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## Numbers and Algebra

### Advanced Algebra - Complex Numbers and the Complex Plane

#### Q.01

'Find the sum and product of the following complex numbers. Also, discuss the square root of negative numbers.\n(1) The sum and product of 5-2i and its conjugate complex number.\n(2) The sum and product of sqrt(2)i and its conjugate complex number.\nPlease point out any specific things to note.'

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'If an n-th degree equation with rational coefficients has p+q√r as a solution, explain another solution and demonstrate its properties.'

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'Find the complex number z such that squaring z equals 3+4i.'

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'Unable to determine if the overall average score of high school students differs from the prefectural average score'

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'When dividing x^{2025} by x^{2}+1, let the quotient be Q(x) and the remainder be a x + b (a, b are real numbers). Then, x^{2025} = (x^{2}+1) Q(x) + a x + b. Substituting x=i on both sides, we get i^{2025} = a i + b. Here, i^{2025} = (i^{2})^{1012} * i = (-1)^{1012} * i = i. Therefore, i = a i + b. Since a and b are real numbers, a=1, b=0. Thus, the required remainder is x.'

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'Please provide the page numbers for terms related to complex numbers.'

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'Math II 59 Solving this gives \\( \\quad x=-2,-\\frac{1+2 i}{1+i}\\left(=-\\frac{3+i}{2}\\right) \\) From [1], [2], the required real solutions and the value of b are \x=0, \\quad b=0 ; x=-2, \\quad b=2\'

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'When a < 1 / 4, M = -3a + b + 1, when 1 / 4 ≤ a < 1, M = 2a√a + b, when a ≥ 1, M = 3a + b - 1'

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'Integers a, b satisfy the equation (a+bi)^{3}=-16+16i. Here, i is the imaginary unit.\n(2) Find the value of i/(a+bi)- (1+5i)/4.'

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'Find the conditions for the function to have extreme values and the range of values for the function to not have extreme values'

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'Example 19 | Discriminant of 2nd degree equations (2)'

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'There are 3 coins of 100 yen each and 3 coins of 50 yen each, totaling 6 coins, and a die. When these 6 coins and 1 die are thrown simultaneously, the prize is obtained by multiplying the absolute value of the product of the coin total amount of the coins showing heads and the die result n minus 2. For example, if all 6 coins show heads and the die shows 6, the total amount of coins showing heads is 450 yen multiplied by 4, resulting in 1800 yen as the prize.'

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'Find the value of the given equation when \ x=1+\\sqrt{2} i \: \\[ P(x)=x^{4}-4 x^{3}+2 x^{2}+6 x-7 \\]'

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'Please calculate the following expression:\n\n\\[\n\egin{array}{l}\n\\alpha-1) + (\eta-1) + (\\gamma-1) = (\\alpha+\eta+\\gamma)-3 = 0-3=-3 \n(\\alpha-1)(\eta-1)+\eta-1)(\\gamma-1)+(\\gamma-1)(\\alpha-1)\\ = (\\alpha \eta+\eta \\gamma+\\gamma \\alpha)-2(\\alpha+\eta+\\gamma)+3 \n=-4-2 \\cdot 0 +3=-1 \n\\end{array}\n\\]\n\nAlso, \\( x^{3}-4 x+2=(x-\\alpha)(x-\eta)(x-\\gamma) \\) holds. Substituting both sides by \ x=1 \ we get \\( 1-4+2 = (1-\\alpha)(1-\eta)(1-\\gamma) \\) therefore \\( (\\alpha-1)(\eta-1)(\\gamma-1)=1 \\) thus the required cubic equation is: \ x^{3} + 3 x^{2} - x - 1 = 0 \'

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'Express the following calculations in the form of a+bi.\n(1) 1/i, 1/i^2, 1/i^3\n(2) \\\frac{5i}{3+i}\\n(3) \\\frac{9+2i}{1-2i}\\n(4) \\\frac{2-i}{3+i}-\\frac{5+10i}{1-3i}\'

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'For a complex number z, find all complex numbers z such that z^2 = i.'

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'Define the sequence {a_n} as {a_1=3, a_{n+1}=(a_n^2-1)/(n+1) (n=1,2,3, ...)}'

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'Define a sequence {a_n} as follows. Let a_1 = 2. For any natural number n, the x-coordinate of the intersection point of the line passing through (0,1), (a_n,0) and the line y = x is denoted as a_{n+1}.'

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'(1) Find the values of the real numbers \ x, y \ that satisfy the equation \\( (3+i) x+(1-i) y=5+3 i \\).'

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'In mathematics, real numbers x, y, z satisfy the system of equations {x+y+z=-1, x^2+y^2+z^2=7, x^3+y^3+z^3=-1} 1). At this time, xy+yz+zx=⧁, xyz=1□. Therefore, the system of equations (1) has ⧁ sets of solutions, among which satisfy x<y<z is (x, y, z)=1□.'

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'Explain the definition and basic properties of complex numbers.'

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'Basic 34: Conjugate complex numbers and their sum/product'

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'Find the general term of the sequence {an} determined by the following conditions.'

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'Determine the range of values for the constant k that satisfy the following conditions: (1) The function f(x)=x^{3}+6 k x^{2}+24 x+32 has critical points. (2) The function f(x)=2 x^{3}+k x^{2}+k x+1 does not have critical points.'

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'Find the sum and product of each of the following numbers and its conjugate complex number.'

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'Find the complex numbers representing the point that divides the line segment connecting two points A(α) and B(β) internally in the ratio m:n and externally.'

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'Let a be a positive real number, and let w=a\\(\\left(\\cos \\frac{\\pi}{36}+i \\sin \\frac{\\pi}{36}\\right)\\). Define the complex number sequence \\\left\\{z_{n}\\right\\}\ as \\(z_{1}=w, z_{n+1}=z_{n} w^{2 n+1}(n=1,2,\andotsandots)\\)'

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'When a complex number z satisfies z+1/z=√2, find the value of z^20+1/z^20.'

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'(1) In the complex plane, let α be a nonreal complex number and β be a positive real number. Let C be the locus of complex numbers z satisfying the relation α*conj(z) + conj(α)*z = |z|^2. Show that C is a circle passing through the origin.'

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Join our Discord#### Q.35

"Let's think about the primitive 6th roots of 1. For example, in the case of n=6, we seek the solutions of z^6 = 1 that become primitive 6th roots. The solutions to z^6 = 1 are the six values z_0, z_1, ..., z_5 as given in the answer to basic example 105 on page 528."

