Argand diagram representation Representation of complex numbers in the Argand diagram. A complex number z = x + i y z = x + iy z = x + i y ( x , y ) (x, y) ( x , y )
Practice questions 1 Solve the equation i z 5 = − 32 i z^5 = -32 i z 5 = − 32 r e i θ re^{i\theta} r e i θ r > 0 r>0 r > 0 − π < θ ≤ π -\pi<\theta\le\pi − π < θ ≤ π 2 One of the roots of the equation 2 z 4 − 14 z 3 + 33 z 2 − 26 z + p = 0 , where p is a constant 2z^4 - 14z^3 + 33z^2 - 26z + p = 0\text{, where } p \text{ is a constant} 2 z 4 − 14 z 3 + 33 z 2 − 26 z + p = 0 , where p is a constant 3 + i 3 + \mathrm{i} 3 + i 3 One of the roots of the equation 2 z 4 − 14 z 3 + 33 z 2 − 26 z + p = 0 , where p is a constant 2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0\text{, where } p \text{ is a constant} 2 z 4 − 14 z 3 + 33 z 2 − 26 z + p = 0 , where p is a constant 4 Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where ∣ z + 3 ∣ = 2 ℜ ( z ) |z+3|=2\,\Re(z) ∣ z + 3∣ = 2 ℜ ( z ) 5 在复平面内，复数z = 1 3 + i z= \dfrac {1}{3+i} z = 3 + i 1 ( ) () ( ) 6 The complex number z z z arg ( z − 1 − 2 i ) = θ \arg(z-1-2i)=\theta arg ( z − 1 − 2 i ) = θ θ \theta θ − π < θ ≤ π -\pi < \theta \le \pi − π < θ ≤ π 7 The equation z 3 − 2 z + k = 0 z^{3}-2z+k=0 z 3 − 2 z + k = 0 k k k z = 1 + a i z = 1 + ai z = 1 + ai a a a 8 Sketch on a single Argand diagram, the loci defined by − π 4 < arg ( z + 1 + 2 i ) ≤ π 4 -\frac{\pi}{4} < \arg(z + 1 + 2\text{i}) \le \frac{\pi}{4} − 4 π < arg ( z + 1 + 2 i ) ≤ 4 π 9 Solve the equation i z 5 = − 32 iz^5=-32 i z 5 = − 32 r e i θ re^{i\theta} r e i θ r > 0 r>0 r > 0 − π < θ ≤ π -\pi<\theta\le\pi − π < θ ≤ π 10 The complex number a a a r r r θ \theta θ r ∈ R + r \in \mathbb{R}^+ r ∈ R + 0 < θ < π 2 0 < \theta < \frac{\pi}{2} 0 < θ < 2 π 11 The point A represents a fixed complex number a such that − π 2 < arg a < 0 -\dfrac{\pi}{2}<\arg a<0 − 2 π < arg a < 0 i a ia ia a + i a a+ia a + ia 12 The point A A A a a a ∣ a ∣ = 1 |a|=1 ∣ a ∣ = 1 θ = arg ( a ) , 0 < θ < π 4 \theta=\arg(a),\; 0<\theta<\dfrac{\pi}{4} θ = arg ( a ) , 0 < θ < 4 π 13 A complex number z z z t t t z = 2 cos t + i ( 3 sin t ) , where 0 ≤ t < 2 π . z = 2\cos t + \mathrm{i}(3\sin t), \quad \text{where } 0 \le t < 2\pi. z = 2 cos t + i ( 3 sin t ) , where 0 ≤ t < 2 π . 