GRADED QUESTIONS ON COMPLEX NUMBER

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1 E /Math-I/ GQ/Comple umer GRADED QUESTINS N CMPEX NUMBER. The umer of the form + i y where ad y are real umers ad i = - i. e.( i ) is called a comple umer ad it is deoted y z i.e. z = + i y.the comple umer z is represeted y a poit P(,y) o XY - plae. is called real part ad is deoted Re z = & y is called imaiary part ad is deoted y Im z y.. If z = + i y e a y comple umer the its cojuate is deoted y z ad z iy.. 6. (Z + z), y (Z - z) i Euler' s ormulae i - i i) e cos i si ii) e cos i si Polar form i r e y ta y / z = r cos + i si Where r =,. 7. r = y is said to e modulus of the comple umer z ad is deoted y z iy y - 8. = ta y / is said to e amplitude or arumet of the comple umer z = + iy ad is deoted y ar z = amp z ta y / The value such that - is said to e priciple value of the arumet. 9. Properties of modulus ad arumet. z z z z. z z z - z z z. zz z z. z z. ar z z 6. ar z z z z z ar ar ar ar z. Hyperolic uctios et e real or comple the e e is called h yperolic sie of ad is deoted y sih ad e e is called h yperolic cosie of ad is deoted cosh sih = e ; cosh = e e e

2 E /Math-I/ GQ/Comple umer tah = sih cosh coth = cosh sih e e sech = e e cosh e e e e cosech = sih = e e e e. Relatios etwee circular fuctio ad hyperolic fuctios. Square Relatios (i) Si i = i sih (iii) ta i = i tah (v) Cosh i = cos (ii) cos i = cosh (i v) Sih i = i si (vi) tah i = i ta (i) Cosh - si h = (ii) sech + tah = (iii) coth - cosech =. Hyperolic Idetities. (i) sih (-) = - sih ( ii) cosh (-) = cosh (iii) tah - = -tah (i v) e cosh sih cosh sih. Additio formulae Sih (A B) = sih A cosh B cosh A sih B Cosh (A B) = cosha coshb sih A sih B tah A B tah A tah B tah A tah B 6. Product ormulae (i) sih ( A + B) + sih (A - B) = sih A cosh B. (ii) sih (A + B) - sih (A - B) = cosh A sih B (iii) cosh (A + B) + cosh (A - B) = cosh A cosh B (iv) cosh (A + B) - cosh (A - B) = sih A sih B C + DI C - D (v) sih C + sih D = sih cosh K J H G I K J C + DI C - D (vi) sih C - sih D = cosh sih K J H G I K J C + DI C - D (vii) cosh C + cosh D = cosh cosh K J H G I K J C + D C - D (viii) cosh C - cosh D = sih sih I K J H G I K J

3 E /Math-I/ GQ/Comple umer 7. ormulae for ad i) sih = sih cosh ii) cosh = cosh sih iii) cosh cosh iv) sih cosh tah v) tah = vi) sih = sih + sih + tah vii) cosh = cosh cosh viii) tah = i) sih = tah - tah ) cosh = tah + + tah tah tah tah 8. Values of Hyperolic fuctios X sih cosh tah EXAMPES: a i a. If iy the provethat y c d c id. If i the prove that a a i. id the comple umers whose sum is ad whose product is 8.. id the comple umer z such that z + i H G I z ad ar z KJ i) ar z+ ad ar z- 6 ii) id z if z + i / iii) id z if ar z+i = / ad ar z-i =. id the loci represeted y ii) z + i z - i = 6 ii) z - z + = iii) z - + i = iv) z -- 6i =

