Complex Numbers (Cambridge International A-Level Mathematics): Exam Questions

$z_{1} = 3 + i$ and $z_{2} = - 5 + 2 i$

(i) Work out $z_{1} + z_{2}$ and $z_{1} - z_{2}$.

(ii) Use your answers to part (i) to write down $\text{Re} ( z_{1} + z_{2} )$ and $\text{Im} ( z_{1} - z_{2} )$.

By first writing $z_{1} \times z_{2} = ( 3 + i ) ( - 5 + 2 i )$, and then expanding brackets and using the fact that $i^{2} = - 1$, show that

$z_{1} \times z_{2} = - 17 + i$

Given that $z_{1} = 3 - 2 i$ and $z_{2} = 4 + 3 i$:

Write down $z_{1}^{\ast}$ and $z_{2}^{\ast}$.

Show that $z_{1} \times z_{2}^{\ast} = 6 - 17 i$ and $z_{2} \times z_{2}^{\ast} = 25$.

Explain why $\frac{z_{1}}{z_{2}} = \frac{z_{1}\times z_{2}^{\ast}}{z_{2}\times z_{2}^{\ast}}$

Use the results of parts (b) and (c) to work out $\frac{z_{1}}{z_{2}}$, giving your answer in the form $a + b i$ where $a$ and $b$ are real numbers.

Given that $z_{1} = - 7 + 6 i$, $z_{2} = a + b i$ and $z_{1} + z_{2} = 1 - 3 i$, find $a$ and $b$ where $a$ and $b$ are real numbers.

$f ( z ) = z^{2} - 12 z + 37$

By first working out $( 6 - i )^{2}$, show that $z = 6 - i$ is a solution to $f ( z ) = 0$.

Show that $( 6 - i )^{\ast}$ is another solution to $f ( z ) = 0$.

Using laws of surds and the fact that $i = \sqrt{-1}$, find both solutions to the equation $z^{2} + 16 = 0$. Your answers should be given as multiples of $i$.

Solve the equation $z^{2} - 4 z + 13 = 0$ for $z$

(i) by using the quadratic formula

(ii) by completing the square.

In each case, give your answers in the form $a + b i$ where $a$ and $b$ are real numbers. Confirm that the two methods give the same results.

Given that $β = 1 + 2 i$,

(i) express $β^{2}$ and $β^{3}$ in the form $a + b i$, where $a$ and $b$ are real numbers

(ii) hence show that $β$ is a root of the cubic equation

$z^{3} + z^{2} - z + 15 = 0$

The other two roots of the cubic equation in part (a)(ii) are $γ$ and $δ$, where $γ$ is a non-real complex number and $δ$ is a real number.

(i) State $γ$ in the form $a + b i$, where $a$ and $b$ are real numbers.

(ii) Show that $( z - β ) ( z - γ ) = z^{2} - 2 z + 5$.

(iii) Hence fully factorise $z^{3} + z^{2} - z + 15$ and determine the value of $δ$.

$f ( z ) = z^{3} - 9 z^{2} + c z + d$, where $c$ and $d$ are real numbers.

The roots of the equation $f ( z ) = 0$ are $z_{1}$, $z_{2}$ and $z_{3}$.

Given that $z_{1} = 3$ and $z_{2} = 3 - i$,

(i) state $z_{3}$ in the form $a + b i$, where $a$ and $b$ are real numbers

(ii) by first writing $f ( z ) = ( z - z_{1} ) ( z - z_{2} ) ( z - z_{3} )$, find the values of $c$ and $d$.

Let $w = a + b i$, where $a$ and $b$ are real numbers, be a general complex number.

Show that $( - w )^{2} = w^{2}$.

For a complex number $z$, a square root of $z$ is a complex number $x + y i$ where $x$ and $y$ are real numbers and where

$( x + y i )^{2} = z$

(i) Given that $z = - 5 - 12 i$, show that $2 - 3 i$ is a square root of $z$.

(ii) Use the result of part (a) to write down the other square root of $z$.

Given that $z_{1} = 2 - 3 i$ and $z_{2} = - 7 + i$, work out the following:

(i) $z_{1} \times z_{2}$

(ii) $\text{Re} ( z_{1} + z_{2} )$

(iii) $\text{Im} ( z_{2} - z_{1} )$

The quadratic equation $z^{2} + b z + c = 0$, where $b$ and $c$ are real constants, has a root $5 - 3 i$.

