9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers.
Complex Numbers Solution
2
Terry Lee Advanced Mathematics Practice Questions Solutions Complex Numbers (a) (i) Let ( ) 2 2 2 2 2 2 2 2 , while ( )( ) z x iy z x y x y zz x iy x iy x y = + = + = + = + − = + . ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 and , . z zz z z x iy x iy x x iy y x x iy y z z x iy x iy x x iy y z z z z ∴ = + = + + + = + + + = + − + + = − + + = + − + ∴ + = + (ii) ( )( ) ( )( ) 2 2 1 2 1 1 2 2 z z z z z z z z z z z z − + − = − − + − − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 2 1 2 2 2 2 1 2 1 2 1 2 2 2 1 2 1 2 2 2 2 5 5 , since 0, so 0 52. z z z z z z z z zz zz zz zz zz zz zz zz zz zz zz zz z zz z z z z zz z zz z z z z z = − − + − − = + − − + + − − = + + − + − + = + + − + − + = + + + = + = = (b) (i) Let ( ) 2 2 2 2 2 , Re( ) Re 2 z x iy z x y xyi x y = + = − + = − . ∴The locus of z is the rectangular hyperbola 2 2 4 x y − = . (ii) 3 3 arg , arg , arg 4 4 4 4 π π π π π π − < <− − < < < ≤ z z z . −2 2
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Transcript of Complex Numbers Solution
Terry Lee Advanced Mathematics
Practice Questions
Solutions
Complex Numbers
(a) (i) Let ( )22 2 2 2 2 2 2, while ( )( )z x iy z x y x y zz x iy x iy x y= + = + = + = + − = + .
( ) ( ) ( )( ) ( )( )
( )
2
1 2 1 1 2 2 1 2 1 2 1 2 1 2
1 2 1 1 2 2 1 2 1 2
1 2 1 2
and ,
.
z zz
z z x iy x iy x x i y y x x i y y
z z x iy x iy x x i y y
z z z z
∴ =
+ = + + + = + + + = + − +
+ = − + + = + − +
∴ + = +
(ii) ( )( ) ( )( )2 21 2 1 1 2 2z z z z z z z z z z z z− + − = − − + − −
( )( ) ( )( )
( ) ( )
( ) ( )
1 1 2 2
1 1 1 1 2 2 2 2
1 1 2 2 1 2 1 2
2 2 21 2 1 2 1 2
2 21 2 1 2
2
2
2 5 5 , since 0, so 052.
z z z z z z z z
zz z z zz z z zz z z zz z z
zz z z z z z z z z z z
z z z z z z z z z
z z z z
= − − + − −
= + − − + + − −
= + + − + − +
= + + − + − +
= + + + = + ==
(b) (i) Let ( )2 2 2 2 2,Re( ) Re 2z x iy z x y xyi x y= + = − + = − .
∴The locus of z is the rectangular hyperbola 2 2 4x y− = .
(ii) 3 3arg , arg , arg4 4 4 4π π π ππ π− < < − − < < < ≤z z z .
−2 2
Terry Lee Advanced Mathematics
(c) cis cis 22 2
i kπ π π⎛ ⎞= = +⎜ ⎟⎝ ⎠
.
13 2cis , where 0,1, 1.
6 35cis ,cis ,cis
6 6 2
3 3, , .2 2
ki k
i i i
π π
π π π
⎛ ⎞∴ = + = −⎜ ⎟⎝ ⎠
⎛ ⎞= −⎜ ⎟⎝ ⎠
+ − += −
(d) (i) ( ) ( )(1 ) 2(1 ) 1 2 2 1u v i i i+ = − + + = + + − .
2(1 ) 2( 1 )iv i i i= + = − + . (ii) From the diagram, ABCD is a trapezium, where
( ) ( )( )
( )( ) ( )
( ) 2(1 ), 2
( ) 1 , 2
(1 ) 2( 1 ) 1 2 (1 ), 2 1 2 .
1Area of 21 2 2 1 2 2 2 2 1 .2
AB u v u v i AB
CB u v v u i BC
AD u iv i i i AD
ABCD BC AD AB
= + − = = + ∴ =
= + − = = − ∴ =
= − = − − − + = + − ∴ = +
∴ = +
= + + = +
A (u)
C (v)
B (u + v)
D (iv)
Re(z)
Im(z)