4 E /Math-I/ GQ/Comple umer. Simplify C.N. Cos - i si cos 7 + i si 7 Cos - i si cos + i si. id the mod ul us ad ar umet of 9 + si + i cos. Pro ve that cos i si + si - i cos. Evaluate. Sho w that + si + i cos + si - i cos + cos + i si + cos - i si DE MIVRE S THEREM Cos + i si cos - i si. Epress i the form a + i + i si /9 I cos K J Cos isi 8 i d i i. E press i the form a+i 8 i d i i. It z = i 0 0 The simplif y z d z i + i i I K J cos + i si /. Show that + cos + i si + cos - i si = cos cos + - cos - + I K J H G I K J + -I + + = si cos K J H G I K J. Show that + si + i cos + + si - i cos = cos 6. Show that cos - cos + i si - si + cos - cos - i si - si C.N.. If z = cos +i si the pro ve that i) ta ii) +z i icot +z -z. Pro ve that () po wer of +7i th is eq ual to (-) where is positi ve iteer (-i) d d i i I K J I K J 0

5 i i z z. If z e ad z e the pro ve that si - i z z. If a + c & ic ( a)z the prove that a + i iz + c iz ta. If si ita the prove that cos isi ta 6. If si ita the prove that cos isi ta H G I K J E /Math-I/ GQ/Comple umer I KJ C.N.. If z ad z are a y t wo comple umers such that z z ad z z the sho w that z z z z is p urel y imaiar y. If z ad z are a y t wo comple umers the pro ve that z z z z z z. Prove that the statemet Rez 0 & z z are equivalet where z iy. If z ad z are a y t wo comple umers such that z + z z z ad the show that arz. Prove that z z arz ar z C.N.. Pro ve that e a i cot -. If i, i & cot prove that.si.cosec. Pro ve that +cosec / a i- i+ i i c e h c e h

6 E /Math-I/ GQ/Comple umer C.N.. If ad are the roots of the eq uatio + = 0 The pro ve that 0 /. If ad are the roots of the eq uatio - +=0 The pro ve that. H G I cos K J 8 8 hece ded uce that. If ad are the roots of the eq uatio 0 The pro ve that cos H G I 6 6 Hece fid the val ue K J of +. If ad are the roots of the eq uatio 0 the pro ve that H G I cos Hece pro ve that 6 K J. If ad are the roots of the eq uatio z si -z si + = 0 the pro ve that cos cosec ) 6. If z=-+i & is iteer, the prove that z z 0 if is ot a mutiple of. 7. If + r = cos, pro ve that cos r r 8. If + = cos ad y + cos the pro ve that oe of the val ues of y m is cos m + ad that of + y is cos m - m y m y m y 9. If + = cos, y cos ad z+ Z cos y the pro ve that i) y z + = cos + + ii) m + y yz y m = cos m - 0. If - s = si, y - i & z- si the pro ve that y Z i) y z + = cos + + ii) the oe of yz m y + m y is cos m - I K J 6

7 C.N.6 E /Math-I/ GQ/Comple umer. If is a positive iteer the show that e j - a + i + a - i = a + cos ta Hece fid the value of + i + - i m m m m - e j. Prove that a + i a i a cos Hece fid the value of + i - i. Prove that + i i cos. Prove that C.N. 7 N M / / d i d i N M I I R S T -+ i i if = k! KJ KJ if = k a I K J ta / a Where k is a positive iteer 6. If is a positi ve iteer ad + p + p p... p the pro ve that. p0 p p... cos. p p p... si. p p p... cos 0 8 H G I K J 0 H G I K J H G I K J C.N. 8. If = e, y = e ad z = e ad + y + z = 0 the pro ve that + y + i i i z = 0. If si + si = 0 ad cos + cos = 0 the pro ve that i. cos + cos = cos + + ii. si + si = si + +. If cos + cos + cos = 0 ad si +si +si =0 the pro ve that i. cos + cos + cos = 0 ii. si + si + si = 0. If cos +cos +cos = si +si +si =0 the pro ve that i. cos cos cos ii. si si si = / si si iii. cos + cos cos 0 i v. si + 0 C.N. 9 Epasio of cos, si i powers of si, cos 7

