COMPLEX NUMBER
BOOSTER - A
1) The cube roots of unity
a) are collinear
b) lie on a circle of radius √3
c) Form an equilateral triangle d) none
2) If z= (√3/2 + i/2)³ + (√3/2 - i/2)³ then
a) Re(z)>0
b) Re(z)> 0, I'm(z)> 0
c) Re(z)> 0, I'm(z< 0)
d) I'm(z)=0
3) If ω = {(z -1)/(1+ is)}ⁿ, n integral, then ω lies on the unit circle for
a) only even n b) only odd n c) only positive n d) all n
4) if the imaginary part of (2n +1)/(iz +1) is -2, then the locus of the point representing z in the complex plane is
a) a circle b) a straight line c) a parabola d) none
5) The region of argand diagram defined by |z -1|+ |z +1| ≤ 4 is
a) interior of an ellipse
b) exterior of a circle
c) interior and boundary of an ellipse d) none
6) The triangle with vertices at the point z₁ , z₂ and (1- i)z₁ + iz₂
a) right angled but not isosceles
b) isosceles but not right angled
c) right angled and isosceles
d) equilateral
7) If z= x + it lies in III quadrant then conjugate of z/z also lies in III quadrant if
a) x >y>0 b) x < y<0 c) y< x <0 d) y> x >0
8) If a+ ib = r(cosθ+ i sinθ) then tan[i log{(a - ib)/(a+ in)}] is equal to
a) ab b) 2ab/(a²- b²) c) (a²- b²)/2ab d) 2ab/a²+ b²)
9) (cos 2θ + i sin2θ)⁻⁵(cos3θ - i sin3θ)⁶(sin θ - i cosθ)³ is equal to
a) cos25θ + i sin25θ
b) i(cos25θ + i sin25θ)
c) i(cos25θ - i sin25θ)
d) cos25θ - i sin25θ
10) The locus represented by |z -1|= |z + i| is
a) a circle of radius 1
b) an ellipse with foci at (1,0) and (0,-1)
c) a straight line through the origin
d) a circule on the line joining (1,0),(0,1) as a diameter
11) let z and ω be two complex numbers such that |z|≤ 1|ω |≤ 1 and|z + iω| = |z - iω|= 2. Then equals
a) 1 or I b) I or - I c) 1 or -1 d) I or -1
12) The smallest positive number n for with (1+ i)²ⁿ = (1- i)²ⁿ is
a) 4 b) 8 c) 2 d) 12
13) If z₁ , z₂ , z₃ are complex numberss such that |z₁|= |z₂|= |z₃|= |1/z₁ + 1/z₂ + 1/z₃|= 1, then |z₁ + z₂ + z₃| is
a) equals to one
b) less than 1
c) greater than 3
d) equals to 3
14) The complex number z= x + it which satisfy the equation|(z - 5i)/(z + 5i)|= 1 lie on
a) the x-axis
b) the straight line y=5
c) a circle passing through the origin
d) none
15) Let z₁ and z₂ be two non-zero complex numbers such that |z₁|= |z₂| and arg(z₁)+ arg(z₂)= π. Then z₁ equals
a) z₁ b) -z₂ c) conj z₂ d) - conj z₂
16) If z is any complex numbers such that z + 1/z =1, then the value of z⁹⁹+ 1/z⁹⁹ is
a) 1 b) -1 c) 2 d) -2
17) If -1+ √-3= r ₑiθ, then θ is equals to
a) 2π/3 b) -2π/3 c) π/3 d) -π/3
18) If [(√3/2+ i/2)/(√3/2 - i/2)]¹²⁰= a + ib, then
a) a= cos20, b= sin20
b) a= -cos20, b= -sin20
c) a= cos20, b= -sin20
d) a= 1, b= 0
19) If a²+ b²=1, then (1+ a+ b)/(1+ a - ib) is equal to
a) a+ ib b) a - ib c) b + is d) b - ia
20) The complex number which satisfies the equation z + √2 |z +1|+ i= 0 is
a) 2+ I b) -2+ I c) -2- I d) 2- i
21) If z₁, z₂ are two complex numbers such that|(z₁ - z₂)/(z₁ + z₂)|= 1 and iz₁ = kz₂, where k ∈ R, then the angle between z₁ - z₂ and z₁ + z₂ is
a) tan⁻¹{2k/(1+ k)} b) tan⁻¹{2k/(1- k²)} c) 3tan⁻¹k d) 3tan⁻¹k
22) Among the complex number z satisfying the condition |z + 1 - i|≤ 1, the number z having least positive argument is
a) 1- I b) -1+ I c) - I d) none
23) If z satisfy |z +1|
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