(a) π (b)π/2
(c) π/3 (d)π/4
2.Suppose ω₁ and ω₂ are two distinct cube root of unity different form 1.Then,what is (ω₁-ω₂)² equal to? (2016 I)
(a) 3 (b) 1
(c) -1 (d)-3
3.Suppose ω is a cube root of unity with ω≠1.suppose ,P and Q are the points on the complex plane defined by ω and ω² . If O is the origin ,then what is the angel between OP and OQ?
(2016 I)
(a) 60° (b) 90°
(c) 120° (d) 150°
.Let z be a complex number satisfying
|z-4/z-8|=1 and |z/z-2|=3/2 (2016 I)
what is |z|equal to?
(a) 6 (b)12
(c) 18 (d)36
.What is |z-6/z+6| equal to? (2016 I)
(a) 3 (b) 2
(c) 1 (d) 0
Let z₁ ,z₂ and z₃ be non - zero complex number satisfying z² =iz̅,where i=√-1.
(2016 I)
.What is z₁+z₂+z₃ equal to ?
(a) i (b) -i
(c) 0 (d) 1
.Consider the following statements
1.z₁z₂z₃ is purely imaginary
2.z₁z₂+ z₂z₃+z₃z₁ is purely real
(a) only 1 (b) only 2
(c)Both 1 and2 (d)neither 1 nor 2
.If 1,ω and ω² are the cube roots of unity then what is the value of (1+ω)(1+ω²)(1+ω²)(1+ω⁸) is (2015 I)
(a) -1 (b) 0
(c) 1 (d) 2
.If z= -2(1+2i)/3+i where i=√-1, then the argument θ(-π<θ≤π) of z is (2015 I)
(a) 3π/4 (b) π/4
(c) 5π/4 (d)-3π/4
.What is the square root of iwhere i=√-1 (2015 I)
(a) 1+i/2 (b) 1-i/2
(c) 1+i/√2 (d)none of these
.what is (sin π/6 + i (1-cos π/6)/sin π/6 - i(1-cos π/6) )³, where i=√-1, equal to ? (2015 I)
(a) 1 (b) -1
(c) i (d) -i
.What is the real part of (sin x+i cos x)³
where i=√-1? (2015 I)
(a) -cos3x (b) -sin3x
(c) sin3x (d) cos3x
.(x³-1) can be factorized as (2015 I)
(a) (x-1)(x-ω)(x+ω²) (b) (x-1)(x-ω)(x-ω²)
(c) (x-1)(x+ω)(x+ω²) (d) (x-1)(x+ω)(x+ω²)
.If z₁ and z₂ are complex numbers with |z₁|=|z₂|,then which of the following is correct ? (2015 II)
I. z₁ = z₂
II. Real part of z₁=real part of z₂
III. Imaginary part of z₁= Imaginary part of z₂
(a) only I (b) only II
(c)Both I andII (d)neither I nor II
.If the point z₁ = 1+i,where i=√-1 is the reflection of a point z₂= x + iy in the line iz̅ - iz=5, then the point z₂ is(2015 II)
(a) 1+4i (b) 4+i
(c) 1-i (d) -1-i
.zz̅ + (3-i)z + (3+i)z̅+1=0 represent a circle with (2015 II)
(a) center (-3,-1) and radius 3
(b) center (-3, 1) and radius 3
(c) center (-3,-1) and radius 4
(d) center (-3,-1) and radius 4
.what is the argument of the complex number {(1+i)(2+i)}/3-i,where i=√-1?
(2014 I)
(a) 0 (b) π/4
(c) -π/4 (d) π/2
.If P and Q are two complex numbers,then the modulus of the quotient of P and Q is (2014 I)
(a) greater than the quotient of their moduli
(b) less than the quotient of their moduli
(c) less than or equal to the quotient of their moduli
(d)equal to the quotient of their moduli
.Let z=x+iy where x,y are real value of variables and i=√-1 .If |2z-1|=|z-3|,then the point z describes (2014 I)
(a) a circle (b) an ellipse
(c) a hyperbola (d) a parabola
.If |z+z̅|=|z-z̅|,then the locus of z is (2014 I)
(a) a pair of staright line
(b)a line
(c) a set of four straight lines
(d) a circle
.What is {(1+i)⁴ⁿ⁺⁵/(1-i)⁴ⁿ⁺³} equal to ,where n is a natural number i=√-1? (2014 II)
(a) 2 (b) 2i
(c) -2i (d) i
.What is (√3+i/√3-i)⁶ equal to ,where i=√-1? (2014 II)
(a) 1 (b) 1/6
(c) 6 (d) 2
.If z is a complex number such that |z|=4 and arg(z) =5π/6,then what is z equal to? (2014 II)
(a) 2√3+2i (b) 2√3-2i
(c) -2√3+2i (d) -√3+i
.What is the value of √-i,where i=√-1 is?