8 E /Math-I/ GQ/Comple umer. Usi De-Moi vre s theorem sho w that si = cos si si. Use De-Moi ver s theorem to pro ve the follo wi result cos = cos 6cos si si. tai the e pasios of cos ad si i terms of po wers of cos ad si. Pro ve that si = si-0 si +6 si. Pro ve that cos = cos-0 cos +6 cos 6. tai the e pasio of cos 6 i terms of po wers of si ad cos 7. Pro ve that cos 6 = -8 si 8si si 6 8. If si6 = A cos si - B cos si + C cos si id the val ues of A, B ad C Prove that si7 = 7cos si - cos si cos si si 0. Prove that si 6cos cos + si si7 6. Prove that 7-6 si si 6si si si7 6. Prove that 6 cos 80 cos cos si. ta Prove that ta = ta -0 ta Hece deduce that ta ta 0 0 ta ta 7 Pro ve that ta 7 = 7t-t t t. where t = ta 6 t t 7t Hece ded uce that -ta 7 6 ta ta 0 C.N. 0 Epasio of cos, si i terms of sies or cosies of multiples of 8

9 E /Math-I/ GQ/Comple umer. Pro ve that si si si 0si tai the e pasios of cos ad si i terms of cosies of multiples of 6 6. Pro ve that cos si 8 cos 6 6. Use De-Moi vrers theorem to pro ve that cos si cos 6 cos 6 7. Pro ve that - si si 7 7si si-si 8 6. E press cos as a series i cosies of m ultiples of Pro ve that cos si cos8 + 8 cos E press cos si as a series i sies of m ultiples of cos si = - 9. Prove that si 8 + si6 - si -6 si 7 0. Pro ve that cos si = - 7 si - si0 - si si 6 +si-0 si 6 7. Pro ve that - cos si si - si - 6 si9 + 6 si7 + si - si - 0 si. Sho w that si cos = cos6 - cos - cos +.. If si cos = A cos A cos A cos A cos7 7 Pro ve that A 9A A 9 A 0 7. Pro ve that +cos7 where = cos +cos. Usi De-Moi vre s theorem sho w that +cos8 where cos 6. Use De-Moi vre s theorem to sho w that e +cos 9 = +cos 6cos 8cos cos cos j c h c h C.N. Roots of Comple Numer. id all the values of i ad show that their cotiued product is -. id all the values of + ad show that their cotiued product is.. id all the values of + i ad show that their cotiued product is + i. Show that the cotiued product of all the values of + i is + i. / / id the cotiued product of all the values of - i I KJ C.N. Solve the followi equatio / 8 / 9

10 E /Math-I/ GQ/Comple umer 6. i 0. i i c h C.N. Roots of Comple Numer. id the cue roots of uity ad show that they ca e epressed as,,. Prove that the roots of uity are i eometric proressio & their sum is zero & product is - th - 6. If is the comple cue root of uit y the sho w that - 7. th id the roots of - ad show that they ca e epressed as,,... Also fid their cotiued product. Sho w that the roots of the eq uatio 0 ca e writte as,,,,. c hc hc h Hece pro ve that If,,... are the roots of the eq uatio 0 the pro ve that 6 - c- hc- h c- hc- hc- h 7 7. Sol ve the eq uatio 0 ad sho w that = - + cos c I K J I h cos K J C.N. 0

11 E /Math-I/ GQ/Comple umer. Pro ve that a+i + a-i has real val ues ad fid those of +i + -i. Sho w that the roots of the eq uatio - are i ve y I K J H G I K J -+i si = cos r r where r = 0,,,,. id the commo roots of 0 & 6 i 0. Sol ve the eq uatio 0 ad fid which of its roots satisf y the eq uatio 6 6. Sho w that e ver y root of + 0 has for its real part. 6 6 p+ 6. Sho w that the roots of + 0 are i ve y -i cot N M Sho w that the roots of the eq uatio + are i ve y k 7 H G I icot K J, k =,, Wh y k 0? 0, p=0,,,,, 8. Sho w that the roots of the eq uatio z- z are i ve y z = i k cot, k =,,, k 0 H G I K J 9. id the three c ue roots of -cos -i si 0. Sol ve the eq uatio z i C.N.. If + i is a root of the equatio id all other roots. If oe of the roots of the equatio is + i id all other roots. C.N. 6 Hyperolic uctios Seperate the real ad imaiary parts of i. si + iy ii. cos + iy iii. sih + iy i v. cosh + iy iv. ta + iy v. tah + iy C.N. 7

