29) If z₁ and z₂ are two complex numbers, prove that
a) |z₁+√(z₁²-z₂²)|+|z₁-√(z₁²-z₂²)| =
|z₁+z₂| +|z₁-z₂|
b) |1- ⃗z₁z₂|² - |z₁ -z₂|²=
(1-|z₁|²)(1-|z₂|²)
30) If iz³+ z² - z +i =0, show that |z|=1
31) z₁, z₂, z₃ are three complex numbers. prove that, z₁Iⱼ( ⃗z₂z₃) + z₂ + Iⱼ (⃗ ⃗z₃ z₁) + z₃ Iⱼ ( ⃗z₁ z₂)= 0 where Iⱼ(W) = imaginary part of W, where W is a complex number.
32) z₁= a+ ib and z₂= c + id are two complex numbers such that
|z₁|=|z₂|=1 and Rₑ(z₁z₂)=0, show that the two complex numbers ω₁=a+ ic and ω₂= b+ id are such that |ω₁|= |ω₂|= 1 and Rₑ (ω₁ω₂)=0
33) If z = x+iy, then show √2 |x+iy| ≥
|x| + |y|
34) Let z₁= 10+ 6i and z₂= 4+ 6i. If z be a complex number such that arg[(z - z₁)/(z-z₂)]= π/4, then prove that, |z-7 - 9i| =3√2
35) If (a₁ +ib₁)(a₂ +ib₂)(a₃ +ib₃)...(aₙ+ibₙ) prove that
a) (a₁² + b₁²)(a₂²+ b₂²)(a₃²+b₃²)... (aₙ²+ ibₙ²)= A²+ B²
b) tan⁻¹(b₁/ a₁) +tan⁻¹(b₂/a₂)+ tan⁻¹(b₃/a₃) ......+ tan⁻¹(bₙ/aₙ) =tan⁻¹(B/A)
36) If W= √(z²-1) and z= x+iy, show amp W= 1/2[tan⁻¹(y/(x+1) +
tan⁻¹(y/(x-1)]
37) If the complex number z satisfies the equations
|(z-12)/(z-8i)|= 5/4 and |(z-4)/(z-8)| = 1, find the value of z
38) If z be a compkex number, solve the equation z² + |z| = 0
39) If α²+α+2, show x³-1= (x-1)(y-α) (x-α²)
40) Show (y-z)² + (z-x)² + (x-y)²= 2(x+yω+zω²)(y+yω²+zω), where ω is an imaginary cube root of 1
41) If α be the real cube root of β, η be the complex cube roits of m, where m is positive real number, then for any a, b, c, show that, (aβ+bη+cα)/(aη+bα+cβ) =ω², where ω is a complex cube root of unity.
42) find value of {-1+i√3)¹⁶/(1+i)²⁰} + {-2-i√(3)¹⁶/(1-i)²⁰}
43) show{(-1+√-3)/2)}ⁿ+{(-2-√-3)/2}ⁿ = 2 when n isbpositive integers and multiple of.3.
= -1 when n is any other positive integer.
44) If ω be a complex cube roit of unity, find the value of 1.(2-ω)(2-ω²) + 2.(3-ω)(3-ω²)+...+(n-1)(n-ω)(n-ω²)
45) If ω be an imaginary cube root of unity and x= a+ bω + cω², show that x³-3ax²+3(a²-bc)x=a³+b³+c³-3abc
46) if x= a+ bω+cω², y= aω+bω² +c, z= aω²+ b + cω, show that x²/yz + y²/zx + z²/xy = 3
47) z₁, z₂, z₃ are three complex numbers such that, z₁+ z₂+ z₃= A,
z₁+ z₂ω+z₃ω²= B and
z₁ +z₂ω²+z₃ω= C, where ω is an imaginary cube root of 1. prove
|A|²+ |B|²+ |C|²= 3{|z₁|²+|z₂|²+|z₃|²
48) If x= α² + 2βη, y=β²+2ηα,
z = η²+2αβ prove that (α³+β³+η³-3αβη)= x³+y³+z³- 3xyz
49) If a and b are real numbers between 0 and 1 and the points represented by complex numbers z₁= a+ i, z₂= 1+bi and z₃ =0 form an equilateral triangle in the complex plane, then find the values of a and b.
50) If m be real, n= u+ iv, z= x+iy and ⃗n⃗, ⃗z conjugate of n and z respectively, then show that the equatiin n( z+ ⃗z)+ ⃗n(⃗z - z) + m=0 represents two straight lines in the complex plane. Find the angle between the straight lines.
51) W= (z+1)/(z-1), where z is a complex number. If z lies on the circle |z - 1|= 2, show that W lies on a circle in the complex plane. Find the centre and radius of that circle.
52) Show that the area of the triangle formed by three complex numbers z, iz and (z+iz) in the complex plane is 1/2 |z|².
53) If the area of the Triangle formed by the complex numbers z, iz and z+ iz is ar² , where |z| = r, then find the value of the constant a.
54) If P,Q,R represent the complex numbers z₁, z₂, z₃, respectively in the complex plane and lz₁+mz₂+n z₃ =0 when l+m+n=0, show that P, Q, R are collinear.
55) If the complex number z satisfies the Equation|z - 6/z|= 2, find the greatest value of |z|.
56) Find the least value of|z + 1/z|, if |z| ≥ 2.
57) In the complex plane the condition |z + 2/z|=2 is satisfied by the moving point which is represented by the complex number z (≠0). Find the maximum distance of the point from the origin.
58) Show that the three cube roots if I, are - I, (I+√3)/2, (i-√3)/2
59) If z= 1+ sinx + I cosx, find amp z
60) If z= 1+ cos2x + isin2x,
π/2<x<3, π/2 find amp. z
61) If az+ ib ⃗z = 0, where z= x+ iy and a+b ≠0, find the value of x+y
62) If z₁, z₂ are conjugate of each other anf z₃, z₄ are conjugate of each other, show that , amp z₁/z₄ = amp. z₃/z₂.
63) show that, amp (z) - amp (-z)
=π, when amp(z) > 0
= -π when amp (z) < 0
64) Explain with reasons which one of the following is correct?
A) 2 + 3i > 1+ 4i
B) 6+2i > 3+ 3i
C) 5+ 8i > 5+ 7i
D) none of these (i² = -1)
65) If |z₁|=| z₂ |= 1 and
amp z₁ + amp z₂ = 0, prove that
z₁z₂ = 1
66) If the complex number z= x+ iy satisfies the Equation
|(z-5i)/(z+5i)| = 1, then z lies on
A) the x-axis. B) the line y=5
C) a circle passing through the origin
67) Solve: |z| + z= 1 +3i, where z is a complex number.
68) For any complex number z the minimum value of |z| + |z -1| is
A) 1. B) 0. C) 1/2. D) 3/2
* Three complex numbers are in A P. Show that they can not be on a circle in the complex plane.
** Prove that two complex numbers z₁ and z₂, which are not zero form an equilateral triangle with the origin in the complex plane if z₁²- z₁z₂ + z₂² = 0.
***Show that the origin and the points in the complex plane which are represented by the roots of the Equation x²+pz+q= 0 form an equilateral triangle if p²= 3q.
*** If the vertices A,B,C of a right angled isosceles triangle, right angled at C, are represented by complex numbers z₁, z₂, z₃, respectively, then show that
(z₁- z₂)² = 2(z₁- z₃)(z₃ - z₂).
** Three complex numbers z₁, z₂, z₃, are such that |z₁|=|z₂|=| z₃| and they represent the vertices of an equilateral triangle in the complex plane; show that z₁+z₂+ z₃=0
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