Download Complex numbers

Complex numbers
1
Complex numbers in algebraic,
trigonometric and exponential form
9000034808 (level 1):
Find the algebraic form of the complex number
z = 2 (cos π + i sin π).
9000034801 (level 1):
Given complex numbers z1 = 4 − i and z2 = 1 − 2i, find z1 − z2 .
3 − 3i
3+i
5 − 3i
−2
3−i
−3 − i
3+i
3−i
−1 − 3i
−1 + 3i
√
2 2
2
10
√
2
π
2
9000034805 (level 1):
Find the complex number z which satisfies 2z = 2 − 3i.
3
1− i
2
−3i
4 − 6i
5
π
9
−
π
2
3
− π
2
3
−1 + i
2
3
2
A
−1
1
i
1
−5
9000034807 (level 1):
Find the polar form of the complex number z = 2i.
2 cos
cos
π
π
+ i sin
2
2
π
π
+ i sin
2
2
3
π
2
9000035701 (level 1):
5
Find the algebraic form of the complex number graphed in the
complex plane.
4
Im
9000034806 (level 1):
Simplify i15 .
−i
3π
9000034810 (level 1):
π
π
and
Given complex numbers z1 = 2 cos + i sin
4
4
√
7π
7π
+ i sin
z2 = 2 cos
, find the angle in the polar form
4
4
z1
of the quotient .
z2
1 − 3i
9000034804 (level 1):
Find the absolute value of the complex number z = 3 − i.
√
2
π
9
3π
2
9000034803 (level 1):
Find the complex conjugate of z = 1 − 3i.
1 + 3i
2i
9000034809 (level 1):
π
π
and
Given complex numbers z1 = 2 cos + i sin
6
6
√
4π
4π
z2 = 3 cos
+ i sin
, find the angle in the polar form
3
3
of the product z2 z2 .
9000034802 (level 1):
Find the opposite number to the complex number z = 3 − i.
−3 + i
−2i
2
−4
−3
−2
−1
1
−1
√ π
π
2 cos + i sin
2
2
−2
−3
2 (cos 0 + i sin 0)
−4
1
−5
2
3
Re
4
5
5
z = −3 + 2i
z = 2 − 3i
4
Im
z = −3 − 2i
z = 2 + 3i
3
5
9000035702 (level 1):
4
Find the absolute value of the complex number graphed in the
complex plane.
3
Im
2
A
1
2
−5
−4
−3
−2
−1
1
−4
−3
−2
−1
1
2
Re
3
4
−4
√ √
π
π
3π
3π
z = 2 2 cos − i sin
+ i sin
z = 2 2 cos
4
4
4
4
−5
−3
√ π
π
z = 2 2 − cos + i sin
4
4
−4
√
−5
3
5
5
−3
−2
5
Re
4
−2
5
−1
A
3
−1
1
−5
2
√
5π
5π
+ i sin
z = 2 2 cos
4
4
4
9000035705 (level 1):
Find the absolute value of the complex number
z = (1 − 2i)(2 + i).
9000035703 (level 1):
Find the absolute value of the complex number graphed in the
complex plane.
Im
5
3
√
10
√
2 2
5
4
A
9000035706 (level 1):
Find the absolute value of the complex number z =
3
√
2 2
2
√
2 5
2
2 + 6i
.
1 − 2i
√
2 3
1
−5
−4
−3
−2
−1
1
2
3
Re
4
9000035707 (level 1):
Find the real part of the complex number 2 + 2i2 + i3 − i4 + 2i5 .
5
−1
√
2 5
√
2 3
4
−2
−1
√
6
1
5
−3
9000035708 (level 1):
Find the imaginary part of the complex number
1 + 2i12 + 3i19 − i22 + 2i105 .
−3
9000035704 (level 1):
−4
Find the polar form of the complex number graphed in the
complex plane.
−5
−1
−5
9000035710 (level 1):
2
1
4
Find the complex conjugate of
2 − 4i
2 + 4i
√
√
− 2+i 2
3+i
+ (i + 1)(2 + i).