12 E /Math-I/ GQ/Comple umer. Sho w that cosh -sih +tah. Pro ve that cosh -tah sih +tah. Pro ve that cosh 6 6 -tah sih cosh +sih. Pro ve that cosh cosh -sih sih. Pro ve that cosec h + coth = coth / 6. If sih cosh, fid tah cosh sih 7. Pr ove that 6cosh cosh cosh 0cosh - 8. If =tah 0. the pro ve that sih = / ad cosh = / 9. If si = tah Pro ve that ta = sih 0. Pro ve that - - -cosh C.N. 8 cosh ( M'96 ). Pro ve that - - +sih sih.. If cosh + i i y the prove that i. y y ii. cosh sih cos si i. sec y cosec ii. sec h y cosech. If si + i y. If cos i y i the prove that i. u cosec v sec ii. u sec h y + v cosech y If sih a + i i. - u i v the prove that i y the prove that y y ii. sih a cosh a si cos

13 E /Math-I/ GQ/Comple umer C.N. 9. If ta + i = + i y the pro ve that i. + y + cot = ii. + y - y coth = -. If tah a + i = + i y the pro ve that i. + + y = coth a ii. + y + y cot =. If + i y = ta. If tah + i 6 C.N. 0 + i the pro ve that + y 6 I K J + = I K J = + i y the pro ve that + y + y = i. If si +i e the pro ve that cos = si i. If sih i e the pro ve that sih cos cos. If cosh +i cos i si the pro ve that si si sih C.N.. If si + i Pcos + isi the prove that i. P cosh - cos ii. ta = tah cos z. If e si u + iv ad z = + iy the prove that e = cosh v - cosu. If + iy = cos + i the epress ad y i terms of ad. Hece show that cos ad. cosh are the roots of the equatio. y If si i iy the show that si ad cosh y are the roots of the equatio p c h p + 0. If cos + i cos i si the prove that cos cosh c h C.N.

14 . If ta +i y i the pro ve that. If +i y = c cot u+i v the pro ve that E /Math-I/ GQ/Comple umer - cos cosh y y si u v = c sih cosh v-cos u. If ta +i y = si u+i v the pro ve that si sih y ta u tah v C.N.. If +i = tah + i I the pro ve that K J. If cosec +i = u+i v the pro ve that u v = u I K J v I K J c h c h If +i y= cosh + i. the pro ve that y C.N.. If fa +i the pro ve that i. = + ta i sec ii. e cot I K JI KJ. If u = lo ta + the pro ve that i. tah u/ = ta ii. cosh u.cos. If ta = tah u the pro ve that i. sih u = ta ii. cosh u = sec N M Q P. If cos +i r e the sho w that = i lo si - si. If lo ta = y the pro ve that i. sih y = ta cot ii. cosh + y + cosh - y = cosh y.cosec 6. If cosh = sec the pro ve that i. tah = ta I ii. sih = ta iii. = lo ta + iv. ta ( e ) K J

15 C.N. +i y-c u+iv. If = e where, y, u ad v are real the pro ve that +i y+c i. = -c sih u cosh u - cos v ii. y = c si v cosh u - cos v E /Math-I/ GQ/Comple umer r e i. If ta /+i v the sho w that r=, ta =sih v ad tah v = ta. If = si cosh ad y = cos sih the pro ve that i y i. cosec -i + cosec +i = ii. cosec -i - cosec +i = y y. If = cos cosh ad y = si sih the pro ve that i y i. sec i + sec -i = ii. sec +i - sec i = y y. If +i y = cos u+i v the pro ve that i. + y cosh v cosu ii. - y cosh v-cosu Pro ve that ta If u- u+ ta tah ta tah + ta ta cot coth ta tah ta tah si +i y the sho w that the ar umet of u is + where cos sih y ta = +si cosh y, - - cos sih y ta = -si cosh y C.N. 6 Pro ve that e j e j - -. sih = lo + +. cosh = lo + - cosech = lo sech = lo tah = lo + coth = - I - - lo K J H G I - K J

