B2C3

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1 - () 60 =60 =60 () = () 60 = ( ) =80 80 () = =0.075( ) () =( ) = Ex. θ=0 θ?θ Ans 0-8 Ex. a=sinb=sinc=sind=cose=cos5 Ansb>a>e>c>d Ex. P(costan6) Ans r θ( )A () S=rθ () =r+rθ () A = r θ Ex Ans 00 Ex5. 60 Ans Ex6. k = k Ans 6 Ex7. 0 Ans6 Ex8. AB a(= AB ) C C Ans( + ) a ( ) a A B ---/8

2 Ex9. Ans + Ex0. 6 A A C Ans6 A B ---/8

3 . y=sinx () R {y -y} () ()sin(-x)=-sinx y= y=-. y=cosx () R {y -y} () ()cos(-x)=cosx y ()y=sinx y=cosx cosx=sin(x+90 o ) y= - 0 y=- - y= y=- ---/8

4 . y=tanx () { x x R, x k +, k Z} () R () () x = k + k Z tanx (5) x = k + k Z (6) k Z k < x < k + (7) tan(-x)=-tanx /8

5 . y=cotx () {x x Rxk k Z} () R () ()x=k k Z cotx (5) k Z k<x<k+ (6) x=k k Z (7) cot(-x)=-cotx (8)y=tanx x y=cotx /8

6 5. y=secx () { x x R, x k +, k Z} () {y y y- } () () y=cosx (5) x = k + k Z secx (6) x = k + k Z (7) sec(-x)=secx y (8) y = sec x = y=cosx y=secx cos x /8

7 6. y=cscx () { x x R, x k, k Z} () {y y y- } () () x=k k Z cscx (5) (6) x=k k Z (7) csc(-x)=-cscx (8) y = csc x = sin x y=sinx y=cscx (9)y=cscx y=secx /8

8 x R T 0 f(x+t)=f(x)f f(x+t)=f(x) T f.f(x+t)=f(x) f(x+nt)=f(x)n Z k.f(x) T f(x)+kk f(x) f ( x T(k ) ) f(ax+b) T (ab a 0) a. sinxcosxsecxcscx tanxcotx. sinx cosx tanx cotx secx cscx x y x=0 x= x= x= x= y=sinx x R y y=cosx x R y 0 0 y=tanx x n+ y R y=cotx x n y R 0 0 y=secx x n+ y ory y=cscx x n y ory y=af(bx+c)+d a a ( )( ) b b ( )( ) c c ( ) d d ( ) Ex. x 0 y= x y=tanx (A)0(B)(C)(D)(E)AnsD Ex. () sinx= x 0 x ()cosx =0 Ans Ex. f(x)=sinx? (A) f(x) (B)f(x)x= (C)f(x) 6 (D)y=f(x) x= (E)f()>0AnsABCD ---8/8

9 Ex. y= x(cosx) AnsB (A) (B) (C) (D) (E) Ex5. f ( x) = cos x + sin x f(x) Ans 6 cos x Ex6. x R cos x + Ans Ex7. x sin x > x 5 Ans x < Ex8. sin=a (A) 0 < a (B) < a (C) < a AnsC n n Ex9. cos > 0, cos( ) < n=ans5 Ex o Ans50 Ex. k r= Ansk, k, Ex ( )Ans9( 70 ) ---9/8

10 Ex. AB = a EF CD, AB CD = a, EF = a Ans ( 6 + ) a A E C D F B Ex. Ans, Ex5. A A A 8 8 A A A A A 8 A 7 Ans + A A A A 5 Ex6. AB AB =6 Ans A B Ex7. ABCD AB = a, AD = a A a a Ans( + ) a A B D C Ex8. OPRQ 5 O, 0 Q R Ans + 75 Ex9. AOB AOB=90 o OA = OB = OB OA, B O P Ans O A ---0/8

11 Ex0. O O AOAC BD O C D () COD()ACCD O D B DB Ans + 6 A C Ex. 0 x ()y=sinx y=cosx ()y=cosx y=tanx ()y= secx y= cscx Ans Ex. ()0sinx=x()cosx=logxAns7 Ex. ( + sin x ) + ( + sin x) + ( + sin x) + (0 x )() x 5 () x=ans0 x< 6 <x 6 Ex. 0 x sin 5 xcosx<0ans0<x< <x< Ex5. f(x)=sin 7 x cosx+ x [, ] f Ans { y y } 6 8 Ex6. 6 θ y= log ( + sinθ ) + log ( sin θ ) Ans0- Ex7. cosθ + = k(cosθ ) k Ansk or k ---/8

