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True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q. 1. For complex number z= 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 True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   (1981 - 2 Marks)

Ans. T

 Sol. Let z = x + iy

then  1 ∩ z ⇒ 1 ≤ x & 0 ≤ y (by def .)

Consider

True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ⇒  True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

and  True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 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, Zand Z3 represent the vertices of an equilateral triangle such that | Z1| = | Z2 | = | Z| then Z1 + Z2 + Z3 = 0. (1984 - 1 Mark)

Ans. T

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

⇒  True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   ⇒   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) 

Ans. F

Sol. If z1, z2, z3 are in A.P. then, True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 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)

Ans. T

Sol. ∵ Cube roots of unity are 1,  True/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEETrue/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Vertices of triangle are

A(1, 0), BTrue/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE cTrue/False: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ AB = BC = CA

∴ Δ is equilateral.

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