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'Consider the sequence of complex numbers \ \\left\\{z_{n}\\right\\} \.'

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'Translate the given question to the following languages.'

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'The transformation from z to w represented by the following equation is called a first-order fractional transformation (or Möbius transformation).'

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"In our daily lives, alternating current electricity is widely used. In the calculation of alternating current circuits, complex numbers and differential equations are sometimes used, so let's take a look at some of that. Please note that the following content includes university-level topics, so a rough understanding is sufficient."

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'For the sequence {a_{n}}, where a_{1}=3 and a_{n+1}=\\frac{3 a_{n}-4}{a_{n}-1}.'

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'There are 4 points A(2+4i), B(z), C(conjugate of z), D(2z) on the complex plane. Find the value of the complex number z when the quadrilateral ABCD is a parallelogram.'

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'On the complex plane, for the 3 points A(1+i), B(3+4i), C, when the triangle ABC is an equilateral triangle, find the complex number z representing point C.'

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'Let z be a complex number. Prove that |z|=1 when z+1/z is a real number. Also, find all complex numbers z that make z+1/z a natural number.'

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'When the two quadratic equations $x^{2}-(m+1) x-m^{2}=0$ and $x^{2}-2 m x-m=0$ have only one common solution, the value of $m$ is $\\qquad$, and the common solution is $x=$ $\\square$.'

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'There are 12 lottery tickets, among which there are n winning tickets (0 ≤ n ≤ 12). When ^3161 tickets are drawn from these tickets, winning tickets score 3 points and losing tickets score -1 point. Find the range of values of n for which the expected score is greater than or equal to 1.'

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'Determine the value of the constant c such that the minimum value of the function f(x) = -x^2 + 4x + c is -50 in the range (-4 ≤ x ≤ 4).'

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'For a complex number \\\alpha=a+bi\ and its complex conjugate \\\overline{\\alpha}=a-bi\, the following properties hold.'

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'Find the values of real numbers x, y that satisfy the following equations:\n(1) x + 2i = 9 - yi\n(2) (2x - 1) + (y + 3)i = 0'

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'The sequence {an} is defined by a1=2, an+1=3an-n²+2n. By considering a quadratic function g(n) to make the sequence {an}-g(n) a geometric sequence with a common ratio of 3, express an in terms of n.'

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'Find the range of values for c such that the equation x^3-6x+c=0 has two distinct positive solutions and one negative solution.'

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'Let i be the imaginary unit, x = \\sqrt{3} + \\sqrt{7}i. Let y be the conjugate of x. Find the following values.'

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'Let the sum of the first to nth terms of a geometric progression with a positive common ratio be Sn. If S2n=2 and S4n=164, find the value of Sn.'

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'The sequence {an} is defined as a1=2 and the recurrence formula an+1=2-an/(2an-1).'

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'Let i be the imaginary unit, and let x=√3+√7i. Let y be the complex conjugate of x. Find the following values.'

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'Perform the following calculations: (1) Decompose three fractional expressions into partial fractions. (2) If (numerator degree) < (denominator degree), rearrange the expression so that the numerator degree is lower than the denominator degree before performing the calculation for smoothness.'

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'Find the two solutions because the ratio of the two solutions is 3:2.'

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'Find the range of real numbers a for which at least one of the equations has complex roots'

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'There are exactly two complex numbers z=x+yi (x, y are real numbers) such that squaring z equals i. Find these z.'

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'Chapter 2\nComplex Numbers and Equations\nFrom D_{3}<0 to a(a+2)<0, therefore -2<a<0. The range of values for a is obtained by combining (1), (2), and (3), so -2<a<5/3.'

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'Regarding logarithmic scales (1), (2), when focusing on the opposite scales a and c, as well as b and d, the relationship a/c = b/d always holds true, showing that the ratio of opposite scales is constant. Also, looking at logarithmic scale (3), the relationship cf = de always holds true.'

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'Find the conditions under which three distinct tangents can be drawn from the point \\((a, b)\\) to the curve \y=x^{3}-x\, and illustrate the range of points \\((a, b)\\) that satisfy this condition. [Kansai University]'

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"The recurrence formula a_{n+1}=2a_{n}-n from the previous page does not seem to fit into any of the top 3 patterns as it is. Even if we substitute a_{n+1} and a_{n} with α and consider the characteristic equation α=2α-n, we still end up with α=n, unable to proceed as in Basic Example 30. Therefore, let's take a closer look at the answer on the left."

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'Find the general term of the sequence \ \\left\\{a_{n}\\right\\} \ determined by the following conditions.'

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'Please find the solution for x = (-(-√2) ± √((-√2)^2 - 4 * 1 * (-4))) / (2 * 1).'

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'What is the equation of the parabola obtained by translating the parabola y=1/2 x^2 in such a way that it passes through the point (1,5) and its vertex lies on the line y=-x+2?'

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'When a point z moves along the circle with center at the origin O and radius 2, what kind of shape does the point w = \\frac{2z-i}{z+i} trace?'

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'Explain how to handle the position vector corresponding to the addition of complex numbers α + β on the complex plane.'

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'Find the complex number representing the point obtained by rotating the point 2+2i around the point i for the following angles.'

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'I would like to summarize what kind of geometric movement complex number calculations represent.'

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'For the points A(6-3i), B(1+7i), C(-2+i) on the complex plane, find the following complex numbers:\n\n1. The point D that divides the line segment AB in the ratio 2:3.\n2. The point E that divides the line segment BC externally in the ratio 2:3.\n3. The midpoint F of the line segment CA.\n4. The centroid G of triangle DEF.'

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'For the points A(-1+i) and B(3+4i), find (2) the midpoint M of the line segment AB.'

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'Find the sum and difference of the complex numbers α=3+4i and β=1+2i, and plot the points they represent on the complex plane.'

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'When a complex number z satisfies |z-1|≤|z-4|≤2|z-1|, illustrate the range of movement of point z on the complex plane.'