(i) State the other root of the equation.

(ii) Find the value of $b$ and the value of $c$.

Given that $z_{1} = 5 + 7 i$ and $z_{2} = 2 - i$:

Work out $z_{1} \times z_{2}^{\ast}$ and $z_{2} \times z_{2}^{\ast}$

Hence or otherwise work out $\frac{z_{1}}{z_{2}}$, giving your answer in the form $a + b i$ where $a$ and $b$ are real numbers.

For a general complex number $z = x + i y$, where $x$ and $y$ are real numbers, show that:

(i) $\text{Re} ( z ) = \frac{z+z^{\ast}}{2}$

(ii) $\text{Im} ( z ) = \frac{z-z^{\ast}}{2i}$

Given that $z_{1} = 3 + a i$, $z_{2} = b - 7 i$ and $z_{1} + z_{2} = 14 + i$, find $a$ and $b$ where $a$ and $b$ are real numbers.

$f ( z ) = z^{2} - 6 z + 34$

(i) Show that $z = 3 - 5 i$ is a root of $f ( z ) = 0$.

(ii) State the other root of $f ( z ) = 0$.

Solve the following equations for $z$, giving your answers in the form $a + b i$ where $a$ and $b$ are real numbers:

(i) $z^{2} + 13 = 0$

(ii) $z^{2} - 14 z + 53 = 0$

Given that $7 - i$ is one of the roots of a quadratic equation with real coefficients,

(i) state the other root of the equation

(ii) find the equation giving your answer in the form $z^{2} + b z + c = 0$ where $b$ and $c$ are real constants.

Given that $β = - 2 - 3 i$,

(i) express $β^{2}$ and $β^{3}$ in the form $a + b i$, where $a$ and $b$ are real numbers

(ii) hence show that $β$ is a root of the cubic equation

$z^{3} - z^{2} - 7 z - 65 = 0$

Find the other two roots of the cubic equation in part (a)(ii).

$f ( z ) = z^{3} - 6 z^{2} + c z + d$, where $c$ and $d$ are real numbers.

Given that $- 2$ and $4 - i$ are roots of the equation $f ( z ) = 0$,

(i) state the other complex root of the equation

(ii) find the value of $c$ and the value of $d$.

For a complex number $z$, a square root of $z$ is a complex number $x + i y$ where $x$ and $y$ are real numbers and where

$( x + i y )^{2} = z$

Given that $z = - 3 + 4 i$, show that $x^{2} - y^{2} = - 3$ and $2 x y = 4$.

(i) Solve the equations in part (a) to find the possible values of $x$ and $y$.

(ii) Hence determine the two square roots of $z = - 3 + 4 i$.

Given that $z_{1} = 1 - 2 i$ and $z_{2} = - 3 + 5 i$, work out the following:

(i) $\text{Re} ( z_{2} - z_{1} )$

(ii) $\text{Im} ( z_{1} z_{2} )$

(iii) \left(\dfrac{z_{1}}{z_{2}}\right)^{*}

For part (iii) give your answer in the form $a + b i$, where $a$ and $b$ are real numbers.

The complex number $z$ satisfies the equation $( 2 + 5 i ) ( z + 2 i ) = - 7 - 32 i$.

Find $z$, giving your answer in the form $a + b i$, where $a$ and $b$ are real numbers.

Given that $- 3 + 2 i$ is one of the roots of the quadratic equation $z^{2} + b z + c = 0$, where $b$ and $c$ are real constants, find the values of $b$ and $c$.

$f ( z ) = z^{2} + 2 i z - 10$

(i) Show that $f ( z )$ can be rewritten in the form $( z + a i )^{2} + b$, where $a$ and $b$ are real numbers to be found.

(ii) Hence find the solutions to the equation $f ( z ) = 0$.

Show that $α = - 1 + 4 i$ is a root of the cubic equation

$z^{3} + 5 z^{2} + 23 z + 51 = 0$

Find the other two roots of the cubic equation in part (a).

$f ( z ) = z^{3} + z^{2} + c z + d$, where $c$ and $d$ are real numbers.

