Edexcel A Level Further Maths
Topic questions Pure mathematics Complex arithmetic
16 questions · 114 marks · about 2.5 hours
The complex number z 
a 
bi where a and b are real constants.
Given that 
is purely imaginary,
show that a2 = b2
Given also that zz* 
50
determine the possible complex numbers z
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Given that
show that
arg(zw) 
arg(z) 
arg(w)
z1 
a 
bi and z2 c di
where a, b, c and d are real constants.
Given that
• z1 
z2 is real
show that a 
2 and determine the value of each of b, c and d
On the same Argand diagram
• sketch the locus of points z which satisfy z 
12 7
• sketch the locus of points w which satisfy w 
5i 4
showing the coordinates of any points of intersection with the axes.
f(z) = z3 + az + 52 where a is a real constant
Given that 2 – 3i is a root of the equation f(z) = 0
write down the other complex root.
Hence
solve completely f(z) = 0
determine the value of a
Show all the roots of the equation f(z) = 0 on a single Argand diagram.
f(z) = z3 + az + 52 where a is a real constant
Given that 2 – 3i is a root of the equation f(z) = 0
write down the other complex root.
Hence
solve completely f(z) = 0
determine the value of a
Show all the roots of the equation f(z) = 0 on a single Argand diagram.
Given that z = a + bi is a complex number where a and b are real constants,
show that zz* is a real number.
Given that
• zz * = 18
determine the possible complex numbers z
Given that z = a + bi is a complex number where a and b are real constants,
show that zz* is a real number.
Given that
• zz * = 18
determine the possible complex numbers z
A student was asked to answer the following:
The student’s attempt is shown below.
The student made errors in line 1 and line 3
Correct the error that the student made in
line 1
line 3
Given that
write down the exact value of
|z1z2|
arg(z1z2)
Given that w = z1z2 and that arg(wn) = 0, where n 
+
determine
the smallest positive value of n
the corresponding value of |wn|
Given that
write down the exact value of
|z1z2|
determine
the smallest positive value of n
the corresponding value of |wn|
Given that |z| < 1, write down the sum of the infinite series
Given that z = 
cosθ + i sinθ),
use the answer to part (a), and de Moivre’s theorem or otherwise, to prove that
show that the sum of the infinite series 1 + z + z2 + z3 + … cannot be purely imaginary, giving a reason for your answer.
Given that |z| < 1, write down the sum of the infinite series
Given that z = 
cosθ + i sinθ),
use the answer to part (a), and de Moivre’s theorem or otherwise, to prove that
show that the sum of the infinite series 1 + z + z2 + z3 + … cannot be purely imaginary, giving a reason for your answer.
f (z) = 3z3+pz2+ 57z+q
where p and q are real constants.
show all the roots of f(z) = 0 on a single Argand diagram,
find the value of p and the value of q.
f (z) = 3z3+pz2+ 57z+q
where p and q are real constants.
show all the roots of f(z) = 0 on a single Argand diagram,
find the value of p and the value of q.
where a, b, c and d are real constants.
Given that í1 + 2i and 3 í i are two roots of the equation f (z
=show all the roots of f (z) = 0 on a single Argand diagram,
find the values of a, b, c and d.
where a, b, c and d are real constants.
Given that −1 + 2i and 3 − i are two roots of the equation f (z) = 0
show all the roots of f (z) = 0 on a single Argand diagram,
find the values of a, b, c and d.
Memes
Laugh now. Cry in the exam hall later. Then come back and do a past paper.
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