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'When (*) has two imaginary solutions that are conjugates of each other, denoted as $z=\\gamma, \\overline{\\gamma}(\\gamma$ is imaginary)$,$ we can determine from the relationship between the solutions and coefficients that $\\gamma+\\overline{\\gamma}=-a$ and $\\gamma \\overline{\\gamma}=b$, which implies $a=-(\\gamma+\\overline{\\gamma}), \\quad b=|\\gamma|^{2}$. In this case, $D=a^{2}-4b=(\\gamma-\\overline{\\gamma})^{2}$, and since $\\gamma-\\overline{\\gamma}$ is purely imaginary, we always have $D=(\\gamma-\\overline{\\gamma})^{2}<0$. By combining $|a| \\leqq 1,|b| \\leqq 1$, we get $|\\gamma+\\overline{\\gamma}| \\leqq 1,|\\gamma|^{2} \\leqq 1$, or equivalently, $\\left.\\left\\lvert\\,(\\text{Real part of }\\gamma)\\right\\rvert\\leqq \\frac{1}{2}\\right|\\gamma\\right.\\right\\rvert\\leqq 1$. Therefore, the range of possible values for the imaginary solution z of (*) is $\\left.\\left\\lvert\\,(\\text{Real part of }z)\\right\\rvert\\leqq \\frac{1}{2}\\right|z\\right.\\right\\rvert\\leqq 1$.'

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'When a point z on the complex plane moves along the circumference excluding the point -1 from the unit circle, what kind of shape does the point w represented by w=\\frac{2z+1}{z+1} draw?'

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"In this chapter, we learned about addition, subtraction, multiplication, and division of complex numbers. Next, let's represent complex numbers as points on the coordinate plane and think about the geometric meaning of complex numbers. Specifically, we will consider the geometric interpretation of the sum, difference, absolute value, and conjugate of complex numbers, as well as learn about polar form to deal with the product and quotient of complex numbers, and explore various figures on the complex plane."

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'From the result of (1), the condition for the existence of a complex number z satisfying (2) is α≤-2, -1≤α. Thus, |α|≤2 implies α=-2, -1≤α≤2.'

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'Here, all characters are considered complex numbers. The transformation from z represented in the equation to w is called a first-order fraction conversion.'

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'Express the following complex numbers in polar form. The argument 𝜃 should satisfy 0 ≤ 𝜃 < 2π.'

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'For a complex number z different from -1, when the complex number w is defined by w= z/z+1, find the shape traced by w as z moves along the imaginary axis. Also, find the shape traced by w as z moves on the circle |z-1|=1 in the complex plane.'

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'When the points O(0), A(3+4i), B(1+2i) are not collinear, and the result of addition is denoted as C(α+β), what will the quadrilateral OACB look like?'

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'In the complex plane, if point P(z) and point Q(w) are symmetric with respect to the line passing through the origin O and point A(α) (α ≠ 0), then express w in terms of α and z.'

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'(2) For a complex number $z$ that is not a real number, prove that $\\alpha \\overline{z}-\\overline{\\alpha} z$ is a purely imaginary number.'

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'Let k be a natural number greater than 2, and let z = cos(2π/k) + i sin(2π/k). Here, i is the imaginary unit. (1) Let m, n be integers. Prove that m-n is a multiple of k if and only if z^m = z^n.'

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'Prove that for any complex number z, the expression z \ar{z}+α \ar{z}+\ar{α} z is a real number.'

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'Consider a sequence of complex numbers defined by the following recurrence relation. Here, i represents the imaginary unit.'

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'Express the conjugate of a complex number z in polar form.'

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'For the function f(x)=(x+1)/(x^2+2x+a), find the range of values for the constant a that satisfy the following conditions:'

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'Let \ z \ be a non-zero complex number. 85 (1) If we represent \ z \ as having an absolute value of \ r \ and an argument of \\( \\theta(0 \\leqq \\theta<2 \\pi) \\), determine the values of \ r \ and \ \\theta \ such that \ \\frac{z}{4}+\\frac{4}{z} \ is a real number.'

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"Problem 12 concerns the validity of the Newton's method for finding the real solutions (approximate values) of the equation f(x)=0 when f(x) is a convex function."

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'(2) On the complex plane, for points \\alpha, \eta, show the following points: (a) \\alpha+\eta (b) \\alpha-\eta (c) 2\\alpha+\eta (d) -(2\\alpha+\eta)'

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'Express the complex number z=a+bi in polar form.'

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'Complex number multiplication and rotation (2)\n(1) For two points z=3+i, w=2-i, find the complex number representing the point obtained by rotating z around the center of w by π/6.'

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'Let a, c be real numbers, and β be a complex number. When a ≠ 0 and |β|² > ac, the equation az¯z+β¯z+βz¯+c=0 represents a circle with center at -β/a and radius √(|β|²-ac)/|α|, and when a = 0 and β ≠ 0, the equation β¯z+βz¯+c=0 represents a line. For more details, refer to solution page p.109. In the exercise, let k be a real number and α=-1+i. A point w moves on the complex plane satisfying the equation w¯α-¯wα+ki=0. Find the maximum value of k when the trajectory of point w has at least one point in common with a circle of radius 1 centered at 1+i.'

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'(1) Let a complex number z be given in the form z=r(cosθ+isinθ), where r is a positive real number and θ is a real number. Prove the necessary and sufficient conditions for both sequences {xn} and {yn} to converge to 0.'

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'Assuming that the equation (1) has a complex solution with an absolute value of 1, z=\\cos \\theta+i \\sin \\theta, then from (2) we have'

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'In the complex plane, let a complex number $\\alpha=a+bi$ represent a point $A(\\alpha)$. Now, on this plane, let the position vector of point $A$ with respect to the origin $O$ be $\\vec{p}$. Therefore, the complex number $\\alpha$ corresponds to the position vector $\\vec{p}$. For two complex numbers $\\alpha, \eta$, let $A(\\alpha), B(\eta)$, $\\overrightarrow{OA}=\\vec{p}$, $\\overrightarrow{OB}=\\vec{q}$, then the equality of complex numbers $\\alpha, \eta$ can be seen as the equality of position vectors $\\vec{p}, \\vec{q}$.'

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'For points A(-1+i) and B(3+4i), find the complex number representing the following point: (1) Point P that divides the line segment AB in the ratio 2:1'

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'For the triangle ABC with vertices at points A(α), B(β), and C(γ) on the complex plane, if the equation 903α^2+β^2+γ^2+βγ=3αβ+3γα holds, what type of triangle is ABC?'

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'Show the conditions when a complex number z or z1, z2, z3, z4 lies on the unit circle.'

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'Translate the given text into multiple languages.'

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'Explain the geometric meaning of the product of complex numbers.'

