Baixe Solução de equações complexas e outras Exercícios em PDF para Matemática, somente na Docsity!
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
1. If the cube roots of unity be 1, ω, ω^2 , then the roots of the equation, (x - 1) 3 + 8 = 0 are : (A) - 1, 1 + 2ω, 1 + 2ω^2 (B) - 1, 1 - 2ω, 1 - 2ω^2 (C) - 1, - 1, - 1 (D) None of these 2. −^2 −^3 =
(A) 6 (B) - (^6)
(C) i 6 (D) None of these
3. The inequality z - 4 < z - 2 represents the region given by : (A) Re (z) > 0 (B) Re (z) < 0 (C) Re (z) > 2 (D) None of these 4. If z 1 and z 2 are two non-zero complex numbers such that, z 1 + z 2 = z 1 + z 2 , then Arg(z 1 ) − Arg(z 2 ) is :
(A) - π (B) -
π 2
(C)
π 2
(D) 0
5. If
1 2
z z
is a purely imaginary
number, then
z z z z
1 2 1 2
is equal to :
(A)
(B) 1
(C)
(D)
6. If z 1 & z 2 are any two complex
numbers then z 1 z 2
2
2 − is equal to :
(A) 2 z^1
2 z (^2)
2 (B) 2 z^1
2
2
(C) z^1
2
2
(D) 2 z^1 z^2
7. If z = x + iy and ω =
i z z i
, then
ω = 1 implies that : (A) z lies on the imaginary axis (B) z lies on the real axis (C) z lies on the unit circle (D) None of these
8. If x + iy =
a i b c i d
, then (x 2 + y 2 ) 2 =
(A)
a b c d
2 2 2 2
(B)
a b c d
(C)
c d a b
2 2 2 2
(D) None of these
9. The points 1 + 3i, 5 + i and 3 + 2i in the complex plane are : (A) vertices of a right angled triangle (B) collinear (C) vertices of an obtuse angled triangle (D) vertices of an equilateral triangle 10. The complex numbers, sin x + i cos 2x and cos x - i sin 2x are conjugate to each other for :
(A) x = nπ (B) x = n^ +
^
π
(C) x = 0 (D) No value of x
Complex Numbers
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
11. The values of x & y satisfying the
equation,
(^1 )^2 3
i x i i
( 2 3 ) 3
i y i i
= i
are : (A) x = - 1, y = 3 (B) x = 3, y = - 1 (C) x = 0, y = 1 (D) x = 1, y = 0
12. If z is a complex number, then the minimum value of z + z - 1 is : (A) 1 (B) 0
(C)
(D) None of these
i i
i i
(A)
− i
(C)
− i (D)
14. The points z 1 , z 2 , z 3 , z 4 in the complex plane are the vertices of a parallelogram taken in order, if and only if : (A) z 1 + z 4 = z 2 + z (^3) (B) z 1 + z 3 = z 2 + z (^4) (C) z 1 + z 2 = z 3 + z (^4) (D) None of these 15. − 8 −6 i =
(A) 1 ± 3 i (B) ± (1 - 3 i) (C) ± (1 + 3 i) (D) ± (3 - i)
16. If α & β are imaginary cube roots of
unity, then α^4 + β^4 +
α β
(A) 3 (B) 0
(C) 1 (D) 2
17. For any two complex numbers z 1 & z 2 and any real numbers a & b ;
( a z^1 b z 2 )
2 − + (^) ( b z 1 a z 2 )
2
(A) (a 2 + b 2 ) (^) ( z 1 +z 2 )
(B) (a 2 + b 2 ) ( z^1 z )
2 2
2
(C) (a 2 + b 2 ) ( z^1 z )
2 2
2 −
(D) None of these
18. If z = 4 and arg z =
π , then z =
(A) 2
− 2i
(B) 2 3 + 2i
(C) − 2 3 + 2i
(D) − 3 + i
19. Area of the triangle formed by the points z, iz and z + iz is :
(A) 2 z^
2 (B)
z 2
(C) z
2 (D) None of these
i i
sin sin
θ θ
will be purely imaginary,
if θ =
(A) 2 nπ ±
π 3
(B) nπ +
π 3
(C) nπ ±
π 3
(D) None of these
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
31. If x = a + b, y = aω + bω^2 , z = aω^2 + bω, then the value of x 3 + y 3 + z 3 equals (A) a 3 + b 3 (B) 3 (a 3 + b 3 ) (C) 3 (a 2 + b 2 ) (D) None of these 32. The value of,
a b c b c a
ω ω ω ω
2 2 +^
a b c a a b
ω ω ω ω
2 2
will be : (A) 1 (B) - 1 (C) 2 (D) - 2
33. The value of
sin sin
(^6) π k π i
k k
^
= (^) is:
(A) - 1 (B) 0 (C) - i (D) i
34. If three complex numbers are in A.P. then they lie on : (A) A circle in the complex plane (B) A straight line in the complex plane (C) A parabola in the complex plane (D) None of these 35. If the complex numbers z 1 , z 2 , z (^3) represent the vertices of an equilateral triangle such that, z 1 = z 2 = z 3 , then z 1 + z 2 + z 3 = (A) 0 (B) 1 (C) - 1 (D) None of these 36. If a & b are real numbers between 0 and 1 such that the points z 1 = a + i , z 2 = 1 + bi and z 3 = 0 form an equilateral triangle, then :
(A) a = b = 2 + (^3)
(B) a = b = 2 - (^3)
(C) a = 2 - 3 , b = 2 + (^3) (D) None of these
