# PRACTICE QUESTIONS ON COMPLEX NUMBERS INVOLVING MODULUS

Practice Questions on Complex Numbers Involving Modulus :

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

## Practice Questions on Complex Numbers Involving Modulus - Questions

(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.

(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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