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'(3) \\( z_{n} = \\left(\\frac{1+\\sqrt{3} i}{2}\\right)^{n} \\cdot (-\\sqrt{3} i) + 1+\\sqrt{3} i \\)'

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'In practicing on the complex plane, let complex numbers a, b, c represent points A, B, C respectively, such that these points are not collinear. Let α, β, γ be complex constants, express β/α and γ/α in terms of a, b, c and their conjugate complex numbers a̅, b̅, c̅ when the complex number z satisfies the equation αz+β z̅+γ=0.'

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'In the complex plane, let the complex numbers representing the vertices of the triangle be 0, α, β for O, A, B respectively.'

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'In the complex plane, find the point represented by the complex number α=3+4i, and find the point represented by the complex number obtained by multiplying it by -2.'

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'In this chapter》 In Mathematics I, we learned about addition, subtraction, multiplication, and division of complex numbers. In this chapter, we will learn about how complex numbers are represented as points on the complex plane and the geometric meanings of sum, difference, absolute value, and conjugate complex numbers. Additionally, to consider the product and quotient of complex numbers, we will learn about the geometric representation of complex numbers in the complex plane in polar form and study various shapes in the complex plane.'

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'Complex Number Multiplication and Rotation (2)\n(1) For two points z=3+i and w=2-i, find the complex number representing the point z rotated clockwise by π/6 around the center of point w.\n(2) When the complex number representing the point 3-2i after rotating around the center of 1+i by an angle θ (0 ≤ θ < 2π) is (4+3√3)/2 + (-1+2√3)i/2, find the value of θ.'

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'Complex Number Multiplication and Rotation (2) What point is represented by the following complex numbers, moved from point z? (A) $\\frac{1}{2} (-1+i) z$ (B) $\\frac{z}{\\sqrt{3}-i}$ (C) $\\overline{iz}$'

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'In what case is a complex number z for which the argument is indeterminate?'

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'Therefore, $\\lim _{n \\rightarrow \\infty}\\left|x_{n}-\\alpha\\right|=0$, hence $\\lim _{n \\rightarrow \\infty} x_{n}=\\alpha$'

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'Express the following complex numbers z in polar form. Note that the argument θ is 0≤θ<2π.'

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'For the triangle ABC with vertices at 3 points A(α), B(β), and C(γ) on the complex plane, what kind of triangle is triangle ABC when the following equations hold true?\n(1) β-α=(1+√3i)(γ-α)\n(2) α+iβ=(1+i)γ'

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'Complex number multiplication and rotation (1) Let z = 2 + √2i. Find the complex number representing the point obtained by rotating z, with the origin as the center, by -3/4 π.'

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'Let n be a natural number greater than or equal to 2, and let i be the imaginary unit.'

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'Find the quotient of the complex number \\\frac{c+d i}{a+b i}\.'

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'Prove that when an nth degree equation with rational coefficients has a solution of p+q√r, the other solution is p-q√r.'

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'Please perform arithmetic operations with complex numbers and calculate the square root of negative numbers.'

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'Chapter 2 Complex Numbers and Equations Reference Finding the solutions of quadratic equations with complex number coefficients'

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'Let k be a real constant, i=√-1 be the imaginary unit. Find the value of k when the equation (1+i)x^{2}+(k+i)x+3-3ki=0 has purely imaginary roots.'

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'Practice\nLet k be a real constant, i be the imaginary unit √(-1). Find the value of k when the quadratic equation of x, (1+i) x^2 + (k+i) x + 3-3k i = 0, has purely imaginary roots.'

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'Let ω be one of the solutions of the equation x^2+x+1=0, and let α be the other solution. Based on the relationship between the solutions and the coefficients, we have ω+α=-1...(1) and ω*α=1.'

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'Find the maximum and minimum values of BC² under the following conditions: Conditions: when \ \\frac{\\pi}{2}<\\theta<\\pi \, then \ \\cos \\theta < 0 \'

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'Find the value of x^5 + x^4 - 2x^3 + x^2 - 3x + 1 when x = (1 - √3i)/2.'

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'There is an infinite geometric series with real numbers as its first term and common ratio, and the sum is 3. Also, there is an infinite geometric series where each term is the cube of the corresponding term in the first series, and the sum is 6. Find the common ratio of the first series.'

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'In the complex plane, let the vertices of the triangle O, A, B be represented by complex numbers 0, α, and β'

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'For a complex number z, let w=(z-i)/(z+i). When the point z moves along the real axis of the complex plane, what kind of figure does the point w trace out?'

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'In the complex plane, let point A represent 6, point B represent 7+7i. Also, for a positive real number t, let point P represent \\(\\frac{14(t-3)}{(1-i)t-7}\\).'

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'nth roots of 1\nThe nth roots of 1 (i.e., the solutions to the equation z^n=1) are the following n complex numbers.\nzk=cos(2kπ/n)+isin(2kπ/n) (k=0,1,2, ..., n-1)'

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'Consider the function of a complex number z, f(z)=z+1/z. When z satisfies 1/3<=|z|<=2 and 0<=arg z<=π/4, find the maximum and minimum values of the real part of f(z).'

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'For the complex number z = cos(2/7π) + i sin(2/7π), find the value of (4) (1) (7) z+z^2+z^3+z^4+z^5+z^6. Also find the value of (1) 1/(1-z) + 1/(1-z^6). If t=(z^2+1)/z, then prove that t is a real number and show that t is a root of the cubic equation t^3+⏋⎹t^2-1␍t-ウ␍.'

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'For three points A(α), B(β), C(γ) on the complex plane\n(1) When α=2, β=1+i, γ=(3+√3)i, find the size of ∠ABC.\n(2) When α=1+i, β=3+4i, γ=a*i (a is a real number), if a=⬜, then points A, B, C are collinear; if a=1⬜, then AB⊥AC.'

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'Let α be a complex number with absolute value 1. In this case, \x0crac{α+z}{1+αz} is real if and only if z is a complex number on the unit circle.'

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"Show the calculation of expressing the complex number z in polar form using De Moivre's Theorem on the complex plane."

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'Express the following complex numbers in polar form. Assume the argument θ satisfies 0 ≤ θ < 2π. (1) 2 - 2i (2) -3 (3) cos(2/3π) - isin(2/3π)'

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'Let α be a complex number with absolute value 1. Under what conditions on z will (α+z)/(1+αz) be a real number?'