12 - ( vs. ) sin(α±β)=sinαcosβ±cosαsinβ cos(α±β)=cosαcosβ sinαsinβ Ex8. cos= 7 cos(α+β)= αβ cosβ=ans Ex9. sinα+sinβ+sinγ=0cosα+cosβ+cosγ=0 cos(α-β) Ans Ex0., < α < < β < sin α =,sin β = 5 ()cos(α+β)()sin(αβ) Ans tanα ± tan β tan( α ± β) = tanα tanβ cot β cotα cot( α ± β) = cot β ± cotα Ex. tanαtanβ x 5x+=0 sin (α+β)+sin(α+β) cos(α+β)+6cos (α+β) Ans6 Ex. α+β= (+tanα)(+tanβ)=ans Ex. tan0 +tan0 + tan0 tan0 Ans ( ) sin(α+β)sin(α-β)=sin α-sin β=-cos α+cos β cos(α+β)cos(α-β)=cos α-sin β=-sin α+cos β Ex. ()sin 75 o sin 5 o =()cos 7.5 o sin 7.5 o =Ans 6 ---/8

13 Ex5. x+y= sin x sin y Ans.α+β+=( ) ()tanα+tanβ+tan=tanαtanβtan ()cotαcotβ+cotβcot+cotcotα= Ex6. ABC ABC ()tana+tanb+tanc=tanatanbtanc ()tana+tanb+tanc ()cotacotb+cotbcotc+cotccota= ()cota+cotb+cotc= ABC Ex7. ABC tana= tanb= tanc C Ans 5 o.α+β+= ()tanαtanβ+tanβtan+tantanα= ()cotα+cotβ+cot=cotαcotβcot Ex8. ABC ABC A B B C C A () tan tan + tan tan + tan tan = () A B C tan + tan + tan = ABC. θ x y=mx+p θ m=tanθ 5. L L m = tan β m = tanα L L y L L L L θ θ tanα tan β m m + tanαtanβ + mm tanθ = tan( α β) = = x ---/8

14 Ex9. x +xyy +5xy+=0 θ sinθ=ans 0 Ex50. L xy7=0l x5y+6=0 (5) Ans9x7y=07x+9y7=0( ) Ex5. ()sin(00 +θ)sin(0 -θ)cos(00 +θ)cos(0 -θ)=ans ()sin(θ+75 )+cos(θ+5 )- cos(θ+5 )=Ans0 ()cos77 o cos6 o sin77 o sin6 o =Ans ()sin9 cos79 -sin7 cos =Ans (5)cos6 o sin66 o sin6 o cos6 o =Ans Ex5. AB sina= sinb= A+B=Ans0 Ex5. tanα+tanβ=cotα+cotβ= tan(α+β)ans- Ex5. tan(α-β)=tan(β-γ)= tan(α-β-γ)=ans Ex55. α+β+γ=ans Ex56. ABC C= A DE BC BD = DE = EC tan( DAE)=Ans B D E C ---/8

15 Ex57. () tan87 o tan7 o tan87 o +tan7 o =Ans ()tanα=tan(α β)= tanβ=ans Ex58. () () y=x+6 5 () () x y + = 0 0 Ansx+y6=0 xy+8=0x y+ ( ) = 0x= Ex59. 6x xyy +ax+by+c=0 () ()(abc)=() Ans(050)5 5 Ex60. ABCD AB =6BC =5CD =5 ABC BCD sin ABC= 5 sin BCD= 5 BD AD Ans0 Ex6. 0 < α, β, γ < tanα=tanβ=tanγ= ()tan(α+β+γ)=()α+β+γ=ans 5 Ex6. α+β= (tanα)(tanβ)=ans ---5/8

16 - ( ) tanθ t () sin θ = sinθcosθ = = + tan θ + t tan θ t () cosθ = cos θ sin θ = cos θ = sin θ = = + tan θ + t tanθ t () tan θ = = tan θ t cot θ () cot θ = cotθ Ex6. sinθx +x =0,cosθ Ans Ex6. 0 < θ < tanθcotθ x 5x+=0 sinθsinθcosθsinθans Ex65. cos0 o cos0 o cos80 o Ans 8 Ex66. sin5 o sin5 o sin5 o sin55 o sin65 o sin85 o Ans 6 Ex67. csc0 o - sec0 o =Ans ( ) () sinθ = sinθ sin θ () cosθ = cos θ cosθ ($.=$.-$) Ex68. sin8 o =cos6 o =Ans 5 5+ Ex69. sinθ cosθ =,sinθ+cosθ=ans /8