2−i
−2 − 4i
−2 + 4i
13 + i
−7 − i
Find the imaginary part of the complex number z =
3
2
−
3
2
1
2
−
5π
5π
3 cos
+ i sin
4
4
2+i
.
1−i
1
2
4
−0.8
Find the absolute value of the complex number z =
1
5
√
7
5
2−i
.
2+i
√
− 3+i
√
5
5
−3 − i
−3 + i
Find the opposite number to the complex number z =
1
−1
−4
√
3−i
√
3+i
√
− 3−i
i + 3i(2 − i)2 − 4(1 − i)3
20 − 18i
3+i
9000031206 (level 2):
−i
√
7π
7π
3 2 cos
+ i sin
4
4
9000035801 (level 2):
Find the complex conjugate of the following complex number.
9000031205 (level 2):
Find the complex conjugate of z = i5 − 3i10 .
3−i
9000031210 (level 2):
√ π
π
Given complex numbers z1 = 2 3 cos + i sin
and
6
6
√
4π
4π
z1
+ i sin
, find the quotient .
z2 = 3 cos
3
3
z2
1
9000031204 (level 2):
√
√
− 2−i 2
π
π
3 cos + i sin
4
4
−4i
4i
2−i
Find the real part of the complex number z =
.
2+i
0.8
√
2−i 2
9000031209 (level 2):
√ π
π
Given complex numbers z1 = 2 2 cos + i sin
and
4
4
√
7π
7π
z2 = 2 cos
+ i sin
, find the product z1 z2 .
4
4
9000031203 (level 2):
0.6
√
√
3π
3π
3 2 cos
+ i sin
4
4
13 + 11i
9000031202 (level 2):
√
2+i 2
9000031208 (level 2):
Find the polar form of the complex number z = −3 + 3i.
9000031201 (level 2):
Given complex numbers z1 = 1 − 2i and z2 = 3 + 5i, find z1 z2 .
13 − i
√
20 − 24i
20 + 18i
−8 + 26i
9000035802 (level 2):
Solve the following equation for z ∈ C.
1+i
.
1−i
3z − 2z = 8 − 10i
i
8 − 2i
9000031207 (level 2):
Find the
of the complex number
algebraic form 3π
3π
z = 2 cos
+ i sin
.
4
4
1 + 5i
8 − 10i
2 + 2i
9000035803 (level 2):
Given the complex number z = −1 + 2i, find the imaginary
1
part of the complex number .
z
3
−
1
2
2
5
2
5
−
Evaluate i50 .
1
2
−1
9000035804 (level 2):
Find the algebraic form of the following complex number.
i
1
−i
i
1
9000037402 (level 2):
Evaluate i7 .
(2 + i) (3 + 2i)
−1
−i
4 + 7i
8 − 7i
8 + 7i
4 − 7i
9000037408 (level 2):
Find the polar form of the complex number
9000035805 (level 2):
Given the complex numbers
√
3π
2π
2π
3π
a = 2 cos
+ i sin
, b = 2 cos
+ i sin
,
3
3
4
4
cos
4π
4π
+ i sin
3
3
cos
3
3
+ i sin
2π
2π
find the product ab.
√
17π
17π
+ i sin
2 2 cos
12
12
√ π
π
2 2 cos + i sin
2
2
√
√
5π
5π
+ i sin
2 2 cos
7
7
5π
5π
+ i sin
2 2 cos
12
12
2
3
cos
5π
5π
+ i sin
6
6
2
3
2
3
cos
cos
cos
π
+ i sin
6
6
7π
7π
+ i sin
6
6
3
2− i
2
8 7
− i
5 5
1
.
cos 7π
+
i sin 7π
6
6
π
π
+ i sin
6
6
π
π
cos −
+ i sin −
6
6
9000037501 (level 2):
Find the absolute value of the following complex number.