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'Let two complex numbers be represented as \ \\alpha=\\cos \\theta_{1}+i \\sin \\theta_{1}, \eta=\\cos \\theta_{2}+i \\sin \\theta_{2} \, where the arguments are such that \ 0<\\theta_{1}<\\pi<\\theta_{2}<2 \\pi \. (1) Express \ \\alpha+1 \ in polar form, where the argument \ \\theta \ satisfies \ 0 \\leqq \\theta<2 \\pi \. (2) Show that \ \eta=-\\alpha \ holds when the real part of \ \\frac{\\alpha+1}{\eta+1} \ is equal to 0.'

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'Express the following complex numbers in polar form. Assume the argument theta is between 0 and 2π.'

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'(A) Express α and β in polar form: α = cos(θ1) + i sin(θ1), β = cos(θ2) + i sin(θ2), where 0 < θ1 < π < θ2 < 2π.'

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'Find the value of z when the complex number z satisfies arg z = π/4 and |(z+i)/(1+2i)| = 1.'

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'A problem concerning the equation of a circle in the complex plane. For the equation representing a circle |z-α|=r, squaring it gives |z-α|^{2}=r^{2}, which when expanded leads to z \ar{z}- \ar{α} z- α \ar{z}+|α|^{2}-r^{2}=0. This equation involves real numbers, hence the need for consideration of a geometric representation.\nNext, by using real numbers a, c, and a complex number β, we explore what shape the equation a z \ar{z}+ \ar{β} z+ β \ar{z}+c=0 represents.\nCase 1: When α ≠ 0, |β|^{2}>a c, and a ≠ 0, this equation represents a circle with center at -β/a and radius |β|^{2}-a c|a.\nCase 2: When a = 0, and \ar{β} z+ β \ar{z}+c=0, the equation represents a line equation A x + B y + c = 0 (interpreting the complex number β as p + qi). Line B is perpendicular to the line connecting the two points (0, β) (refer to line 261).'

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'When three distinct points A(α), B(β), C(γ) satisfy the following conditions, find the measures of the three angles of triangle ABC.'

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'Assume there are three distinct complex numbers α, β, γ such that the equation α^3 - 3α^2β + 3αβ^2 - β^3 = 8(β^3 - 3β^2γ + 3βγ^2 - γ^3) holds.'

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'On the complex plane, for the 3 points A(1+i), B(3+4i), and C, if AB = AC and ∠BAC = π/3, find the complex number representing point C.'

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'Find the complex number z representing the vertex C of an equilateral triangle ABC with segment AB as one side on the complex plane with points A (-1+i) and B (√3-1+2i).'

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'Find the complex number representing the centroid of the triangle ABC with vertices A(α), B(β), and C(γ).'

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'Find the complex number z representing the vertex C of an equilateral triangle ABC with segment AB as one side, where A(-1+i) and B(√3-1+2i) are two points on the complex plane.'

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'What is the shape of the set of all points that satisfy the following equations:\n(1) |z - 2i| = |z + 3|\n(2) 2|z - 1 + 2i| = 1\n(3) (2z + 1 + i)(2 conj(z) + 1 - i) = 4\n(4) 2z + 2conj(z) = 1'

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'Practice: Express the following complex numbers in polar form. The argument θ should satisfy 0 ≤ θ < 2π. (1) 2-2i (2) -3 (3) cos(2π/3)-isin(2π/3)'

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'Prove that all solutions to the equation zⁿ = α can be given as α₀, ωα₀, ω²α₀, ..., ωⁿ⁻¹α₀.'

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'Suppose that between three distinct complex numbers α, β, γ, the equation α³ - 3α²β + 3αβ² - β³ = 8(β³ - 3β²γ + 3βγ² - γ³) holds.'

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'Express the following complex numbers in polar form. Assume that the argument θ satisfies 0 ≤ θ < 2π.'

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'For points A(-1+4i), B(2-i), C(4+3i), find the complex numbers representing the following points:\n(1) Point P that divides the line segment AB in the ratio 3:2\n(2) Point Q that divides the line segment AC externally in the ratio 2:1\n(3) Midpoint M of the line segment AC\n(4) Vertex D of the parallelogram ABCD\n(5) Centroid G of triangle ABC'

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'On the complex plane, there are three points O(0), A(-1+3i), and B. When △OAB is a right-angled isosceles triangle, find the complex number z representing point B.'

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'Practice\nConsider a complex number α=cosθ+i sinθ where the angle θ is greater than 0 and less than π/2. Let z0=0 and z1=1, and define the sequence {zk} by zk-zk-1=α(zk-1-zk-2) for k=2,3,4,... in the complex plane, where zk(k=0,1,2,...) represents the point Pk.\n(1) Express zk in terms of α.\n(2) Let A(1/1-α), then show that the points Pk(k=0,1,2,...) lie on a circle with A as the center.'

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'For a non-zero complex number d, what kind of shape does the equation dz(𝞍⁻+1)=𝞍⁻dz(z+1) represent in the complex plane?'

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'Let a be a positive real number, w=a(cos(π/36) + i sin(π/36)). Define a sequence of complex numbers {z_n}, z_1 = w, z_(n+1) = z_nw^(2n+1) (n=1,2,...). (1) Find the argument of z_n.'

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'When a complex number z satisfies z - 3\x08ar{z} = 2 + 20i, use the properties of conjugate complex numbers to find z.'

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'Let α and β be complex numbers. (1) When |α|=|β|=1 and α-β+1=0, find the values of αβ and α/β+β/α. (2) When |α|=|β|=|α-β|=1, find the value of |2β-α|.'

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'In the complex plane, the complex number α = a + bi is corresponded to the point (a, b) on the coordinate plane. This plane is called the complex plane. So, which point on the complex plane does the complex number α = 3 + 4i correspond to?'

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'When a complex number z satisfies 3z + 2\x08ar{z} = 10 - 3i, find z using the properties of conjugate complex numbers.'

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'Consider the sequence of complex numbers defined by the following recurrence relation: z1 = 1, z_n+1 = (1 + √3 i)/2 z_n + 1 (n=1,2, ...). Here, i is the imaginary unit. (1) Find z_2, z_3. (2) Express the above recurrence relation as z_n+1 - α = ((1+√3 i)/2)(z_n - α) and find the complex number α. (3) Find the general term z_n. (4) Find all natural numbers n for which z_n = -(1 - √3 i)/2 holds true.'

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'Find the product αβ and quotient α/β of the following complex numbers.'

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'In the complex plane, let 2 points A(α), B(β) be connected by a line segment AB which divides in the ratio m:n. The point that divides internally in the ratio m:n is C(γ), and externally is D(δ).'

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'Multiply the complex number with absolute value of 2 and argument of π/3 by z.'

