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 z1 and z2 , such that |z1| = |z2| = 1 and z1 z2 ≠ -1, then show that z1 + z2/(1 + z1 z2) 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 ≤ |z2 - 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 z1, z2 and z3 are three complex numbers such that |z1| = 1, |z2| = 2, |z3| = 3 and |z1 + z2 + z3| = 1, show that |9 z1z2 + 4 z1 z3 + z2 z3| = 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 z3 + 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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