Test: Complex Numbers IV - Ambitious

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Question 1:   Find 3 cube roots of z=8( cos81°+isin81° )
{ z 1 =2 27° z 2 =2 54° z 3 =2 81°
{ z 1 =2 27° z 2 =2 147° z 3 =2 267°
{ z 1 =8 27° z 2 =8 54° z 3 =8 81°
{ z 1 =8 27° z 2 =8 147° z 3 =8 267°
Question 2:   Find 5 fifth roots of z=14 305°
{ z = 1 2 61° z = 2 2 111° z = 3 2 161° z = 4 2 201° z = 5 2 251°
{ z = 1 1.695 61° z = 2 1.695 133° z = 3 1.695 205° z = 4 1.695 277° z = 5 1.695 349°
{ z = 1 1.695 60° z = 2 1.695 120° z = 3 1.695 180° z = 4 1.695 240° z = 5 1.695 300°
{ z = 1 2 61° z = 2 2 133° z = 3 2 203° z = 4 2 274° z = 5 2 345°
Question 3:   Find the principal root of the 3 cube roots of the z=5( cos270°+isin270° )
1.71 90°
1.71 330°
5 3 90°
1.71 210°
Question 4:   Find the expression for sin4θ in terms of sinθ (De Moivre’s theorem)
2cosθ3cosθ+5
3 cos 4 θ+8 cos 3 θ+1
8 cos 4 θ8 cos 3 θ+2
8 cos 4 θ8 cos 2 θ+1
Question 5:   Find the expression for sin 3 θ
sin 3 θ= 1 7 isin5θ+ 1 6 isin3θ
sin 3 θ= 2 5 isin3θ 1 4 isin2θ
sin 3 θ=2isin3θ 1 2 isinθ
sin 3 θ= 1 4 isin3θ 3 4 isinθ
Question 6:   Find the locus defined as | z |=8 , where z=x+iy
x 2 + y 2 +8=0
x 2 + y 2 64=0
x 2 + y 2 8=0
2 x 2 +3 y 2 2x+y=0
Question 7:   Find the locus argz= π 3 , where argz= tan 1 { y x }
y=1.5 x 2 2x
y= 1 2 x
y=6 x 2 +7
y= 3 x
Question 8:   Find the equation of the locus | z |= x 2 + y 2 , where | z+2 z2 |=3
8 x 2 +9 y 2 40x+32=0
2 x 2 2 y 2 +36x9=0
7 x 2 +8 y 2 12x+13=0
4 x 2 +4 y 2 12x+9=0
Question 9:   Determine the equation of the loci | z3 z |=4
2 x 2 3 y 2 +5x+2=0
3 x 2 y 2 +8x+1=0
7 x 2 +3 y 2 +5x3=0
5 x 2 +5 y 2 +2x3=0
Question 10:   Find the locus of the z=x+iy , where arg( z+2 )= π 4
y=x2
y=x+2
y=x+1
y=x+1