√
3 + 2i
π
√
11
√
3
13
√
3 2
9000037502 (level 2):
Find the total sum of the complex numbers a, b and c.
√
√
a = 3 + 2i, b = 1 − 4i, c = 3 − 3i
9000035807 (level 2):
Given the complex numbers a = 2 − 3i, b = 1 + 2i, find the
a
quotient .
b
4 7
− − i
5 5
3
3
− i sin
2π
2π
5π
5π
cos −
+ i sin −
6
6
5π
5π
cos
+ i sin
6
6
a
.
b
2
11π
11π
cos
+ i sin
3
6
6
cos
Find the polar form of the complex number
9000035806 (level 2):
Given the complex numbers
5π
5π
11π
11π
a = 2 cos
+ i sin
, b = 3 cos
+ i sin
,
3
3
6
6
find the quotient
4π
4π
cos −
+ i sin −
3
3
9000037409 (level 2):
1
.
cos 2π
+
i sin 2π
3
3
4 7
+ i
3 3
9000037401 (level 2):
4
4+
√
√
3 + i( 2 − 7)
4+
√
√
2 + i( 3 − 3)
√
4+i 3
4+
√
√
3 − i( 2 − 7)
9000037503 (level 2):
Given complex numbers
√
√
a = 2 + 3i,
b=
√
2−
√
9000037508 (level 2):
Find the absolute value of the following complex number.
√ π
π
2 cos + i sin
3
3
3i,
find the product ab.
5
√
2
√
2+i 3
√
√
√
2−i 3
a = 5 + 2i,
b = 3 − i,
π
π
,
a = 3 cos + i sin
3
3
c=i
find the product abc.
1 − 17i
−1 − 17i
1 + 17i
√
2 3−i
11
√ π
π
3 2 cos + i sin
2
2
√ π
π
3 2 cos − i sin
2
2
√ π
π
−3 2 cos + i sin
2
2
10i
find the quotient
a
find the quotient .
b
b=2−i
7
1
+i
5
5
9000037507 (level 2):
Given complex numbers
√
a = 3 + 2i,
find the quotient
2π
2π
2 cos
+ i sin
3
3
√
−3 2
π
π
a = cos + i sin
,
3
3
a = 3 + 5i,
13
1
+i
3
3
b=
√
9000037510 (level 2):
Given complex numbers
9000037506 (level 2):
Given complex numbers
1
13
+i
5
5
2−2
find the product ab.
9000037505 (level 2):
Find the complex conjugate of the following complex number.
√
−2 3 − i
√
−2 3 + i
√
2
2+2
9000037509 (level 2):
Given complex numbers
9000037504 (level 2):
Given complex numbers
−1 + 17i
√
2
a
.
b
b=
√
2π
2π
2 cos
+ i sin
3
3
a
.
b
√ π
π 2
cos −
+ i sin −
2
3
3
√ π
π 2
cos −
− i sin −
2
3
3
7
1
+i
3
3
√ π
π 2
−
cos −
− i sin −
2
3
3
b=
√
√ π
π 2
−
cos −
+ i sin −
2
3
3
2−i
√
√
√
6−2
2 2+ 3
+i
3
3
√
√
√
6−2
2 2+ 3
−i
3
3
√
√
√
6−3
2 2+ 3
+i
2
2
√
√
√
6−2
2 2+ 3
−i
2
2
9000038601 (level 2):
Find the polar form of the following complex number.
√
1
3
− +i
2
2
5
cos
2π
2π
+ i sin
3
3
π
π
+ i sin
3
3
√
2π
2π
5 cos
+ i sin
3
3
3π
3π
+ i sin
2
2
√
2π
2π
5 cos
+ i sin
5
5
cos
π
π
+ i sin −
cos −
3
3
cos
9000038602 (level 2):
Find the polar form of the following complex number.
