# 2023 Algebra and roots for complex numbers

```SHEET # 1
1. Evaluate the following: (a) (4 + i )(3 - 2i), (b) (3 + 2i )(5 - 6i) (c) 1 + i (d) 3 + 5i (e)
2 - 3i
2-i
4
4 + 2i
(2) Given that z1 = 1 + 2i, z2 = 3 - 4i, z3 = 2 - i find (a) z1 z2* (b) z2 z3 (c) z3 z2
z1
(3) Find the square roots of the following (a) 5 + 12i (b) 3 - 4i (c) 2i (d) 21 - 20i
(4) Given that u + iv = 2 + 5i , find the value of u and of v .
1- i
(5) One root of a quadratic equation with real coefficients is z = 1 + 2i , find the equation.
(6) One root of a quadratic equation with real coefficients is z = 1 - 3i , find the equation.
(7) 3 + i is a root of the equation z 2 + (a + bi ) z - (9 + 23i ) = 0
.
&gt; Find the real numbers a and b
(8) Given that 1 + i is a root of the equation z 3 + 4 z 2 - 10 z + 12 = 0, find the other roots.
(9)
One root of the equation 2 z 3 - 9 z 2 + 30 z - 13 = 0is 2 + 3i . Find the other two roots.
(10)
One root of the equation z 3 - 10 z 2 + 33z - 34 = 0 is 4 + i . Find the other two roots.
(11)
One root of the equation z 4 + 3z 3 + 12 z - 16 = 0 is 2i . Find the other roots.
(12)
One root of the equation 2 z 4 - 11z 3 + 27 z 2 - 25z + 7 = 0 is 2 - i 3 . Find the other roots.
(13) Given that 3 + 5𝑖 is a root of the quadratic equation 𝑧 ! + 𝑝𝑧 + 𝑞 = 0, determine the
values of 𝑝, 𝑞 ∈ ℛ.
(14) Find the complex numbers 𝑢 = 𝑥 + 𝑖𝑦 such that 𝑥 and 𝑦 are real and 𝑢! = −15 + 8𝑖.
Hence, or otherwise, solve the equation 𝑧 ! − (3 + 2𝑖)𝑧 + (5 + 𝑖) = 0.
(15) A complex number 𝑣 = 𝑥 + 𝑖𝑦 is such that 𝑣 ! = 2 + 𝑖. Show that 𝑥 ! =
!&quot;√\$
!
.
(16) One root of a quadratic equation is given as 4 − 7𝑖. Determine the quadratic equation
with real coefficients which has the root 4 − 7𝑖.
(17) Given that
has root
, determine the values
where
```