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'Find the complex number w2 obtained by rotating the point z = 4 - 2i by π/3 radians counterclockwise around the origin and scaling the distance from the origin by a factor of 1/2.'

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'Find the complex number z that satisfies |z|=5 and |z+5|=2√5, then calculate the following values. 31 (1) z bar{z} (2) z+bar{z} (3) z'

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'In this chapter, we learn about complex numbers being represented as points on the complex plane, as well as the geometric meaning of complex number addition, subtraction, absolute value, and conjugate complex numbers. Additionally, we learn about polar form, the geometric representation of complex numbers, and study various shapes on the complex plane.'

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'When the complex number z satisfies |z-3-4i|=2, find the maximum value of |z| and the corresponding value of z.'

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'Prove that for any natural number \ n \, the inequality \ 2 \\sqrt{n+1}-2<1+\\frac{1}{\\sqrt{2}}+\\frac{1}{\\sqrt{3}}+\\cdots +\\frac{1}{\\sqrt{n}} \\leqq 2 \\sqrt{n}-1 \ holds.'

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'Prove that for two distinct lines l and m on the complex plane that do not pass through the origin, the points on line l always satisfy the equation α z + ᾱ z = |α|².'

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'Assume that the point z on the complex plane lies on the unit circle. Show that z can be represented as z = e^(iθ).'

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'Express the following complex numbers in polar form. The argument should be within the range 0 ≤ θ < 2π.'

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'Prove that for complex number z, where z^3 is not a real number, z^3 - (conjugate of z)^3 is purely imaginary.'

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'It was the 19th-century Gauss (C. F. Gauss, 1777-1855) who accurately pointed out that this was simply a psychological problem. He used the concept of the complex number plane extensively to clearly illustrate the arithmetic of complex numbers.'

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'Prove the inequality |(α-β)/(1-αβ)|<1 holds for complex numbers α, β with absolute value smaller than 1.'

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'In the complex plane, representing the points as -1+2i, 3+i. If we consider segment AB as one side, find the complex number representation of the vertices C, D of the square ABCD.'

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'Prove that if the 4th degree equation ax^4+bx^2+c=0 has a complex solution α, then the conjugate of α is also a solution of this equation.'

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'Let i be the imaginary unit and k be a real number. Given α=-1+i, point z moves on the complex plane along the unit circle with the origin as the center.'

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'Symmetric movement about a line\nIn the complex plane, when the point P(z) and the point Q(w) are symmetric with respect to a line passing through the origin O and the point A(α) (α ≠ 0), express w in terms of α and z.'

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'Derive the polar form of a complex number z=a+bi and explain its multiplication and division.'

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'Express the following complex number in polar form, where the argument θ satisfies 0 ≤ θ < 2π. 1 + cos α + i sin α (0 ≤ α < π).'

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'Practice problem: In the complex plane, let complex numbers a, b, c represent points A, B, C respectively, where the points are not collinear. Consider α, β, γ as constants, express β/α and γ/α in terms of complex numbers a, b, c and their conjugates a, b, c when the complex number z satisfies the equation αz+βz+γ=0 representing the following figures: (1) Line AB (2) Line passing through point C and perpendicular to line AB'

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'Find the conditions for z to be real when given in the form H|NT \\( 100 z=r(\\cos \\theta+i \\sin \\theta) \\) or \ z=x+yi \, and substitute it into the inequality.'

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'Basic question 106 Shape of a triangle (1) For triangle ABC with vertices A(α), B(β), and C(γ) on the complex plane, if the equation β-α=(1+√3i)(γ-α) holds true, find the sizes of the three interior angles of triangle ABC.'

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'Let α be a complex number with absolute value 1. For which complex number z does (α+z)/(1+αz) become a real number?'

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'In the complex plane with origin O, let the points representing complex numbers α, β be A, B respectively. Where α ≠ 0, β ≠ 0. Choose two of the following equations that ensure triangle △OAB is always a right-angled isosceles triangle.'

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'Determine the value of a so that the points A(α=2+i), B(β=3+2i), and C(γ=a+3i) are collinear on the complex plane.'

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'Find the cube of the complex number z = 1 + i and express it in polar form.'

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'For complex number z different from -1, define complex number w as w=z/(z+1). When point z moves along the circle with radius 1 centered at the origin, find the shape formed by point w.'

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'Consider the sequence {an} determined by the conditions (i)(ii).'

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'For the points A(-2-2i), B(5-3i), C(2+6i), find the complex numbers representing the following points.'

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'Consider the four points A(α), B(β), C(γ), D(δ) on the complex plane forming a quadrilateral ABCD. Assume that the quadrilateral ABCD is a convex quadrilateral with all internal angles less than 180 degrees. Also, assume that the vertices of the quadrilateral ABCD are arranged in counterclockwise order as A, B, C, D. Construct right isosceles triangles APB, BQC, CRD, DSA with sides AB, BC, CD, DA as their hypotenuses on the outside of the quadrilateral ABCD. (1) Find the complex number representing point P. (2) The necessary and sufficient condition for quadrilateral PQRS to be a parallelogram is what kind of quadrilateral ABCD? (3) If quadrilateral PQRS is a parallelogram, prove that quadrilateral PQRS is a square.'

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'Let α=2+i and β=4+5 i. Find the complex number γ representing the point obtained by rotating β about α by π/4.'

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'Explain the addition of complex numbers α and β as α+β using position vectors and illustrate it using a parallelogram.'

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'Complex plane\nLet w = (1 + α)z + 1 + α. When the lines OW and OZ are perpendicular, answer the following questions:\n(1) Find the value of |z-α|.\n(2) Find the complex number z for △OAZ to form a right triangle.\n[Type: Yamagata University]'

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'Find the general term of the complex number sequence {zn} that satisfies the following recurrence relation.'

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'Prove that the inequality |1+z| ≥ (1+|z|)/√2 holds when the complex number z=x+yi(x, y are real numbers) with x≥0. Also, determine when the equality holds.'

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'Please explain the relationship between complex numbers and vectors in the plane, and prove that it is a one-to-one correspondence.'

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'Find the complex number representing the following points A(-1+i) and B(3+4i).'

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'Express the following complex numbers in polar form, where the argument \ \\theta \ is such that \ 0 \\leqq \\theta<2 \\pi \.\n(1) \ 1+i \\n(2) \ i \\n(3) -2'

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'Let a, b be real numbers, and suppose that the cubic equation x^{3}+ax^{2}+bx+1=0 has an imaginary root α. Prove that the conjugate complex number of α is also a root of this equation. Furthermore, express the third root β and the coefficients a, b using α and the conjugate of α.'

