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Q. 1. For complex number z1 = x1+ iy1 and z2 = x2+ iy2 , we write z1 ∩ z2 , if x1 ≤ x2 and y1 ≤ y2 . Then for all complex numbers z with 1 ∩ z , we have (1981 - 2 Marks)
Sol. Let z = x + iy
then 1 ∩ z ⇒ 1 ≤ x & 0 ≤ y (by def .)
⇒ 1- x2 - y2 ≤ 0 and -2y≤0
⇒ x2 + y2 ≥ 1 and y≥0
which is true as x ≥ 1 &y≥0
∴ The given statement is true ∀ z∈C .
Q. 2. If the complex numbers, Z1, Z2 and Z3 represent the vertices of an equilateral triangle such that | Z1| = | Z2 | = | Z3 | then Z1 + Z2 + Z3 = 0. (1984 - 1 Mark)
Sol. As | z1 | = | z2 | = | z3 |
∴ z1, z2, z3 are equidistant from origin.
Hence O is the circumcentre of ΔABC.
But according to question ΔABC is equilateral and we know that in an equilateral Δ circumcentre and centriod coincide.
∴Centriod of ΔABC = 0
⇒ ⇒ z1 + z2 +z3= 0
∴ Statement is true.
Q. 3. If three complex numbers are in A.P. then they lie on a circle in the complex plane. (1985 - 1 Mark)
Sol. If z1, z2, z3 are in A.P. then,
⇒ z2 is mid pt. of line joining z1 and z3.
⇒ z1, z2, z3 lie on a st. line
∴ Given statement is false
Q. 4. The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle. (1988 - 1 Mark)
Sol. ∵ Cube roots of unity are 1,
∴ Vertices of triangle are
A(1, 0), B c
⇒ AB = BC = CA
∴ Δ is equilateral.