√
1
3
+i
2
2
cos
cos
π
π
+ i sin
3
3
3π
3π
+ i sin
2
2
cos
4π
4π
+ i sin
2 cos
3
3
√
3π
3π
5 cos
+ i sin
2
2
9000038606 (level 2):
Find the algebraic form of the following complex number.
cos
2π
2π
+ i sin
3
3
√
√
2
2
+i
2
2
√
√
2
2
−i
2
2
π
π
+ i sin
4
4
√
√
√
3
3
+i
2
2
√
3
3
−i
2
2
π
π
+ i sin −
cos −
3
3
9000038607 (level 2):
Find the algebraic form of the following complex number.
π
π
3 cos + i sin
2
2
9000038603 (level 2):
Find the polar form of the following complex number.
√
√
2
6
+i
2
2
√ π
π
2 cos + i sin
3
3
√ π
π
5 cos + i sin
3
3
√
3i
2π
2π
2 cos
+ i sin
3
3
−3i
8(cos π + i sin π)
−8
9000038604 (level 2):
Find the polar form of the following complex number.
√
√
3
3
√ + i√
2
2
√
√ π
π
2 cos + i sin
3
3
√
3 − 3i
9000038608 (level 2):
Find the algebraic form of the following complex number.
3π
3π
+ i sin
2 cos
2
2
√ π
π
3 cos + i sin
4
4
3 + 3i
8 + 8i
8
8 − 8i
9000038609 (level 2):
Find the algebraic form of the following complex number.
3π
3π
5 cos
+ i sin
4
4
3π
3π
3 cos
+ i sin
4
4
√
√
5 2
5 2
−
+i
2
2
2π
2π
2 cos
+ i sin
3
3
√
√
5 2
5 2
−i
2
2
5
5
−i
2
2
9000038605 (level 2):
Find the polar form of the following complex number.
√
√
5
15
+i
−
2
2
9000038610 (level 2):
6
5
5
+i
2
2
Find the algebraic form of the following complex number.
3π
3π
2 cos
+ i sin
4
4
√
√
√
− 2+i 2
√
1 1
− i
3 3
3 6
z=− + i
5 5
z =1+i
1 3
z=− − i
5 5
9000039109 (level 2):
Assuming z ∈ C, solve the following equation.
2 − 2i
2 + 2i
2−i 2
z=
2z − iz = 1 − i
9000039101 (level 2):
Find the polar form of the complex number
√
3π
3π
+ i sin
2 cos
4
4
√ π
π
2 cos + i sin
4
4
√
i14 − 1
.
i9 + 1
z =1−i
√
1
3
−
i
2
2
3 4
− − i
5 5
√
1
3
z=− −
i
2
2
−i
z=−
x = 13, y = 8
√
3 1
z=
+ i
2
2
1 1
z=− + i
3 3
√
3 1
+ i
2
2
x = 8, y = 13
−2i
x = 2, y = 3
2
9000039104 (level 2):
Assuming x ∈ R, y ∈ R, solve the following equation.
(3 − 2i)x + (5 − 7i)y = 1 + 3i
x = −1, y = 2
√
1
3
z=− +
i
2
2
9000070110 (level 2):
5
5
Given z1 = 4 cos π + i sin π and
3
3
1
1
z1
z2 = 2 cos π + i sin π , evaluate .
6
6
z2
(2 + 5i)x + (1 − i)y = 13i + 8
x = 2, y = −1
1 1
− i
3 3
9000039110 (level 2):
Assuming z ∈ C, solve the following equation.
√
√ 1+i 3 z =1−i 3
√
7π
7π
2 cos
+ i sin
4
4
9000039103 (level 2):
Assuming x ∈ R, y ∈ R, solve the following equation.
x = 3, y = 2
z=
5π
5π
+ i sin
2 cos
4
4
9000039102 (level 2):
Identify a complex number with absolute value different from
1.
1+i
z =1+i
4i
i
1
− i
2
Moivre’s theorem
9000035709 (level 1):
Simplify (1 − i)−3 .
x = −2, y = −1
1 1
− + i
4 4
x = −1, y = −2
1 + 3i
−2 − 2i
1 1
+ i
2 2
9000035808 (level 2):
Evaluate (1 − i)10 .
