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The Principal Argument Of 2 Root 3 2i Is

Solution: Given complex number = -2√3-2i Let r cos θ = -2√3 r cos θ = -2 Squaring and adding... View Article

The Area Of The Triangle In The Argand Diagram Formed By The Complex Number Z Iz And Z Plus Iz Is

Solution: Since i = eiπ/2 iz = zeiπ/2 Vector iz is obtained by rotating z in anticlockwise direction through 900.... View Article

If Z1 2 I And Z2 Equal 1 Plus I Then Z1 Z2 1 By Z1 Z2 I Equals

Solution: Given z1 = 2-i and z2 = 1+i (z1– z2+1)/(z1+ z2-i) = ((2-i)-(1+i)+1)/((2-i)+(1+i)+i) = (2-2i)/(3+i) = (2-2i(3-i))/(3+i)(3-i) [multiply numerator... View Article

If 1 Omega And Omega 2 Are The Cube Roots Of 1 Then 1 Plus Omega Equals

Solution: Use ω3 = 1 and (1+ω) = -ω2, 1+ω2 = ω (1+ω)(1+ω2)(1+ω4)(1+ω8) = (1+ω)( -ω2)(1+ ω)(1+ ω2) = (1+ω)2(... View Article

If W Not Equal 1 Is Cube Root Of Unity And 1 W2 N Equal 1 W 4 N Then Smallest Possible Value Of N Is

Solution: We know 1+w+w2 = 0 1+w2 = -w 1+w4 = 1+w3w = 1+w = -w2 (1+w2)n/(1+w4)n = (-w/-w2)n =... View Article

Let Alpha Beta Be Real And Z Be A Complex Number If Z Alpha Z Beta Equal Has Two Distinct Roots On The Line Then It Is Necessary That

Solution: z2 + αz + β = 0 Let z1 and z2 be the roots. z1 = x+iy z2 =... View Article

1 Plus I 2 Ai 2 3i 3 Plusi Equal X Plus Iy X Y Belongs To R And X Equal Y Then A Equal

Solution: (1+i)(2+ai)+(2+3i)(3+i) = x+iy (2+2i+ai-a+6+9i+2i-3 = x+iy 5-a+(13+a)i = x+iy Equate real and imaginary parts 5-a = x 13+a =... View Article

The Number Of Complex Numbers Z Such That Z 1 Equal Z Plus 1 Equal Z I Equal

Solution: Given |z-1|=|z+1|= |z-i| Let z = x+iy |z-1| = |x+iy-1| = √[(x-1)2+y2] |z+1| = |x+iy+1| = √[(x+1)2+y2] |z-i| =... View Article

If 1plus2i By 2 Plus I R Cos Theta Plus I Sin Theta Then R And Theta Equal

Solution: Given (1+2i)/(2+i) = r (cos θ+i sin θ) (1+2i)(2-i)/(2+i)(2-i) = r (cos θ+i sin θ) (2+4i-i+2)/(4+1) = r (cos... View Article

If Z 4 By Z Equal 2 Then The Maximum Value Of Z Is 1

Solution: We know |a+b| ≤ |a|+|b| Given |z-4/z| = 2 |z| ≤ |z -(4/z)|+|(4/z)| |z| ≤ 2+|(4/z)| |z|- 2+|(4/z)| ≤... View Article

If W Z Z 1 By 3 I And W 1 Then Z Lies On

Solution: Given w = z/(z-⅓)i and |w| = 1 w = 3z/(3z-i) |w| = 3|z|/|(3z-i)| 3|z| = |(3z-i)| Put z... View Article

Let Zw Be Two Complex Numbers Such That Bar Z I Bar W 0 And Arg Zw Pi Then Arg Z Equal

Solution: z = iw w = -iz arg (zw) = π arg (iz2) = π arg (-i)+2 arg(z) = π... View Article

If Z X Iy And Z 1 By 3 P Plus Iq Then X P Plus Y Q By P 2 Q2 Equal

Solution: Given z = x-iy and z1/3 = p+iq x-iy = (p+iq)3 x-iy = p3-3pq2+i(3p2q-q3) Equating real and imaginary part... View Article

If 1 Plus I X 2i By 3 I Plus 2 3i Y Plus I By 3 I I Then X Y Equal

Solution: Given ((1+i)x-2i)/(3+i) +((2-3i)y+i)/(3-i) = i [((1+i)x-2i)(3-i)/(3+i)(3-i)]+ [((2-3i)y+i)(3+i)/(3-i)(3+i)] = i [(x+xi-2i)(3-i)+(2y+(1-3y)i)(3+i)]/(9+1) = i [(x+xi-2i)(3-i)+(2y+(1-3y)i)(3+i)] = 10i [(3x+3xi-6i-xi+x-2)+(6y+3i-9yi+2yi-1+3y)] = 10i (4x+9y-3)+i(2x-7y-13)... View Article

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