37. A point z moves on Argand diagram in such a way that z - 3 i = 2, then its locus will be : (A) y - axis (B) A straight line (C) A circle (D) None of these 38. If A, B, C are represented by 3 + 4 i , 5 - 2 i , - 1 + 16 i , then A, B, C are: (A) collinear (B) vertices of equilateral triangle (C) vertices of isosceles triangle (D) vertices of right angled triangle 39. If - 1 + − 3 = re iθ^ , then θ is equal to
(A)
π 3
(B) -
π 3
(C)
π (D) -
π
40. For any complex number z,
z
z
^
if an only if : (A) z is a pure real number (B) z = 1 (C) z is a pure imaginary number (D) z = 1
41. Let z 1 & z 2 be two complex numbers with α & β as their principal arguments such that α + β > π, then principal arg (z 1 z 2 ) is given by : (A) α + β + π (B) α + β − π (C) α + β − 2 π (D) α + β
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
42. The locus of z given by
z z i
= 1, is
(A) a circle (B) an ellipse (C) a straight line (D) a parabola
43. The true statement is : (A) 1 - i < 1 + i (B) 2 i + 1 > - 2 i + 1 (C) 2 i > 1 (D) None of these 44. If z 1 = 10 + 6 i , z 2 = 4 + 6 i and z is a complex number such that,
amp
z z z z
2
π 4
, then the value of
z - 7 - 9 i is equal to :
(A) 2 (B) 2 (^2)
(C) 3 2 (D) 2 (^3)
45. If ω is a complex cube root of unity, then the positive integral value of n, the product of ω, ω^2 , ω^3 , .....ωn^ , will be :
(A) 1 3
− i (^) (B) - 1 3 2
− i
(C) 1 (D) (B) & (C) both
46. The value of (1 + i) (^5) × (1 - i) 5 is : (A) - 8 (B) 8 i (C) 8 (D) 32 47. If x = - 5 + 2
, then the value of the expression, x 4 + 9 x 3 + 35 x 2 − x + 4 is : (A) 160 (B) - 160 (C) 60 (D) - 60
48. Suppose Z 1 , Z 2 , Z 3 are the vertices of an equilateral triangle inscribed in the
circle Z = 2. If Z 1 = 1 + i 3 , then values of Z 3 and Z 2 are respectively :
(A) − 2, 1 − i (^3)
(B) 2, 1 + i (^3)
(C) 1 + i 3 , - 2 (D) None of these
49. The values of z for which, z + i = z - i are : (A) Any real number (B) Any complex number (C) Any natural number (D) None of these 50. If ω (≠ 1) is a cube root of unity and (1 + ω) 7 = A + B ω, then A and B are respectively, the numbers : (A) 0, 1 (B) 1, 0 (C) 1, 1 (D) - 1, 1 51. If ω (≠ 1) is a cube root of unity, then
1 1 1 1 1 1 1
2 2 2
i i i i
ω ω ω ω
equals:
(A) 0 (B) 1
(C) ω (D) i
52. Let z & ω be the two non-zero complex numbers such that z = ω and arg z + arg ω = π. Then z is equal to : (A) ω (B) − ω (C) ω (D) −
ω
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
65. If
4
cos sin sin cos
θ θ θ θ
i i i
cos nθ + i sin nθ, then n is equal to : (A) 1 (B) 2 (C) 3 (D) 4
66. The common roots of the equations, x 12 - 1 = 0 , x 4 + x 2 + 1 = 0 are : (A) ± ω (B) ± ω^2 (C) ± ω, ± ω^2 (D) None of these 67. For positive integers n 1 , n 2 value of
the expression, (^1 +^ i^ )^1
n
- i^7 n^2 , where i = − 1 is a real number if and only if : (A) n 1 = n 2 + 1 (B) n 1 = n 2 - 1 (C) n 1 = n 2 (D) n 1 > 0, n 2 > 0
68. The value of the expression, 1. (2 − ω) (2 − ω^2 ) + 2.(3 − ω) (3 − ω^2 ) + ........ + (n − 1).(n − ω) (n − ω^2 ), where ω is anb imaginary cube root of unity, is :
(A) 21 (n - 1) n (n 2 + 3n + 4)
(B) 41 (n - 1) n (n 2 + 3n + 4)
(C) 21 (n + 1) n (n 2 + 3n + 4)
(D) 41 (n + 1) n (n 2 + 3n + 4)
69. The equation,
z z + (2 - 3i) z + (2 + 3i)
z
represents a circle of radius : (A) 2 (B) 3 (C) 4 (D) 6
70. If z 1 , z^2 , z^3 , z^4 are the roots of the
equation z 4 =1, then value of
z (^) i i
3 1
4
=
ˇ
is:
(A) 0 (B) 1 (C) i (D) 1 + i
71. If
c i c i
= a + ib, where a, b, c are
real, then a 2 + b 2 = (A) 1 (B) - 1 (C) c 2 (D) - c 2
72. If i 2 = - 1, then the value of n =
ˇ 1
200 = i n^ is
(A) 50 (B) - 50 (C) 0 (D) 100
73. If α is an inaginary cube root of unity, then for n ∈ N, the value of α3n + 1^ + α3n + 3^ + α3n + 5^ is : (A) - 1 (B) 0 (C) 1 (D) 3
ANSWERS
1. B 2. B 3. D 4. D 5. B 6. B
- B 8. A 9. B 10. D 11. B 12.A
- D 14. B 15. B 16. B 17. B 18.C
- B 20. C 21. D 22. B 23. C 24.C
- C 26. B 27. D 28. C 29. C 30.B
- B 32. B 33. D 34. B 35. A 36.B
- C 38. A 39. C 40. B 41. C 42.C
- D 44. C 45. D 46. D 47. B 48.A
- A 50. C 51. A 52. D 53. C 54.D
- D 56. C 57. A 58. C 59. A 60.B
- C 62. C 63. C 64. A 65. D 66.C
- D 68. B 69. B 70. A 71. A 72.C
- B
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