Parameter from real/imaginary condition

Questions involving a complex expression with a real parameter where the condition is that the expression is real, purely imaginary, or equates to a specific form, requiring the student to find the parameter value.

4 questions · Moderate -0.1

4.02a Complex numbers: real/imaginary parts, modulus, argument 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

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CAIE P3 2023 November Q8

7 marks Standard +0.3

8 It is given that $\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )$, where $a$ and $\lambda$ are real constants.

Show that $3 a ^ { 2 } + 4 a - 4 = 0$.
Hence find the possible values of $a$ and the corresponding values of $\lambda$.

Edexcel FP1 2016 June Q7

13 marks Standard +0.3

7. A complex number $z$ is given by $$z = a + 2 i$$ where $a$ is a non-zero real number.

Find $z ^ { 2 } + 2 z$ in the form $x +$ iy where $x$ and $y$ are real expressions in terms of $a$. Given that $z ^ { 2 } + 2 z$ is real,
find the value of $a$. Using this value for $a$,
find the values of the modulus and argument of $z$, giving the argument in radians, and giving your answers to 3 significant figures.
Show the points $P , Q$ and $R$, representing the complex numbers $z , z ^ { 2 }$ and $z ^ { 2 } + 2 z$ respectively, on a single Argand diagram with origin $O$.
Describe fully the geometrical relationship between the line segments $O P$ and $Q R$.

OCR FP1 2013 June Q1

6 marks Moderate -0.5

1 The complex number $3 + a \mathrm { i }$, where $a$ is real, is denoted by $z$. Given that $\arg z = \frac { 1 } { 6 } \pi$, find the value of $a$ and hence find $| z |$ and $z ^ { * } - 3$.

Edexcel FP1 Q46

7 marks Moderate -0.3

The complex number $z$ is defined by $$z = \frac{a + 2i}{a - 1}, \quad a \in \mathbb{R}, a > 0 .$$ Given that the real part of $z$ is $\frac{1}{2}$ , find

the value of $a$, [4]
the argument of $z$, giving your answer in radians to 2 decimal places. [3]