**Question 1.Find the value of**

Solution:

**Question 2.**

**Question 3.**

**Question 4.If α and β are the complex cube roots of unity, show thatSolution:**α and β are the complex cube roots of unity.

**Question 5.If x = a + b, y = αa + βb and z = aβ + bα, where α and β are complex cube roots of unity, show that xyz = a ^{3} + b^{3}.Solution:**

x = a + b, y = αa + βb, z = aβ + bα

α and β are the complex cube roots of unity.

**Question 6.Find the equation in cartesian coordinates of the locus of z if(i) |z| = 10Solution:**

Let z = x + iy

|z| = 10

|x + iy| = 10

**(ii) |z – 3| = 2Solution:**

Let z = x + iy

|z – 3| = 2

|x + iy – 3| = 2

|(x – 3) + iy| = 2

**(iii) |z – 5 + 6i| = 5Solution:**Let z = x + iy

|z – 5 + 6i| = 5

|x + iy – 5 + 6i| = 5

|(x – 5) + i(y + 6)| = 5

**(iv) |z + 8| = |z – 4|Solution:**Let z = x + iy

|z + 8| = |z – 4|

|x + iy + 8| = |x + iy – 4|

|(x + 8) + iy | = |(x – 4) + iy|

16x + 64 = -8x + 16

24x + 48 = 0

∴ x + 2 = 0

**(v) |z – 2 – 2i | = |z + 2 + 2i|Solution:**

Let z = x + iy

|z – 2 – 2i| = |z + 2 + 2i|

|x + iy – 2 – 2i | = |x + iy + 2 + 2i |

|(x – 2) + i(y – 2)| = |(x + 2) + i(y + 2)|

-4x – 4y = 4x + 4y

8x + 8y = 0

x + y = 0

y = -x

Solution:

Let z = x + iy

18y – 27 = 0

2y – 3 = 0

**Question 7.Use De Moivre’s theorem and simplify the following:Solution:**

Solution:

Solution:

**Question 8.Express the following in the form a + ib, a, b ∈ R, using De Moivre’s theorem.(i) (1 – iSolution:**

**(ii) (1 + iSolution:**

**(iii) (1 – √3 iSolution:**

**(iv) (-2√3 – 2iSolution:**