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Q. 1. For complex number z_{1 }= x_{1}+ iy_{1} and z_{2} = x_{2}+ iy_{2} , we write z_{1} ∩ z_{2} , if x_{1} ≤ x_{2} and y_{1} ≤ y_{2} . Then for all complex numbers z with 1 ∩ z , we have (1981  2 Marks)
Ans. T
Sol. Let z = x + iy
then 1 ∩ z ⇒ 1 ≤ x & 0 ≤ y (by def .)
Consider
⇒
and
⇒ 1 x^{2}  y^{2} ≤ 0 and 2y≤0
⇒ x^{2} + y^{2} ≥ 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, Z_{1}, Z_{2 }and Z_{3} represent the vertices of an equilateral triangle such that  Z_{1} =  Z_{2}  =  Z_{3 } then Z_{1} + Z_{2} + Z_{3} = 0. (1984  1 Mark)
Ans. T
Sol. As  z_{1}  =  z_{2}  =  z_{3} 
∴ z_{1}, z_{2}, z_{3} 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
⇒ ⇒ z_{1} + z_{2} +z_{3}= 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)
Ans. F
Sol. If z_{1}, z_{2}, z_{3} are in A.P. then,
⇒ z_{2} is mid pt. of line joining z_{1} and z_{3}.
⇒ z_{1}, z_{2}, z_{3} 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)
Ans. T
Sol. ∵ Cube roots of unity are 1,
∴ Vertices of triangle are
A(1, 0), B c
⇒ AB = BC = CA
∴ Δ is equilateral.
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