER α = + β = α* = β* = + A αβ* = ( + ) ( + ) = = MA C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of

15 IB Math Hgh Leel: Complex Nmbers 0708 & SP - MarkScheme βα* = ( )( ) = + = + A αβ* + βα* = A OR Snce α* = β and β* = α αβ* = βα* = π αβ* + βα* = π e e e e π e e π π π e π e π MA π π π π = sn sn A π = 8 8 A n πn (g) α n = e MA Ths s real when n s a mltple of R +.e. n = N where N A Ale - Desert Academy [] C:\Users\Bob\Docments\Dropbox\Desert\HL\ AlgFnc\6ComplexNmbers\HL.ComplexPractce0708SP.docx on 5//6 at : PM Page of

(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1)

(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1) . (a) (cos θ + sn θ) = cos θ + cos θ( sn θ) + cos θ(sn θ) + (sn θ) = cos θ cos θ sn θ + ( cos θ sn θ sn θ) (b) from De Movre s theorem (cos θ + sn θ) = cos θ + sn θ cos θ + sn θ = (cos θ cos θ sn θ) +