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| Incomplete
BOOSTER - B
1) The value of i⁵⁷ + 1/i¹²⁵ is
a) 0 b) -2i c) 2i d) 2. 0
2) (3+ 2i sinθ)/(1- 2i sinθ) will be purely imaginary, if θ=
a) 2nπ ± π/3, n ∈ I
b) nπ + π/3, n ∈ I
c) nπ ± π/3, n ∈ I d) none. c
3) Determine least positive value of n for which {(1+ i)/(1- i)ⁿ = 1.
4) Find the value of the sum ⁵ₙ₌₁∑ (iⁿ + iⁿ⁺²), where i= √-1.
5) The value of x and y satisfying the equation {(1+ i)x - 2i}/(3+ i) + {(2- 3i)y + i}/(3- i)= 1 are
a) x =-1, y= 3
b) x = 3, y= -1
c) x =0, y= 1
d) x = 1, y= 0
6) Find the square root of 7+ 24i. ±(4+ 3i)
7) If x= -5+ 2 √-4, find the value of x⁴ + 9x³+ 35x² - x + 4. -160
8) Find the value of x³ + 7x² - x + 16, where x= 1+ 2i.
9) If a+ ib = (c + i)/(c - i), where c is a real number, then show that: a² + b² = 1 and b/a = 2c/(c² -1).
10) Find the modulus, argument, principal value of argument, least positive argument of complex numbers
a) 1+ i √3. 2, 2nπ+ π/3, n ∈ I, π/3, π/3
b) -1+ i √3. 2, 2nπ+ π/3, n ∈ I, 2π/3, 2π/3
c) 1- i √3. 2, 2nπ- π/3, n ∈ I, 5π/3, -π/3
d) -1- i √3. 2, 2nπ- 2π/3, n ∈ I, 4π/3, -2π/3
11) Find modulus and argument for z= 1- sinα + i cosα, α ∈ (0,2π).
12) Find the modulus and amplitude of following complex numbers.
a) -2+ 2√3 i
b) -√3 - i.
c) -2i.
d) (1+ 2i)/(1- 3i).
e) (2+ 6√3 i)/(5+ √3 i)
13) Find the locus of:
a) |z -1|²+ |z +1|²= 4. x²+ y²=1, circle, centre in origin, radius is unity
b) Re(z²)= 0. y= ± x. Straight lines passing through origin
14) If z is complex number such that z² = (conjugate of z)², then
a) z is purely real
b) z is purely imaginary
c) either z is purely real or purely imaginary d) none.
15) Among the complex number z which satisfies |z - 25i|≤ 15, find the complex numbers z having
a) least positive argument
b) maximum positive argument
c) least modulus
d) maximum modulus
16) Find the distance between two complex numbers z₁ = 2+ 3i and z₂ = 7- 9i on the complex plane.
17) Find the locus of |z - 2 - 3i|= 1.
18) If z is a complex number, then z² + (conjugate of z)² = 2 represents
a) a circle b) a straight line c) a hyperbola d) an ellipse
BOOSTER - C
1) Express the following complex numbers in polar and exponential form:
a) (1+ 3i)/(1- 2i). √2(cos(3π/4) + i sin(3π/4)), √2 ₑ3π/4
b) (i -1)/(cos(π/3) + i sin(π/3)). √2(cos(5π/4) + i sin(5π/4)), √2 ₑ5π/12
2) xₙ = cos(π/2ⁿ) + i sin(π/2ⁿ) then x₁, x₂, x₃.....∞ is equal to
a) -1 b) 1 c) 0 d) ∞. a
3) Express the following complex number in polar form and exponential form:
a) -2+ 2i
b) -1- √3 i
c) (1+ 7i)/(2- i)².
