**Practice Questions on Complex Numbers Involving Modulus :**

Here we are going to see some example problems to understand properties of modulus of complex numbers.

(1) Find the modulus of the following complex numbers

(i) 2/(3 + 4i)

(ii) (2 - i)/(1 + i) + (1 - 2i)/(1 - i)

(iii) (1 - i)^{10}

(iv) 2i(3− 4i)(4 − 3i) Solution

(2) For any two complex numbers z_{1} and z_{2} , such that |z_{1}| = |z_{2}| = 1 and z_{1} z_{2} ≠ -1, then show that z_{1} + z_{2}/(1 + z_{1} z_{2}) is a real number. Solution

(3) Which one of the points 10 − 8i , 11 + 6i is closest to 1 + i Solution

(4) If | z |= 3, show that 7 ≤ | z + 6 − 8i | ≤ 13. Solution

(5) If |z| = 1, show that 2 ≤ |z^{2} - 3 | ≤ 4 Solution

(6) If | z - (2/z) | = 2, show that the greatest and least value of | z | are √3 + 1 and √3 − 1 respectively Solution

(7) If z_{1}, z_{2} and z_{3} are three complex numbers such that |z_{1}| = 1, |z_{2}| = 2, |z_{3}| = 3 and |z_{1} + z_{2} + z_{3}| = 1, show that |9 z_{1}z_{2} + 4 z_{1} z_{3} + z_{2} z_{3}| = 6 Solution

(8) If the area of the triangle formed by the vertices z, iz , and z + iz is 50 square units, find the value of z . Solution

(9) Show that the equation z^{3} + 2z bar = 0 has five solutions. Solution

(10) Find the square roots of (i) 4 + 3i (ii) −6 + 8i (iii) −5 −12i . Solution

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