9000039108 (level 2):
Assuming z ∈ C, solve the following equation.
−32i
2z − i z = 1 − i
7
32
32i
−32
Evaluate the following complex number.
π
π 3
cos + i sin
4
4
9000035809 (level 2):
Given the complex number z = −1 + i, find the angle in the
polar form of the number z 6 .
π
2
3π
2
3π
4
√
2
2
−
+i
2
2
√
9 9i 3
+
2
2
3
2
−
3
2
√
1
3
− −i
2
2
9000037404
2):
√ (level
π
π
Given z = 2 cos − i sin
, find z 2 .
3
3
√
−1 − i 3
√
1+i 3
√
−2 − i 2
7 − 7i
√
3 1
−
− i
2
2
2+i 2
1+i
−i
1 + 2i
1
−i
i
11
i
−i
1−i
1−i
1
i
i13
1
cos
1+i
−1
9000070105 (level 2):
Evaluate the following complex number.
9000037410 (level 2):
3
Evaluate (1 − i) .
2 + 2i
√
1
3
− +i
2
2
3 1
−
+ i
2
2
(sin 2π + i cos 2π)
cos
−2 − 2i
√
9000070104 (level 2):
Evaluate the following complex number.
9000037407
2):
(level
π
π 13
Evaluate cos + i sin
.
2
2
i
√
2
2
+i
2
2
9
9000037406
2):
(level
π 40
π
Evaluate cos + i sin
.
4
4
1
√
(cos π + i sin π)
−1
1
√
2
2
−i
2
2
9000070103 (level 2):
Evaluate the following complex number.
√
9000037405 (level 2):
7
Evaluate (1 + i) .
8 − 8i
√
√
2
2
−
−i
2
2
9000070102 (level 2):
Evaluate the following complex number.
π
π 10
cos + i sin
3
3
9000037403
2):
√ (level
π
π
Given z = 3 cos + i sin
, find z 4 .
3
3
√
9 9i 3
− −
2
2
√
√
7π
4
π
π
+ i sin
2
2
cos
π
π
− i sin
2
2
3π
3π
+ i sin
2
2
i
9000070106 (level 2):
Evaluate the following complex number.
9000070101 (level 2):
(1 − i)8
8
sin
π
π
+ i cos
2
2
−16i
16
−16
16i
9000035602 (level 2):
Find the values of the parameter m ∈ C which guarantee that
the following quadratic equation has a double solution.
9000070107 (level 2):
Evaluate the following complex number.
1
π
+ cos + i cos 2π
2
3
−4 + 4i
−4 − 4i
4 − 4i
mx2 − 2x − 1 + i = 0
5
1 1
m=− − i
2 2
4 + 4i
m = −1
m = −1 + i
1 1
m=− + i
2 2
9000070108 (level 2):
Evaluate the following complex number.
9000035603 (level 2):
Find the solution set of the following equation.
√ !6
3
1
+
i
2
2
−1
1
4x2 + 9 = 0
−i
i
3 3
− i; i
2 2
2 2
− i; i
3 3
9 9
− i; i
4 4
3 3
− ;
2 2
9000070109 (level 2):
Evaluate the following complex number.
√
−8i
3
3−i
x2 − 2ix + 3 = 0
−8
8
9000035604 (level 2):
Solve the following equation.
3
8i
{−i; 3i}
Quadratic equations with complex
roots
3
−∞; −
4
∪
3
;∞
4
3
;∞
4
9000035605 (level 2):
7
7
The number cos π + i sin π is a solution of a quadratic
6
6
equation with real valued coefficients. Find the second
solution.