Given that $3$ and $- 2 - 3 i$ are roots of the equation $f ( z ) = 0$, find the value of $c$ and the value of $d$.

For a general complex number $z = x + i y$, where $x , y \in ℝ$ and $z \neq 0$, show that

(i) \text{Re}\left(\dfrac{1}{z} + \dfrac{1}{z^{*}}\right) = \dfrac{2x}{x^{2} + y^{2}}

(ii) \text{Im}\left(\dfrac{1}{z} + \dfrac{1}{z^{*}}\right) = 0

Given that $z_{1} = 3 - 4 i$, $z_{2} = a + b i$ and $\frac{2z_{1}}{z_{2}} = - 1 - 7 i$, where $a$ and $b$ are real numbers, find the values of $a$ and $b$.

Given that $z_{1} = a - 6 i$, $z_{2} = 1 + b i$ and $z_{1} z_{2} = - 17 - 9 i$, where $a$ and $b$ are real numbers, find the possible values of $a$ and $b$.

Using your answers to part (a), state values for $c$ and $d$ that will satisfy the equation

$- ( 3 + i ) ( c + d i ) = - 17 - 9 i$

The equation $z^{2} + b z + 18 = 0$, where $b \in ℝ$, has distinct non-real complex roots.

Find the range of possible values of $b$.

For a complex number $z$, a square root of $z$ is a complex number $x + i y$ where $x$ and $y$ are real numbers and where

$( x + i y )^{2} = z$

By expanding $( x + i y )^{2}$ and solving the resultant equations for $x$ and $y$, determine the two square roots of the complex number $z = - 21 - 20 i$.

Find all the complex numbers $z$ for which $z^{2} = 4 z^{\ast}$.

Given that $3 + q i$ is one of the roots of the quadratic equation $z^{2} - 12 p z + 58 = 0$, where $p$ and $q$ are positive real constants, find the values of $p$ and $q$.

Work out the solutions to the equation $z^{2} - 6 i z - 14 = 0$.

You are given that the complex number $α = 1 - 3 i$ satisfies the cubic equation

$z^{3} + 4 z^{2} + k z + m = 0 ,$

where $k$ and $m$ are real constants.

By first calculating $α^{2}$ and $α^{3}$, find the values of $k$ and $m$.

Find the other two roots of the cubic equation.

$α$, $β$ and $γ$ are the three roots of the cubic equation

$z^{3} + ( b - 3 ) z^{2} + ( 7 - 3 b ) z - 21 = 0$

where $b \in ℝ$.

Given that $α = 3$ and that $β$ and $γ$ are distinct non-real complex numbers, find the range of possible values of $b$.

$f ( z ) = z^{4} - 2 z^{3} + 6 z^{2} + p z + 125$, where $p$ is a real constant.

Given that $3 - 4 i$ is a root of the equation $f ( z ) = 0$, show that $z^{2} + 4 z + 5$ is a factor of $f ( z )$.

Hence find the value of $p$ and solve completely the equation $f ( z ) = 0$.

The principal square root of a complex number $z$ is defined as

$\sqrt{z} = x + i y$

where $x$ and $y$ are real numbers and $x \geq 0$. If $x = 0$ then the value for $y$ is chosen such that $y \geq 0$. Note that the other square root of $z$ will then be given by $- \sqrt{z} = - x - i y$.

Show that

x = \sqrt{\dfrac{\text{Re}(z) + \sqrt{\left(\text{Re}(z)\right)^{2} + \left(\text{Im}(z)\right)^{2}}}{2}}

Given that $x > 0$, derive a formula for $y$ in terms of $x$ and $\text{Im} ( z )$, and explain why $y$ in this case will always have the same sign (positive, negative, or zero) as $\text{Im} ( z )$.

Hence show that in general

y = \pm\sqrt{\dfrac{-\text{Re}(z) + \sqrt{\left(\text{Re}(z)\right)^{2} + \left(\text{Im}(z)\right)^{2}}}{2}}

with the choice of the positive or negative value being dependent on the properties of $z$.

Explain what must be true of $z$ for each of the following to be true:

(i) $x = 0 , y \neq 0$

(ii) $x \neq 0 , y = 0$

(iii) $x = 0 , y = 0$