17 Ex70. () cos5θ = a cosθ + bcos θ + ccos 5 θ,abc=ans5-06 () cos o Ex7. x+sin 8x -6x+ Ans 8 θ θ ( )( ) θ cosθ () sin =± θ + cosθ () cos =± θ cosθ sinθ cosθ + sinθ cosθ () tan =± = = = + cos θ + cosθ sinθ + sinθ + cosθ! Ex7. sin67.5 o + =Ans Ex7. () cos 5 o =Ans + ()cos.5 o sin.5 o =Ans ()sin.5 o cos.5 o =Ans Ex7. cos x+sin x Ans 5 Ex75. sin + sin + sin + sin =Ans θ 7 Ex76. 5sinθ=8sin,cosθAns 5 or Ex o <α<60 o sinα+cosα= 5,sinαcos α Ans /8

18 Ex78. tan θ = sinθ=ans 5 Ex79. sin sin (60 o ) sin o θ + + θ + (60 θ) =Ans Ex80. 0 << cosθ= sinθ+cosθ=ans 7 Ex8. sin sin + sin + sin =Ans t Ex8. t=tan t sin cos Ans + t 6t + t + t + t Ex8. cos cos cos =Ans Ex8. sec o sec o sec8 o sec8 o =Ans6 5 Ex85. 0<x< sinx=cosx Ans 6 6 Ex86. sinθ= cosθ+sinθ=ans5 7 Ex87. tan= tan=tan5=ans 8, Ex88. sinx=cosx cosx=ans /8

19 - sinαcosβ=sin(α+β)+sin(αβ) cosαsinβ=sin(α+β)sin(αβ) cosαcosβ=cos(α+β)+cos(αβ) sinαsinβ=cos(α+β)cos(αβ) Ex89. ()cos7.5 sin7.5 =Ans + ()sin7.5 sin7.5 =Ans ()sin0 sin0 sin60 o sin80 =Ans 6 sina+sinb=sin A+ B cos A B sinasinb=cos cosa+cosb=cos A+ B sin A+ B cos A B A B cosacosb=sin A+ B sin A B Ex90. ()sin5 sin5 +sin5 =Ans0 ()cos5 +cos5 +cos5 =Ans0 ()sin50 +sin70 +sin90 o =Ans0 ()cos55 o cos65 o +cos65 o cos75 o +cos75 o cos55 o =Ans 5 Ex9. cos cos + + cos =Ans [HINT]sin ( ) 7 Ex sin + sin + sin + sin =Ans0[HINT]sin /8

20 Ex9. ()cos θ+cos (0 +θ)+cos (0 +θ)=ans [HINT] ()sin θ+sin (0 +θ)+sin (0 +θ)=ans Ex9. ABC sinb sinc=cos A ABC Ans Ex95. x + y = sinxsiny Ans Ex96. sina+sinb=acosa+cosb=ba +b >0 a,b ()cos(ab)=ans ( a + b ) [HINT] b a ()cos(a+b)=ans a + b ()sin(a+b)=ansab a + b [HINT] tan cos Ex97. sinα+sinβ= cosα+cosβ= cos(α β)sin(α+β) Ans7 5 o o sin55 sin5 Ex98. o cos80 cos0 =Ans o + Ex99. 0 o x90 o,tanx= o o cos85 + sin7 o o sin85 cos 7,x=Ans69o Ex00. sin 7θ sin θ + sinθ () =Anstanθ cos7θ cos θ + cosθ (cosθ cosθ )(sin8θ + sin θ ) () =Ans (sin 5θ sinθ )(cos6θ cos θ ) sin 5θ sinθ () =Anstanθ cos5θ + cosθ ---0/8

21 Ex0. ()cos6 o cos o cos66 o cos78 o =Ans 6 ()cos5 o cos5 o cos5 o cos55 o cos65 o cos85 o =Ans 6 ()cos0 o cos0 o cos0 o cos0 o cos50 o cos60 o cos70 o cos80 o =Ans 56 Ex0. 6 () cos + cos + cos =Ans () cos cos cos cos cos =Ans Ex0. 0x cosx+cosx+cosx=0ans, 5 7,,,, Ex0. α = 7 cos α + cos5α =Ans cosα cosα cos A + cos B + cosc Ex05. ABC A=60 Ans sin A + sin B + sin C Ex06. a=sin0,b=sin0,c=sin80 ()a+bc()a +b +c ()a +b c Ans0 8 Ex07. sinα+sinβ=cosα+cosβ= cosα+cosβ Ans0 ---/8