16 E /Math-I/ GQ/Comple umer C.N. 7 Pro ve the follo wi relatios I - -. tah si = cosh sec + KJ I - KJ ta I = lo ta + / tah cos = cosh cosec K J. cosh + = tah - -. sech si = lo cot. tah = sih sih cosh + = sih 8. sih = cosec - + C.N. 8. Seperate the real ad Imaiar y parts of ta. c h i i Pro ve that ta e lo ta c h - i i. Pro ve that ta e lo ta c h - i. Pro ve that si e cos si i lo si si c h - i. Pro ve that cos e si si i lo si si - i ce h 6. Seperate the real ad imaiar y parts of i. ta - iy ii. tah - iy I K J I K J C.N. 9 Pro ve that. si z= - - lo iz z. si i i lo i -.cos i = i lo +. cos z = -i lo z+ z. sih 7. ta z = i lo e j e j e j e j i cosh i 6. cosh i sih i - - i+zi 8. si cosec i-z K J i lo cot 6

17 C.N. 0 E /Math-I/ GQ/Comple umer Epress i the form a + i I K J i i. cos - ii. cos - i iii. si - i / i v. si - i C.N. I. If + iy = i + i e lo the show that = ad y = lo sec + ta i i KJ i e I. If cos cosh + the show that = lo + K J H G I ia i KJ d i. Pr ove that tah(lo ). If ta +i y i where ad y are real the pro ve that is idetermiate ad y is ifiite. I K J - a. Pro ve that ta i lo a/ i a C.N. GARITHMS CMPEX NUMBER E press i the form a+ i i. lo - ii. lo - iii. lo + i i v. lo + i H- I K H - i I K m+ v. Prove that lo i i = + C.N. si +i y -. Pro ve that lo M P i ta cot tah y si iy N a+i -. Pro ve that lo i ta / a a-i. Pro ve that ta i lo. Pro ve that cos i lo. Pro ve that si i lo Q I K J a-ii a K J a a+ii a-i K J a a a-ii a+i K J a a 7

18 C.N. I -e cosec i K J H G I K J I lo i K J. Pro ve that lo p iy y ta / a. If a+i m the pro ve that loca h iy +i -. If i the pro ve that ta / y lo iy. Pro ve that lo +cos i si lo cos i E /Math-I/ GQ/Comple umer C.N.. Seperate the real ad imaiary parts y cosideri the pricipal values oly. C.N. 6 d i d i i I i K J i i i) + i ii) iii) i - i i i iv) i v) + i vi) -i lo. id the pricipal val ue of i i - /8 ad sho w that its real part is e cos lo. If lo lo +i y p iq the pro ve that y= ta taq lo y. d i H G I K J d ii Seperate i ito real ad imaiar y parts y cosideri pricipal val ues ol y C.N. 7. If i i the pro ve that e +i. i A ib, Pro ve that i) ta. A i-- i - B H G I K J A i B If i i the pro ve that i A B + = B A ii) A + B = e i. Pro ve that i cos i si where = +/ e. id the pricipal val ue of +i ta i m 6. Pr ove that the eeral value of ( i ta ) is e cos(lo ) i si(lo ) 7. id the pricipal val ue of +i d i d i i i - m+/ 8