d) (1- cosθ + i sinθ), θ∈ (0,π)
4) Find amplitude z and |z| if z=[{(3+ 4i)(1+ i)(1+ 3i)}/{(1- i)(4- 3i)(2i)}]². 1, -π/3
5) If |(z - i)/(z + i)|= 1, then locus of z is
a) x-axis b) y-axis c) x= 1 d) y= 1. a
6) If |z₁ + z₂|²= |z₁|² + |z₂|² then (z₁/z₂) is
a) zero or purely imaginary
b) purely imaginary
c) purely real d) none b
7) If z₁ and z₂ are two complex numbers such that (z₁ - 2z₂)/(2- z₁z₂) is unimodular (whose modulus is one), while z₂ is not unimodular. Find |z₁|. 2
8) The locus of the complex number z in argand plane satisfying the inequality
log₁/₂ {(|z - 1| +4)/(3 |z -1| -2)}> 1 (where |z -1≠ 2/3) is
a) a circle
b) interior of a circle
c) exterior of a circle d) none c
9) If |z - 4/z|= 2, then the greatest value of |z| is
a) 1+√2 b) 2+√2 c) √3+1 d) √5+1. d
10) Shaded region is given by
a) |z + 2|≥6, 0≤ arg(z) ≤ π/6
b) |z + 2|≥6, 0≤ arg(z) ≤ π/3
c) |z + 2|≤ 6, 0≤ arg(z) ≤ π/2 d) none. c
11) The inequality |z -4|< | z -2| represents region given by
a) Re(z)> 0 b) Re(z)< 0 c) Re(z)> 3 d) none
12) If z= reⁱᶿ, then the value of|eⁱᶻ| is equal to
a) e⁻ʳᶜᵒˢᶿ b) eʳᶜᵒˢᶿ c) eʳˢᶦⁿᶿ d) e⁻ʳˢᶦⁿᶿ
13) ⁻¹₁₂₃₁₂²₁₃₃₂₁₂₃₄₃₄₁₂₁ᵢ₁₂₃₁₂₃∈₁₂₃₂₁₃₁₂₃αβγαβγαβγαβγαβγαβγαβγαβγᵣ₁₂₃₄₅⁴ⁿⁿαβαⁿβⁿαβγ³²ωαββγγαωωωωωωωωωωωω⁶ₖ₌₁∑ ₁₂ₙₖ²ᵏ₁₂₁₂²₁²⁻¹₂²θ∈⁴₁³₂²₃₄₅₁₂₃₁₂₃₁₂₃₁₂ααα₁α₂¹³ₙ₌₁ⁿⁿ⁺¹²³²₁₂₁₂αβαβαβαβ²²ₑ∑ⁱᶿᶜᵒˢᶿᶜᵒˢᶿˢᶦⁿᶿˢᶦⁿᶿ ωθωωωωω²³¹⁷²³¹³₁₂₃₁₂₁₃₂₃₁₂₃²∈ωωωωωωωω³³⁴³⁶⁵₁₂₃₂₃₁₂₁₂₂₁₂₂₂₁⁶⁶⁵⁵ωωωω₁₂₁₁₂₂₁₁₂₂∈²²₁₂₃₁₂₃₁₂₃₁₂₃₁₂₃₁₂₃₂₁₂₁₂₁₂₁₂₁₂₁₂₁₂₁₂₁₁₂₁₂₁₂¹⁾³₁₁₁₁ωωωωωωωωαβααβαβαβⁿ⁻ⁿ∈ⁿ²ⁿ²ⁿⁿ²ⁿⁿ²ⁿⁿ²ⁿⁿⁿ²ⁿⁿ²ⁿⁿ²ⁿ³²¹⁹⁹³¹⁹⁹⁴ωωωω⁹⁸¹ᵣʳ∈lim ₙ→∞ ⁿᵣ₌₁ ∀⇒Πᵣ₁₂₁₂₂₁₂₁₂₂₁αβαβαβαβαⁱᶿθ∈ωωωωω₁λλ₂μμωωωω³³²²ⁿⁿ²ⁿⁿ²ⁿ³²₁₂₃₄⁴³²⁴ᵢ₌₁Π₁⁴³²¹⁴³²¹²¹⁴³²¹⁴³⁴¹²¹²³⁴³⁴¹²¹²³⁴³¹³₄₃₂₁₄₃₂₁₂₁₄₃₂₁₄₃₄₁₂₁₂₃₄₃₄₁₂₁₂₃₄₃₁₃₂₄₁₂₃₀₁₃₂₃₁₀₂₀₁₂₃₀²³³²₁₂₁₂₁₂₁₂₁₂₃₁₂₃₁₂₃₁₂₃₁₂₃₁²₂²₃²₁₂₂₃₃₁ⁿⁿ₁₂ₙ₋₁ⁿ₁₂ₙ₋₁ⁿⁿ⁷ₘ₌₀₇⁷²²²⁵⁵²²²⁵²ⁿ⁺¹²²ₙ₁₂ₙ₁₂ₙₐᵥ꜀θθθθθᵥⁱᴮ²²²²₁₂₁₂₁₂₂₁³²
No comments:
Post a Comment