22 cosα + cos β + cosγ = 0 Ex08. sinα + sin β + sinγ = 0 ()cos(αβ)=ans ()cos(αβ)cosγ=ans0 ()sin α+sin β+sin γ=ans ()cosαcosβ+cosβcosγ+cosγcosα=ans Ex09. ABC, ()sina+sinb+sinc=cos A cos B cos C ()cosa+cosb+cosc=+sin A sin B sin C ()sina+sinb+sinc=sina sinb sinc ()cosa+cosb+cosc= cosacosbcosc (5)sin A+sin B+sin C=+cosAcosBcosC (6)cos A+cos B+cos C= cosa cosb cosc (7) sin A + sin B + sin C A B = sin sin sin C Ex0. ABC sinc= sin A+ sin B cos A+ cos B ABC Ans Ex. ABC sina=sinbcosc ABC Ans Ex. ABC sin A=sin B+sin C sina=sinbcosc ABC Ans Ex. cos7 o cos o (A)cos60 o (B)sin0 o sin o (C)cos0 o cos o (D)sin6 o sin76 o (E)sin6 o +cos66 o AnsDE ---/8

23 -5 ab R a +b >0. y=f(x)= asin x+ bcos x = a a + b ( sinx+ co x b a + b a + b s ) = a b (sin xcosθ cos xsin θ ) + + = a + b sin( x+ θ ) cosθ = a a + b sinθ = a b + b a + b θ. y=f(x)=asinx+bcosx y = a + b cos( x φ) sinφ= a a + b cosφ= a b + b. a + b a sin x + bcos x a + b x R Ex. y = f ( x) = (sin x + cos x) (A)y=f(x) (B) y=f(x) (C) y=f(x) y (0 ) (D) y=f(x) x (E) y=f(x) AnsCD Ex5. f(x)=sinx+ cosx Ans- Ex6. 0 x sinx+cosx Ans- --5-/8

24 Ex7. f ( x) = cos( x) cos x Ans Ex8. x Rf(x)=cosx+cosx Ans5- Ex9. x f(x)=cos x-sinxcosx+sin x Ans + sin x Ex0. y= Ans 0 + cos x Ex. f(x)=(sinx+cosx) +(sinx+cosx) f(x) Ans sinθ cosθ Ex. f ( θ ) = t=sin+cos + sinθ + cosθ t ()t f()f()=ans ()0 x f() Ans 0 Ex. f(x)=sinx cosx+5 Ans00 Ex. 0 x sinθ+cosθ Ans7 Ex5. sinθ+ cosθ= Ans 6 Ex6. x= f(x)=cosx-sinx tan=ans Ex7. f ( x) = cos x + sin( x + ) Ans0 --5-/8

25 Ex8. 0 x f(x)=sinx+cos(x6 ) ()x=f(x)ans 6 ()x=f(x)ans Ex9. x R sinx(cosx sinx) Ans± Ex0. f()=sin+cos f() Ans Ex. 0 θ cos θ + sin θ == Ans +,,0, 5cosx 5 Ex. < x y = f( x) = Ansy = 5 or y 0 cosx + Ex. f(x)=(+sinx)(+cosx) f(x) Ans 0 Ex. x R cos x+(sinx+cosx) sinxcosx Ans Ex5. 0x f(x)=+(sinx+cosx)sinx ()t=sinx+cosx t Ans t ()x=f(x) Ans0 ()x=f(x) Ans /8

26 Ex6. ()y=5sin xsinxcosxcos xx RAns + 0, 0 ()y=cos xcosxsinxsin x0 x Ans ()y=+cosx-cos xx RAns- ()y=cos x+sin xx RAns (5)y=cos 6 x+sin 6 xx RAns cos x + sin x Ex7. y= cos x + sin x + Ans± --5-6/8

27 -6 ().. siny=x sin x y ( arc-sine) x [ ]y [ ] sin(sin x)=x. {x - x }. {y y }. x [ ] sin(sin x)=x y [ ] sin - (siny)=y y=sin x y=sinx x=y cosy=x cos x y ( arc-cosine) x [ ]y [0] cos(cos x)=x. {x - x }. {y 0 y }. x [ ] cos(cos x)=x y [0] cos - (cosy)=y y=cos x y=cosx x=y tany=x tan x y ( arc-tangent) y ( ) tan(tan x)=x. R. {y <y< } (vs. sin). x R tan(tan x)=x y ( ) tan - (tany)=y y=tan x y=tanx x=y --6-7/8