5
5
cos π + i sin π
6
6
3 3
− ;
4 4
3 3
− ;
4 4
{1 − 2i; 1 + 2i}
{−3i; i}
9000035601 (level 2):
Find the values of the parameter t ∈ R which guarantee that
the following quadratic equation has solutions with nonzero
imaginary part.
tx2 − 3x + 4t = 0
{−4i; 4i}
7
7
cos π + i sin π
6
6
1
1
cos π + i sin π
6
6
cos
11
11
π + i sin π
6
6
9000035606 (level 2):
Find the quadratic equation with
√ real valued coefficients and
one of the solutions x1 = −1 + i 3.
3 3
R\ − ;
4 4
9
x2 + 2x + 4 = 0
x2 − 2x + 4 = 0
x2 − 2x − 2 = 0
x2 + 2x + 2 = 0
Find the value of the parameter a which guarantees that the
quadratic equation
x2 + 2x − 4 = 0
x2 + 2ax + a = 0
has a pair of complex conjugate solutions with a nonzero
imaginary part.
9000035607 (level 2):
Identify the quadratic equation with solutions x1 = 2i,
x2 = −i.
x2 − ix + 2 = 0
x2 + ix + 2 = 0
x2 + ix − 2 = 0
x2 − ix − 2 = 0
x2 = 1 − 4i, q = 7 − 4i
x2 = 1, q = −1 − 2i
x2 + 4 = 0
√
x2 = 3 + i 2, p = −2i
x2 − 4i = 0
2
5
x2 − 4 = 0
x2 + 4i = 0
3x2 + 4x + 2 = 0
3x2 − 4x + 2 = 0
9000064503 (level 2):
Find the values of the real coefficients a, b and c such that the
quadratic equation
ax2 + bx + c = 0
√
x2 = −3 − i 2, p = −2i 2
has zeros x1,2 = ±i
9000039105 (level 2):
Find the quadratic equation with real coefficients such that
one of the solutions is the complex number x1 = 1 + 2i.
x2 − 2x + 3 = 0
−
x2 + 4x + 6 = 0
√
x2 − 2x + 5 = 0
2 4
+ i
5 5
x2 − 4x + 6 = 0
√
with a parameter p ∈ C has a solution x1 = 3 − i 2. Find the
second solution x2 and the parameter p.
√
x2 = −3 − i 2, p = 6
2
5
9000064502 (level 2):
√
Find the quadratic equation with the solution x1,2 = 2 ± i 2.
9000035609 (level 2):
The equation
x2 + px − 11 = 0
√
x2 = 3 + i 2, p = 6
such an a does not exist
9000064501 (level 2):
Find the quadratic equation with the solution x1,2 = ±2i.
x2 = −1, q = 1 + 2i
√
√
x2 = −3 − i 2, p = 2i 2
a ∈ (−∞; 0) ∪ (1; ∞)
2
with a parameter q ∈ C has a solution x1 = 1 + 2i. Find the
second solution x2 and the parameter q.
x2 = −1 − 4i, q = 9 − 6i
a ∈ [0; 1]
9000039107 (level 2):
Find the sum of the reciprocal values of the solutions of
5x2 − 2x + 1 = 0.
9000035608 (level 2):
The equation
x2 − 2ix + q = 0
x2 = −1, q = −1 − 2i
a ∈ (0; 1)
x2 + 2x + 5 = 0
√
5
.
3
a = 9, b = 0, c = 5
a = 5, b = 0, c = 9
a = 9, b = 0, c = −5
a = 5, b = 0, c = −9
9000064504 (level 2):
Find the values of the real coefficients a, b and c such that the
quadratic equation
x2 + 2x − 3 = 0
ax2 + bx + c = 0
9000039106 (level 2):
10
i
has solutions x1,2 = 1 ± .
2
Solve the following quadratic equation in the complex plane.