19 C.N. 8. If lo si +i y = a+i the pro ve that a i) e cosh y - cos ii) ta = cot tah y I K J E /Math-I/ GQ/Comple umer -. Sho w that lo ta i i ta sih a. If ta lo+i y a i the pro ve that ta lo c y h where a a i ta sec e j. If +i ta ca ha ve real val ue the sho w that it is sec Eamples P. ANSWERS. i 6. i. z i ii z i. y y i ii iii 7 y DE MIVRE S THM I K J H G I K J cos + i si 6. modulus = i, arumet = / 6 C.N.. Modulus = cos sec C.N. I KJ I K J. i z i iii. z.. - Arumet = C.N. 6 Cos. -. cos ta C.N. 9. cos = cos 0 cos si cos si si = cos si - 0 cos si + si cos 6 = cos - cos si + cos si - si 8. A = 6, B = 0, C = 6 9

20 C.N si cos 6-6 cos + cos -0 6 cos cos cos + cos cos cos cos 6 8cos 6cos 8 8. cos Si si 8 + si 6 - si - 6 si 8 E /Math-I/ GQ/Comple umer C.N.. = cos k + + i si k +, k = 0,,,,,. = cos k + + i si k +, k = 0,, 6 6. =, i. = cos k + i si k +, k = 0,,,,. = cos k + + i si k +, k = 0,,,, ad ki k = cos + i si k = 0,,,, K J H G I K J 6. = cos k + + i si k +, k = 0,,,,, ad 6 6 = cos k + + i si k +, k = 0,, 7. = cos k + + i si k +, k = 0,,,,, ad 6 6 = cos k + + i si k +, k = 0,, 8. = cos k + - i si k +, k = 0,,,, ad 8 8 = cos k + + i si k +, k = 0,, 0

21 E /Math-I/ GQ/Comple umer. = cos k + i si k, k =,,,, k 0. = cos k + i si k, k =,,,, k 0 9. = cos k + + i si k +, k = 0,,, ad / = cos k + + i si k + k = 0,,, 0 = cos k + + i si k +, k = 0,,,, ad / = 0 cos k + + i si k + k = 0,,,,. = cos k + i si k +, k = 0,,,, k. = cos k + i si k +, k = 0,,,,, 6, k 7 7. = cos k + i si k +, k = 0,,,, k. = cos k + i si k, k =,,,, k 0. = cos k + i si k, k =,,,, k 0 6. = k + i si k cos, k =,,,, k 0 7. =, k = 0,,,,,, 6, 7 cos k + i sik = cos k + i si k +, k = 0,,, k, 6 6 C.N. I K J / k + /. cos, k 0,,. = cos k + i si k +, k = 0,,,,, ad 6 6 = cos k k i si, k = 0,,,,, commo roots are = k + i si k, k =,,, k 0,

22 E /Math-I/ GQ/Comple umer / k - k - 8. si / cos isi, k = 0,, Z = cos k + i si k + k = 0,,,, 0 0 C.N.. -i, i. -i, i C.N. 6. Si cosh y + i cos sih y. cos cosh y - i si sih y. sih cos y + i cosh si y. cosh cos y + i sih si y. si + i sih y cos + cosh y C.N. 8 i + si. lo - si 6.. i. C.N. 0 H G I K J N M Q P M P N Q ta - + i lo y + y y ii. lo + y - ta M P i y y y 6. N M sih + i si y cos + cos y ilo. ilo. i lo. lo + Q P d i d i C.N. a f. lo + i k + lo lo + am + fa + f i am f lo - a + f lo. lo a f a f lo i lo -. lo + i k + ta /. 6 lo 6

23 E /Math-I/ GQ/Comple umer C.N.. e - / lo - / I K J I K J I K J I K J I cos lo i si lo K J I K J. e Cos lo i si lo. e. ie. e cos lo i si lo 6. C.N. 6 - / / / i / - /. e cos i si C.N. 7 k +. e cos lo cos - i si lo cos I K J e

Chapter 13: Complex Numbers

Chapter 13: Complex Numbers Sectios 13.1 & 13.2 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber 1. Comple umbers Comple umbers are of the form z = + iy,, y R, i 2 = 1. I the above defiitio, is the real part