Mr. J GallagherName:______
Complex Numbers –Revision Handout
Leaving Cert - Ordinary Level
Syllabus
You should be able to:
Explain / understand the following key words
- Complex number
- Argand diagram
- Real / imaginary
- Translation
- Modulus
- Conjugate
Plot complex numbers on argand diagram
Add / subtract / multiply complex numbers
Calculate the modulus of a complex number
Multiplying complex numbers
Calculate the conjugate of complex numbers
Divide complex numbers using the conjugate
Solving quadratic equations with complex roots
Question 1
Simplify each of the following:
a)1 +
b)
c)
d)
e)
f)2
Question 2 – Adding, Subtracting & Multiplying Complex Numbers
If z1 = 1+3i, z2 = 2 – 4i and z3 = 3 + 2i, evaluate the following:
a)z1 + z2
b)z1 – z2
c)z1 + z3
d)z2 – z3
e)3z1
f)– 2z2
g)3z1 + 6z2
h)3z3 – 2z2
i)z3(z2 – z1)
j)z1z2
k)z3z2
Question 3
a)Solve z2 + 2z + 2 = 0 and write the two complex numbers in the form a bi
b)Solve z2 + 4z + 5 = 0 and write the two complex numbers in the form a bi
c)Solve z2 – 6z + 34 = 0 and write the two complex numbers in the form a bi
d)Solve z2 + 6z + 13 = 0 and write the two complex numbers in the form a bi
Question 4
Find the conjugate of each of the following complex numbers:
a)z1 = 3 + 5i
b)z2 = 3 – i
c)z3 = 5i
d)z4 = – 1 – 3i
e)z5 = 1 + 3i
f)z6 = 2 – 4i
Question 5
If z1 = 1 + 3i, z2 = 2 – 4i and z3 = 3 + 2i, write each of the following in the form
a + bi, where a, b R.
a)
b)
c)
d)
e)
Question 6
Represent each of the following complex numbers on an Argand Diagram and hence find the modulus and argument of each. (Separate diagram for each complex number).
a)z1 = 3 + 5i
b)z2 = 3 – i
c)z3 = 5i
d)z4 = – 1 – 3i
e)z5 = 1 + 3i
f)z6 = 2 – 4i
Question 7
a)If z1 = – 2 – 3i and z2 = 3 + i. Investigate if =
b)Find the values of a for which |a + 8i| = 10. (i.e. modulus)
c)If z1 = 2 + 3i and z2 = 2 – 3i, express in the form a + bi.
Now find the value of k such that|z1|= .
Question 8
Find the values of x and y in the following questions:
a)(x + 4) + (y – 2)i = 3 + 2i
b)(3x + 1) + (2 – y)i = 4 – i
c)x(2 + 3i) – 2y = 3 + 6i
d)(4x – 2) + (a – 4)i = (4 – 2b) + 2bi
e)x(3 + 4i) + 5 = y(1 + 2i)
f)(x + yi) (5 + i) = 3 – 2i
2011 Exam Question (25 Marks)
2012 Exam Question (25 Marks)
2013 Exam Question (25 Marks)
2014 Exam Question (25 Marks)
2015 Exam Question (25 Marks)
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