2x2 + x + 1 = 0
a = 4, b = −8, c = 5
a = 1, b = −4, c = 5
a = 4, b = 8, c = 5
a = 1, b = 4, c = 5
x1,2
9000064505 (level 2):
Find the factorization of the following quadratic polynomial in
the set of polynomial with complex valued coefficients.
x1,2
2x2 + 32
2(x + 4i)(x − 4i)
2(x − 4i)2
√
−1 ± i 7
=
4
x1,2
√
−1 ± i 7
=
2
x1,2
√
1±i 7
=
4
√
1±i 7
=
2
9000069901 (level 2):
Solve the following quadratic equation in the complex plane.
x2 + 4x + 5 = 0
(x + 4i)(x − 4i)
2(x + 4i)2
9000064506 (level 2):
Find the factorization of the following quadratic polynomial in
the set of polynomial with complex valued coefficients.
x1 = −2 + i, x2 = −2 − i
x = −2
x1 = 2 + i, x2 = 2 − i
x1 = −3, x2 = −1
9000069902 (level 2):
Solve the following quadratic equation in the complex plane.
2
2x + 4x + 5
3x2 + 2x + 2 = 0
√ !
√ !
6
6
2 x+1+
i x+1−
i
2
2
√
√
1
1
5
5
x1 = − +
i, x2 = − −
i
3
3
3
3
√ !
√ !
6
6
i x−1−
i
2 x−1+
2
2
x1 = −
√
√
1
5
1
5
x1 = +
, x2 = +
3
3
3
3
√ !
√ !
6
6
x+1−
i x+1+
i
2
2
√ !
√ !
6
6
x−1−
i x−1+
i
2
2
x1 =
x2 + 2x + 2
4x2 + 12 = 0
x1,2 = ±3
x1,2 = ±3i
√
√
1
5
1
5
+
i, x2 = −
i
3
3
3
3
9000069903 (level 2):
Find the factorization of the quadratic polynomial
9000064507 (level 2):
Solve the following quadratic equation in the complex plane.
√
x1,2 = ±i 3
1
3
in the set of polynomials with complex valued coefficients.
√
x1,2 = ± 3
9000064508 (level 2):
11
(x + 1 + i)(x + 1 − i)
(x − 1 + i)(x − 1 − i)
(x − i)(x + i)
(x − 1 + i)(x + 1 − i)
9000069904 (level 2):
Find the factorization of the quadratic polynomial
9000069909 (level 2):
One of the solutions of the quadratic equation
x2 + 2x + 5
9x2 − 6x + p = 0
in the set of polynomials with complex valued coefficients.
(x + 1 − 2i)(x + 1 + 2i)
(x − 1 − 2i)(x − 1 + 2i)
(x + 1 − 2i)(x − 1 + 2i)
(x − 1 − 2i)(x + 1 + 2i)
with a real parameter p is
1
+ i.
3
Find the value of p.
9000069905 (level 2):
Find the sum of all the complex solutions of the following
quadratic equation.
4
5
−
24
i
5
4
10
−1
3
9000069910 (level 2):
Find the values of the parameter p ∈ R which guarantee that
the equation
x2 + 2px + 16 = 0
5x2 + 4x + 8 = 0
−
−10
10
0
has solution with a nonzero imaginary part.
p ∈ (−4; 4)
9000069906 (level 2):
Find the sum of all the complex solutions of the following
quadratic equation.
4
p ∈ (−∞; 4)
p ∈ (4; ∞)
p ∈ {}
Binomial equations
2
x − 8x + 17 = 0
8
4
9000034301 (level 2):
Find the solution set of the following equation in the set of
complex numbers.
x3 − 1 = 0
0
4i
9000069907 (level 2):
Find the quadratic equation with real valued coefficients and
one of the solutions x1 = −5 + i.
x2 + 10x + 26 = 0
x2 − 10x + 26 = 0
x2 − 10x − 24 = 0
2
x + 10x + 24 = 0
9000069908 (level 2):
One of the solutions of the quadratic equation
√
√
1
3
3
1
{1; − + i
; − −i
}
2
2
2
2
√
√
{1; −1 + i 3; −1 − i 3}
√
1
3
{1; − + i
}
2
2
√
1
3
{1; − − i
}
2
2
9000034302 (level 2):
Find the solution set of the following equation in the set of
complex numbers.
x3 + 8 = 0
2x2 + px + 5 = 0
with a real parameter p is
x = −1 +
√
√
√
{−2; 1 + i 3; 1 − i 3}
6
i.