28 . ( ) sin(sin - x)=x sin(sin 5 )= 5 sin - (sinx)=x ( ) sin (sin( 7 7 )) sin sin( 7 )=. y=x( )( ).. y=sinx x=sin - y sin - ( )= sin - ( ) y=tanx y=tan - x 5. y=cosx x=cos - y cos - ( ) cos - ( ) Ex8. (A)sin - (B)cos - (C)tan - (D)sin - 0(E)tan - 0AnsCDE Ex9. () sin () cos 0 Ans ()tan ( 6 Ans ) Ans6 ()tan +tan +tan ( )Ans (5)cos +sin +cos ( )+sin ( 7 )Ans 6 Ex0. () sin(sin 0.) Ans0. ()cos(cos ( )) Ans ()tan(tan ) Ans ()sin (sin( 6 ))Ans6 (5)sin (sin( 7 ))Ans --6-8/8

29 Ex. ()sin (sin) Ans- ()sin (sin(-)) Ans- ()cos (cos) Ans- ()cos (cos(-6)) Ans-6 Ex. ()tan[sin ( )+cos (- )]=?Ans 5 ()cos[cos ( )-sin (- )]=Ans 5 65 Ex. ()sin(cos 0.6) Ans0.8 ()cos(tan ( )) Ans 0 ()tan ( cos ) Ans () tan ( sin ) Ans 6 + (5) sin ( tan ( + )) Ans (6)tan ( sin )Ans Ex. [0) () cos 5 x+sinx= Ans () tanx+cotx=5cscxans Ex5. cos x = x=?ans Ex6. f(x)=ax 5 +tan x a f()=0 f( ) Ans /8

30 Ex7. (A) sin sinx = x (B) sinsin x = x (C) cos cosx = (D) coscos x = x (E) tantan x = x (F) tan tanx = x AnsBDE x Ex8. abcd () a = sin ( ) b = sin ( ) c = sin ( ) d = sin ( ) Ansd>b>a>c 5 5 () a = cos ( ) b = cos c = cos ( ) d = cos ( ) Ansa>d>b>c 5 5 () a = tan b = tan c = tan ( + ) d = tan ( + ) Ansd>c>a>b --6-0/8

31 -7 ( ). z=a+biab R (ab). I z a z=a+bi b R z=a+bi(ab R) z = () z z O () z z z z. z=a+bi(ab R) P(ab) a + b z x OP θ z arg(z) 0θ< θz Arg(z) b a b tanθ = cosθ = sinθ = a a + b a + b Ex9. z +i - +i z? Ans Ex50. +i+i θφθ+φ=ans z=a+bi= r( a + b i ) =r(cosθ+isinθ) r r r= z = a + b θ z Ex5. () i Ans (cos5 +isin5 ) () i Ans(cos90 +isin90 ) () sin0 +isin00 Anscos0 +isin0 ()-5(sin7 +icos7 ) Ans5(cos5 +sin5 ) --7-/8

32 z z Ex5. z C = Arg( z z ) = z=ansi.z =cos+isin z =cos+isin α β α + β α + β z +z = (cos+ cos)+i(sin+isin)= cos (cos + isin ) α β cos =0.0 +cos+isin cos α (cos α + isin α ) 7 7 Ex5. α = ( + cos + i sin ) α ArgαAns7 6 + cosθ + i sinθ Ex5. f ( θ ) = + cosθ i sinθ () f(θ)=cosθ+isinθ () [ f ( )] n n=ans. z = a + bi a, b R z = a bi + Z = Z Z + Z Z Z Z Z Z Z = Z Z Z Z = Z Z Z Z = Z Z = Z Z ( Z ) = Z Z Z n n = Z, Z Z = n ( + i) ( i) Ex55. + i 5 Ans /8