14 Solve the equation z 5 + 16 = 16 3 i z^5 + 16 = 16\sqrt{3}\, i z 5 + 16 = 16 3 i r e i θ r e^{i\theta} r e i θ r > 0 r>0 r > 0 − π < θ ≤ π -\pi < \theta \le \pi − π < θ ≤ π 15 The complex numbers a a a b b b a = − ( 1 + 3 i ) a = -(1 + \sqrt{3}\text{i}) a = − ( 1 + 3 i ) b = 1 2 ( 1 − i ) b = \frac{1}{2}(1 - \text{i}) b = 2 1 ( 1 − i ) 16 On a single Argand diagram, sketch the locus of points representing the complex number z z z ∣ z − 4 − 2 i ∣ ≤ 2 and ∣ z − 3 ∣ ≤ ∣ z − 5 ∣ . |z-4-2i| \le 2\quad\text{and}\quad |z-3| \le |z-5|. ∣ z − 4 − 2 i ∣ ≤ 2 and ∣ z − 3∣ ≤ ∣ z − 5∣. 17 The complex number z z z ∣ z ∣ ≤ 2 |z|\le 2 ∣ z ∣ ≤ 2 π 6 ≤ arg z ≤ π 2 \dfrac{\pi}{6}\le \arg z \le \dfrac{\pi}{2} 6 π ≤ arg z ≤ 2 π 18 Given that z = λ − 4 i 1 − λ i z = \frac{\lambda - 4i}{1 - \lambda i} z = 1 − λi λ − 4 i λ ∈ R \lambda \in \mathbb{R} λ ∈ R arg ( z ) = π \arg(z) = \pi arg ( z ) = π z z z 19 Do not use a calculator in answering this question. The complex number z z z θ \theta θ π 2 < θ < π \dfrac{\pi}{2}<\theta<\pi 2 π < θ < π 20 The complex number z z z ∣ z − 4 − 3 i ∣ ≤ 2 and π < arg [ ( z − 4 ) 2 ] < 2 π |z-4-3\text{i}| \le 2 \quad \text{and} \quad \pi < \arg\left[(z-4)^2\right] < 2\pi ∣ z − 4 − 3 i ∣ ≤ 2 and π < arg [ ( z − 4 ) 2 ] < 2 π 21 The points P and Q are represented by the complex numbers p and q respectively, where p = 2 + 2 i , arg ( q ) = 2 3 π and ∣ q ∣ = 2. p=\sqrt{2}+\sqrt{2}i\,,\ \arg(q)=\dfrac{2}{3}\pi\text{ and }|q|=2. p = 2 + 2 i , arg ( q ) = 3 2 π and ∣ q ∣ = 2. 22 One of the roots of the equation 2 z 4 − 14 z 3 + 33 z 2 − 26 z + p = 0 , 2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0, 2 z 4 − 14 z 3 + 33 z 2 − 26 z + p = 0 , p p p 3 + i 3 + i 3 + i 23 The point A on the Argand diagram below represents the complex number w 1 w_1 w 1 r r r θ \theta θ 24 The three distinct roots of the equation x 3 − 1 = 0 x^3 - 1 = 0 x 3 − 1 = 0 1 , ω 1, \omega 1 , ω ω 2 \omega^2 ω 2 25 A complex number z z z t t t z = 2 cos t + i ( 3 sin t ) , where 0 ≤ t < 2 π . z = 2\cos t + i(3\sin t),\quad \text{where } 0 \le t < 2\pi. z = 2 cos t + i ( 3 sin t ) , where 0 ≤ t < 2 π . 