(2013 I)
(a) ±1-i/√2 (b) ±1+i/√2
(c) ±1-i/2 (d) ±1+i/2
.What is the argument of the complex number (-1-i),where i=√-1? (2013 I)
(a) 5π/4 (b) -5π/4
(c) 3π/4 (d) none of these
.What is one of the square root of 3+4i,where i=√-1? (2013 II)
(a) 2+i (b) 2-i
(c) -2+i (d) -3-i
.If A+iB={4+2i/1-2i},where i=√-1,then what is the value of A ? (2012 I)
(a) -8 (b) 0
(c) 4 (d) 8
.If z=-z̅,then which one of the following is correct ? (2012 I)
(a) The real part of z is zero
(b) The imaginary part of z is zero
(c) The real part of z is equal to imaginary part of z
(d) Ths sum of real and imaginary part of z is z
.What is the modulus of {√2+i/√2-i}, where i =√-1? (2012 II)
(a) 3 (b) 1/2
(c) 1 (d) none of these
.Consider the following statesment (2012 II)
I. (ω¹⁰+1)⁷+ω=0
II. (ω¹⁰⁵+1)¹⁰ =p¹⁰ for some prime number p,where ,ω≠1 is a cubic root of unity .
(a) only I (b) only II
(c)Both I andII (d)neither I nor II
. The value of the sum Σ(iⁿ+iⁿ⁺¹),where i=√-1 is (2012 II)
(a) i (b) -i
(c) 0 (d) 1-i
.What is the modulus of {(1/1+3i)-(1/1-3i)} (2011 I)
(a) 3/5 (b) 9/25
(c) 3/25 (d) 5/3
.If ω is the cube root of unity, then what is (2-ω+2ω²)²⁷ equal to ?(2011 I)
(a) 3²⁷ω (b) -3²⁷ω²
(c) 3²⁷ (d) 3²⁷
.{(i+√3/-i+√3)²⁰⁰+(i-√3/i+√3)²⁰⁰+1} is equal to (2010 I)
(a) -1 (b) 0
(c) 1 (d) 2
.If ω is the cube root of unity and x=ω²-ω-2,then x²+4x+7 is equal to (2010 I)
(a) -2 (b) -1
(c) 0 (d) 1
.If x²+y²=1 then (1+x+iy/1+x-iy) is equal to (2010 I)
(a) x-iy (b) x+iy
(c) 2x (d) -2iy
.The modulus of { 1+2i/1-(1-i)² } is
(2010 I)
(a) 1 (b) √5
(c) √3 (d) 5
.for (1+i/1-i)ⁿ=1 the minimum value of n is (2010 I)
(a) 16 (b) 12
(c) 8 (d) 4
.What is the conjugate of (1+2i/2+i)² ?
(2010 II)
(a) 7/25+i 24/25 (b) -7/25-i 24/25
(c) -7/25+i 24/25 (d) 7/25-i 24/25
.What is (√3+i/√3-i)⁶ equal to?(2010 II)
(a) -1 (b) 1
(c) 0 (d) 2
.If ω is the complex cube of unity,then what is ω¹⁰+ω⁻¹⁰ equal to ?(2010 II)
(a) 2 (b) -1
(c) -2 (d) 1
.what is the value of (-1+i√3)⁴⁸ ?(2010 II)
(a) 1 (b) 2
(c) 2²⁴ (d) 2⁴⁸
.What is the value of i+i²+i⁴+i⁶.....+i¹⁰⁰?
(2010 II)
(a) 0 (b) 1
(c) -1 (d) None of the above
.If 2x=3+5i,then what is the value of
2x³+2x²-7x+72? (2009 I)
(a) 4 (b) -4
(c) 8 (d) -8
.What is the value of (-1+i√3/2)³⁰⁰+ (-1-√3i/2)³⁰¹?(2009 I)
(a) 1+i (b) 1-i
(c) √3(1-i)/2 (d) (√3-i)/2
.What is (√3+i)/(1+√3i) equal to?(2009 I)
(a) 1 (b) ω
(c) -ω² (d) ω²
.What is the modulus of 1+2i/1-(1-i)²?
(a) 5 (b) 4
(c) 3 (d) 1
.(-√-1)⁴ⁿ⁺³ +(i⁴¹+i⁻²⁵⁷)⁹ is equal to ?
(a) 0 (b) 1
(c) i (d) -i
.If ω is the cube root of unity then what is the conjugate of 2ω²+3i?
(a) 2ω-3i (b) 3ω+2i
(c) 2ω+3i (d) 3ω-2i
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