PRACTICE QUESTIONS ON COMPLEX NUMBERS INVOLVING MODULUS

(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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