26 Solve the equation z 4 = 3 − i z^4 = \sqrt{3} - i z 4 = 3 − i r e i θ re^{i\theta} r e i θ r > 0 r > 0 r > 0 − π < θ ≤ π -\pi < \theta \le \pi − π < θ ≤ π 27 The complex numbers z z z w w w z = 1 + i a , w = − b − i z = 1 + i a, \quad w = -b - i z = 1 + ia , w = − b − i a a a b b b 28 Do not use a calculator in answering this question. The complex number z z z θ \theta θ 29 Do not use a calculator in answering this question. The complex numbers z z z w w w r r r 30 On a single Argand diagram, sketch the following loci: ∣ z − 3 ∣ = 5 |z - 3| = 5 ∣ z − 3∣ = 5 arg ( z − 8 ) = π − tan − 1 ( 1 2 ) \arg(z - 8) = \pi - \tan^{-1}\left(\frac{1}{2}\right) arg ( z − 8 ) = π − tan − 1 ( 2 1 ) 31 In a single Argand diagram, sketch the following loci, labelling each locus clearly. ∣ z ∣ = 5 , |z| = 5, ∣ z ∣ = 5 , ∣ z + 8 ∣ = ∣ z − 8 i ∣ . |z + 8| = |z - 8i|. ∣ z + 8∣ = ∣ z − 8 i ∣. 32 On a single Argand diagram sketch the loci given by (a) ∣ z − 12 − 12 e x t i ∣ = 12 |z - 12 - 12 ext{i}| = 12 ∣ z − 12 − 12 e x t i ∣ = 12 ∣ e x t i z − 1 − 12 e x t i ∣ = 5 | ext{i}z - 1 - 12 ext{i}| = 5 ∣ e x t i z − 1 − 12 e x t i ∣ = 5 33 A complex number z varies with t such that z = 2 cos t + i ( 3 sin t ) , where 0 ≤ t < 2 π . z = 2\cos t + i(3\sin t),\quad\text{where } 0 \le t < 2\pi. z = 2 cos t + i ( 3 sin t ) , where 0 ≤ t < 2 π . x = R e ( z ) x = \mathrm{Re}(z) x = Re ( z ) 34 It is known that w w w ∣ 2 w − 1 ∣ ≤ ∣ w + 1 ∣ |2w - 1| \le |w + 1| ∣2 w − 1∣ ≤ ∣ w + 1∣ w ∈ R w \in \mathbb{R} w ∈ R w w w 35 The complex numbers z z z w w w z = 1 + i a , w = − b − i z = 1 + i a,\quad w = -b - i z = 1 + ia , w = − b − i a a a b b b 36 The complex number w w w 2 \sqrt{2} 2 1 4 π \tfrac{1}{4}\pi 4 1 π z z z 2 \sqrt{2} 2 37 The complex number z z z ∣ z − 2 i ∣ < 4 |z - 2\mathrm{i}| < 4 ∣ z − 2 i ∣ < 4 ∣ z + 4 − 6 i ∣ ≤ ∣ z − 2 i ∣ |z + 4 - 6\mathrm{i}| \le |z - 2\mathrm{i}| ∣ z + 4 − 6 i ∣ ≤ ∣ z − 2 i ∣ 38 The complex number z z z ∣ z ∣ 2 ≤ 4 and − π 6 ≤ arg ( z + 3 − i ) ≤ 0. |z|^2 \le 4 \quad \text{and} \quad -\frac{\pi}{6} \le \arg(z + \sqrt{3} - \mathrm{i}) \le 0. ∣ z ∣ 2 ≤ 4 and − 6 π ≤ arg ( z + 3 − i ) ≤ 0. 39 In a single Argand diagram, sketch the following loci, labelling each locus clearly. ∣ z ∣ = 5 |z| = 5 ∣ z ∣ = 5 ∣ z + 8 ∣ = ∣ z − 8 i ∣ |z + 8| = |z - 8i| ∣ z + 8∣ = ∣ z − 8 i ∣ 40 The sequence of complex numbers { w n } \{w_n\} { w n } w n = [ 1 + ( n − 1 ) i ] [ 1 + ( n + 1 ) i ] ( 1 + n i ) 2 for n ∈ Z + . w_n = \dfrac{[1+(n-1)i][1+(n+1)i]}{(1+ni)^2}\text{ for }n\in\mathbb{Z}^+. w n = ( 1 + ni ) 2 [ 1 + ( n − 1 ) i ] [ 1 + ( n + 1 ) i ] for n ∈ Z + . 41 A complex number z z z z = r e i π n z = r \mathrm{e}^{\mathrm{i}\frac{\pi}{n}} z = r e i n π 0 < r < 1 0 < r < 1 0 < r < 1 n n n