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'Solve the recurrence relation of complex numbers\n\ z_{1}=3 \ and the recurrence relation \\( z_{n+1}=(1+i) z_{n}+i(n \\geqq 1) \\) that defines a sequence of complex numbers \ \\left\\{z_{n}\\right\\} \, and answer the following questions:\n(1) Find \ z_{n} \.\n(2) Find \ z_{21} \.'

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'Let 𝛼 = -2 + 2𝑖, 𝛽 = -3 - 3√3 𝑖. Where the argument is 0 ≤ 𝜃 < 2𝜋. (1) Express 𝛼𝛽 and 𝛼 / 𝛽 in polar form respectively. (2) Find 𝑎𝑟𝑔 𝛼³ and |𝛼³ / 𝛽|.'

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'The condition for three points including the origin O to be collinear is (1) ○○○) α=3+(2 x-1) i, β=x+2-i. Find the value of the real number x when points A(α), B(β), and the origin O are collinear.'

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'Express the complex numbers 1+i and sqrt(3)+i in polar form, and calculate the values of cos(5/12π) and sin(5/12π) for each of them.'

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'Understanding geometric shapes using complex numbers'

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'Chapter 3 Complex Plane 417\nAlternative solution 2 A(3 i), B(-3), P(z) Let |z-3 i|=2|z+3| hence AP=2BP therefore AP:BP=2:1 Point C(α) divides line segment AB in the ratio 2:1 and point D(β) divides it externally, so the locus of point P is a circle with C and D as the diameter. α=(1⋅3 i+2(-3))/(2+1)=-2+i β=(-1⋅3 i+2(-3))/(2-1)=-6-3 i Therefore, the locus of point z is a circle with -2+i and -6-3 i as the diameter. |z-3 i| represents the distance between points A and P, and |z+3| represents the distance between points B and P. Chapter 3'

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'The point obtained by rotating point z about the origin by an angle θ is (cos θ + i sin θ) z'

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'The complex number \ \\alpha \ satisfies \ \\alpha^{5}=1, \\alpha \\neq 1 \.'

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'Investigate the convergence and divergence of the following infinite series, and find their sum if they converge.'

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'The relationship between complex numbers and vectors in geometry'

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'Given real numbers x, y, z satisfying x+y+z=√5+2, xy+yz+zx=2√5+1, xyz=2, find the values of the following expressions: (1) 1/x+1/y+1/z (2) x^2+y^2+z^2 (3) x^3+y^3+z^3 (4) x^4+y^4+z^4'

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'For the equation f(x)=0 to have two distinct negative roots, the graph of y=f(x) must intersect the negative part of the x-axis at two different points. Therefore, the following must all be true simultaneously.'

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'Examples of Catalan Numbers 3... Ways to Divide a Polygon into Triangles'

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Advanced 40 Unwrapping $\sqrt{A^{2}}$
Advanced 41 Rationalizing the Denominator (2)
Advanced 42 Problems regarding Integer and Decimal Parts
Advanced 43 Removing Double Radicals
Advanced 44 Solving Inequalities Involving Absolute Values by Case Analysis
Advanced 45 Inequality Involving Two Absolute Value Signs
Advanced 46 Conditions for Systems of Inequalities to Have Solutions

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Calculate the following expressions.
1. (cos π/60 + i sin π/60)^{20}
2. (√3 + i)^{-12}
3. (1 + i)^{17}

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Please explain the polar form representation of complex numbers.

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TRAINING Practice 3 There are 6 points \( \mathrm{A}\left(z_{1}
ight), \mathrm{B}\left(z_{2}
ight), \mathrm{C}\left(z_{3}
ight), \mathrm{D}\left(z_{4}
ight), \mathrm{E}\left(z_{5}
ight) \), \( \mathrm{F}\left(z_{6}
ight) \) on the complex plane. When the hexagon $\mathrm{ABCDEF}$ is a regular hexagon as shown in the figure: egin{\overlineray}{l} z_{3}=\square ア \ z_{2}=\square ext { ウ } z_{1}+\square ext { イ } z_{5}, \ z_{6}=\square ext { オ } z_{1}+\square ext { カ } z_{5}, \end{\overlineray} \(\mathrm{D}\left(z_{4}
ight)\) is ア $\square$ Select the corresponding answer (you can select the same option multiple times): (0) rac{3+\sqrt{3} i}{3} (1) rac{1+\sqrt{3} i}{2} (2) rac{\sqrt{3}}{3} i (3) rac{1-\sqrt{3} i}{2} (4) rac{3-\sqrt{3} i}{6} (5) rac{3+\sqrt{3} i}{6}

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Find the center (in polar coordinates) and radius of the circles represented by the following polar equations.
1. $r^{2}-4 r \cos heta+3=0$
2. \( r^{2}-r(\cos heta-\sqrt{3} \sin heta)-8=0 \)

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Calculate the following expressions. (1) \( \left(\cos rac{\pi}{12} + i \sin rac{\pi}{12}
ight)^{6} \) (2) \( \left(rac{1+i}{2}
ight)^{15} \) (3) \( (\sqrt{6} - \sqrt{2} i)^{-6} \) (4) \( \left(rac{1+\sqrt{3} i}{1+i}
ight)^{12} \) (5) \( (\sqrt{3} + i)^{10} + (\sqrt{3} - i)^{10} \)

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Chapter 3 Complex Plane 13 Complex Plane 14 Polar Form of Complex Numbers 15 De Moivre's Theorem 16 Complex Numbers and Figures

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TRAINING 74 Given α=2(cos 11/12π + i sin 11/12π) and β=3(cos π/4 + i sin π/4), find αβ and α/β.

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Given that the complex number $z$ satisfies $z + \frac{1}{z} = \sqrt{2}$, find the value of $z^{15} + \frac{1}{z^{15}}$.

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Find the value of z^{15}+rac{1}{z^{15}} when the complex number z satisfies z+rac{1}{z}=\sqrt{2}.

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What point \( (-1+i) z \) represents as the location of point $z$ after it has been moved. Assume the range of rotation angle $heta$ is $0 \leqq heta<2 \pi$.