2
Find the value of p.
4
−4
8
√
√
{−2; −1 + i 3; −1 − i 3}
−8
12
√
√
3 1
3
1
{−2; + i
; −i
}
2
2 2
2
x1,2 = 2(1 ± i), x3,4 = −2(1 ± i)
√
√
1
3
3
1
{−2; − + i
; − −i
}
2
2
2
2
x1,2 =
x6 − 64 = 0
√
3 1
3 1
− i; −
− i}
2
2
2
2
√
√
√
x1,2 = ±2, x3,4 = 1 ± i 3, x5,6 = −1 ± i 3
√
3 1
3 1
+ i; −
− i}
{−1; −
2
2
2
2
√
{−1;
{i; −
√
2
2
(1 ± i), x3,4 = −
(1 ± i)
2
2
9000034306 (level 2):
Solve the following equation in the set of complex numbers.
9000034303 (level 2):
Find the solution set of the following equation in the set of
complex numbers.
x3 + i = 0
{i;
√
x1,2 = ±2, x3,4
√
3 1
3 1
− i; −
− i}
2
2
2
2
√
√
√
x1,2 = ±4, x3,4 = 1 ± i 3, x5,6 = −1 ± i 3
√
√
√
x1,2 = ±8, x3,4 = 2 ± 2i 3, x5,6 = −2 ± 2i 3
√
3 1
3 1
+ i; −
− i}
2
2
2
2
9000034307 (level 2):
Let x be any of the complex solutions of the following
equation.
√
x5 + 3 − i = 0
9000034304 (level 2):
Find the solution set of the following equation in the set of
complex numbers.
x4 − 1 = 0
Find the absolute value of x.
√
5
{1; −1; i; −i}
{1 − i; −1 − i}
are
√
10
2
x4 + 16 = 0
√
6
5
5
2 cos π + i sin π ,
12
12
√
13
13
6
x2 = 2 cos π + i sin π .
12
12
x1 =
9000034305 (level 2):
Solve the following equation in the set of complex numbers.
2(1 ± i), x3,4
4
x3 + 1 + i = 0
{1 + i; 1 − i; −1 + i; −1 − i}
√
√
5
2
2
9000034308 (level 2):
Two solutions of the equation
{1 + i; −1 + i}
x1,2 =
√
√
1
3
3
1
= ±i
, x5,6 = − ± i
2
2
2
2
Find the third solution.
√
= − 2(1 ± i)
x1,2 = 1 ± i, x3,4 = −1 ± i
13
x3 =
√
6
21
21
2 cos π + i sin π
12
12
x3 =
√
6
9
9
2 cos π + i sin π
12
12
x3 =
√
6
17
17
2 cos π + i sin π
12
12
x3 =
√
6
19
19
2 cos π + i sin π
12
12
9000034309 (level 2):
Find the angle ϕ such that the angles in the polar form of any
two solutions of the equation
√
x5 − 1 + i 3 = 0
differ by a integer multiple of ϕ.
ϕ=
2
π
5
ϕ=
3
π
5
ϕ=
4
π
5
ϕ=π
9000035810 (level 2):
Given
the complex number z = −2 + 2i, find all the roots of
√
3
z.
√
π
π
6
w0 = 8 cos + i sin
4
4
√
11π
11π
6
+ i sin
w1 = 8 cos
12
12
√
19π
19π
6
w2 = 8 cos
+ i sin
12
12
π
π
w0 = 2 cos + i sin
4
4
11π
11π
+ i sin
w1 = 2 cos
12
12
19π
19π
w2 = 2 cos
+ i sin
12
12
√
3
−2 +
√
3
2
π
π
w0 = 2 cos + i sin
3
3
w1 = 2 (cos π + i sin π)
5π
5π
w2 = 2 cos
+ i sin
3
3
14