33 Ex56. α(+i) + α(+i) ()α C Ans ()α=aa R a=ans5 ()α=bib R b= 5. ( ) Z = r (cosθ + isinθ) Z = r (cosθ + isinθ ) R r r Z Z = r r (cos( θ + ) + i sin( θ + θ )) θ Z Z (cos7 + isin07 )(sin5 icos ) Ex57. Ans + cos50 isin 0 = r (cos( θ θ) + isin( θ θ)) r i Ex58. (sin5 +icos5 )(sin0 +icos0 )(sin5 +icos5 )Ans i (DeMoivre stheorem) Z= r(cosθ+isinθ) n n Z = r (cosnθ + isin nθ ) n Z n (cosθ ± isin θ) n = cosnθ ± isinnθ n Z + i Ex59. ( ) 0 = A(B) 5 (C) (D) 5 (E) 5 5 AnsE Ex60. a = + i a n = n Ans Ex6. ()z C + = cos n z θ n Z z + nθ z n = cos z () + = 0 z z + z z 0 Ans Ex6. f(x)=x 00 x 50 + f(x) x+-i Ansi --7-/8

34 a n. n (n N) Z n k k = n z k = cos + i sin k=0 n n n Z n = w cos isin = + n,... n ww w, w n n n w = + w + n Z n w w = 0 = ( Z )( Z w)( Z w )...( Z Z n + n Z Z + = ( Z w)( Z w n ) w )...( Z w n. n Ex6. w = cos + isin, 5 5 () +w+w +w +w =Ans0 () (w)(w )(w )(w )=Ans5 () (+w)(+w )(+w )(+w ) =Ans () ( + w )( + w )( + w )( + w ) =Ans (5) =Ans + w + w + w + w ) n Ex6. θ = n N n ()cosθ+cosθ+.+cos(n)θ=ans ()sinθ+sinθ+.. +sin(n)θ=ans0 Ex65. x 6 +x +x +=0 Ans+.a n (n N) Z n θ + k θ + k =a n Z = n a (cos + i sin ) k=0 n k n n a= a (cosθ+isinθ) n θ θ n n w 0 = r(cos + isin ), w= cos( ) + isin( ) Z = Z0 n w0, w0w, w0w..., w0w n n n n. n n r Ex i Ans+i i --7-/8

35 Ex67. z + ( i 5) z + 8 i = 0 Ans+i-i. O O OX O P[rθ]P r = OP = O P θ = OX, OP. P [rθ] r(cosθ+isinθ) (rcosθrsinθ) Ex68. A[0 o ]B[00 o ] AB Ans Ex69. A[ ]B[ ] AOB AB C OC Ans y O OP θoq P z Q z = z (cos θ + isinθ ) O Q ( z ) P ( z ) x z z =(z z )(cosθ+isinθ) y z z z O x --7-5/8

36 Ex70. OA(+ i)a O BC BCAnsB(-+ i)c() Ex7. α=+cos00 isin00 α ArgαAnscos cos + i sin 00 Ex7. ( ) = + cos i sin Ans Ex7. α=+i =5+5iz=x+yixy R z= z α + z β Ans+i0 + i tan.5 Ex7. Ans 0 + i tan.5 0 i Ex (cos + isin ) (cos5 + isin5 ) (cos isin ) =a+bi,a,b R (a,b)=ans(0,) Ex76. z + = z + = z z 50 Ans 50 ( i ) ( i Ex ) = Ans Ex78. α=sin5 icos5 β=cos+isin α 50 β 00 Ans Ex79. +i +i Ans6i --7-6/8

37 Ex80. w x = () w + w=ans () ( w + w ) + (w + w ) =Ans 6 () ( + 5w+ w ) =Ans79 8 () ( w)( w )( w )( w ) =Ans9 Ex8. x x + x + x + x + x + x + = 0 Ans n + w + n Ex8. w x = + w + w + αβ αβansα= 6 7 β= 7 α+βw Ex8. i Ans i, + i, + i Ex8. Z 5 -+i(z 5 +)=0 Z=Ans k cos( ) sin( k + + i + ) k=0,, Ex85. ( + i) w w, w, w, w,, 5 w6, Ans6 Ex86. x -x +6x -x+8=0 Ans8 Ex87. z i ()z Ans 6 ()z Ans ()z x +αx+β=0 αβans /8

38 Ex88. OABC B(+i)C A Ans 7 + i 7 i y C B Ex89. αβ C α αβ + β = 0 0αβ α () AOB=Ans [HINT z= β z Arg(z)] ()OAB Ans ()OAB Ans O αβ =OAB A x --7-8/8

B3C1

B3C1 - B(. AB. A( ( 3. AA PP 0 a a a 4. ( 5. Ex. ABCDEF Ans8305 Ex. ABCDE Ans00. a+ b a+ b b. a+ b = b + a a b a ( a+ b + c = a+ ( b + c a+ 0= a = 0+a a + ( a = 0 = ( a + a b a b 3. a b = a+ ( b a 4.(P AB =