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Real Multiples of Complex Numbers
For a real number $k$ and a complex number $\alpha = a + b i$, as shown in the following diagram, when $\alpha \neq 0$, the point $k \alpha$ lies on the line $\ell$ passing through the two points 0 and $\alpha$.
Conversely, the points on this line $\ell$ represent real multiples of the complex number $\alpha$.
Consider the following cases:
1. $\alpha = 1 + i$, $k = 2$
2. $\alpha = -1 + 2i$, $k = -3$
For each case, find $k \alpha$ and describe its position from 0 on the complex plane.

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Given that the point lpha is rotated by rac{\pi}{3} around the origin to obtain point eta, where eta = 2 + 2i, find the complex number representing point lpha.

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Given three different points O(0), A(α), B(β) on the complex plane, where α and β satisfy the following equations. What type of triangle is △OAB? (1) α^{2}+β^{2}=0 (2) 3α^{2}+β^{2}=0

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(1) Plot the points representing the following complex numbers on the complex plane.
(a) $5-2 i$
(b) $-1+3 i$
(c) -2
(d) 1
(e) $-3 i$
(f) $2 i$
(2) Answer the complex numbers corresponding to the following points on the coordinate plane.
(a) \( (-3,1) \)
(b) \( (4,0) \)
(c) \( (0,-2) \)

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Given lpha=2+3i, eta=-6+xi . If the two points \( \mathrm{A}(lpha), \mathrm{B}(eta) \), and the origin $\mathrm{O}$ are collinear, find the value of the real number $x$.

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Express the following complex numbers in polar form. The range of the argument θ is 0 ≤ θ < 2π.
(1) $-1+\sqrt{3} i$
(2) $5 i$

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Given lpha = e^{i\pi/6} and eta = e^{-i\pi/4} , find $m$ and $n$.
(1) $m=6, n=12$

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Given three distinct complex numbers lpha, eta, \gamma such that the equation \( \sqrt{3} \gamma-i eta=(\sqrt{3}-i) lpha \) holds, answer the following questions.
(1) Calculate rac{\gamma-lpha}{eta-lpha} .
(2) Determine the angles ngle \mathrm{A}, ngle \mathrm{B}, ngle \mathrm{C} of the triangle $riangle \mathrm{ABC}$ with vertices at the points lpha, eta, \gamma .

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Mathematics C
EX When the point \( \mathrm{P}(z) \) on the complex plane moves on a circle centered at the point $2 i$ with a radius of 1, determine the shape drawn by the point \( \mathrm{Q}(w) \) that satisfies \( w=(1+i)(z-1) \) 27.

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When the point rac{z}{z-2} on the complex plane lies on the imaginary axis, what kind of curve does the point $z$ trace as it moves?

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Given that the complex numbers lpha, eta satisfy the equation rac{eta}{lpha}=rac{1+\sqrt{3} i}{2} , determine the angles of the triangle $riangle \mathrm{OAB}$ with vertices at points \( \mathrm{O}(0) \), \( \mathrm{A}(lpha) \), and \( \mathrm{B}(eta) \) on the complex plane.

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Calculate the following expressions.
(1) \( \left(\cos rac{\pi}{12}+i \sin rac{\pi}{12}
ight)^{6} \)
(2) \( \left(rac{1+i}{2}
ight)^{15} \)
(3) \( (\sqrt{6}-\sqrt{2} i)^{-6} \)
(4) \( \left(rac{1+\sqrt{3} i}{1+i}
ight)^{12} \)
(5) \( (\sqrt{3}+i)^{10}+(\sqrt{3}-i)^{10} \)

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Given Example 73: α=4(cos 5/12π + i sin 5/12π), β=2(cos π/4 + i sin π/4), find α β and α/β.

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For natural numbers $n$ satisfying $1 \leqq n \leqq 100$, how many such $n$ exist such that for the imaginary number lpha=rac{\sqrt{3}+i}{2} , the equation lpha^{n}+rac{1}{lpha^{n}}=-2 holds?

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Let $z=2 \sqrt{2}+\sqrt{2} i$. Find the complex number $w$ representing the point $z$ rotated about the origin by -rac{\pi}{4} .

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Show that for a complex number $z$ , if $|z|=\sqrt{2}$ , then z+rac{2}{z} is a real number.

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Given complex numbers lpha, eta, \gamma, \delta such that lpha + eta + \gamma + \delta = 0 and |lpha| = |eta| = |\gamma| = |\delta| = 1 , find the value of |lpha - eta|^{2} + |lpha - \gamma|^{2} + |lpha - \delta|^{2} .

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Prove the following statements about the complex numbers z, lpha . (1) If $k$ is a positive number and $|z|=k$, then z+rac{k^{2}}{z} is a real number. (2) z \overline{z}+lpha \overline{z}+\overline{lpha} z is a real number.

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Find the complex number representing the point obtained by rotating the point 2+2i about the point i by the following angles: (1) π/6 (2) π/4 (3) π/2 (4) -π/2

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Given three distinct points in the complex plane O(0), A(α), B(β), where α and β satisfy the following equations. What kind of triangle is ΔOAB?
(1) α² + β² = 0
(2) 3α² + β² = 0

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Represent the product of complex numbers $z_1 \cdot z_2$ geometrically, and explain it. As an example, specifically calculate the product \( z_1 = 2(\cos heta + i \sin heta) \), \( z_2 = 3(\cos \phi + i \sin \phi) \).

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Please explain the properties of complex conjugates on the complex plane.

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In the complex plane, when point z is moving on the circle with center O and radius 1, what kind of figure will the point w, represented by the following equations, describe? (1) w=rac{z+2}{z-1} (where z
eq 1) (2) w=rac{z+1}{2z-1}

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On the complex plane, let point A represent the complex number $z = a + bi$.
(1) Let point B represent its conjugate complex number $\overline{z} = a - bi$. Find the coordinates of point B.
(2) Find the distance between points A and B.

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In the complex plane, there are three points \( \mathrm{O}(0), \mathrm{A}(3-2 i), \mathrm{B} \). If $\triangle \mathrm{OAB}$ is a right isosceles triangle, find the complex number $z$ that represents point $\mathrm{B}$.

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Find the values of lpha and eta for 34.
lpha=rac{1}{2}+rac{\sqrt{3}}{2} i, \quad eta=-i

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(1) Plot the points representing the following complex numbers on the complex plane.
(a) $4+2 i$
(b) $-2-3 i$
(c) 3
(d) -4
(e) $4 i$
(f) $-i$
(2) Answer the complex numbers corresponding to the following points on the coordinate plane.
(a) \( (5,-2) \)
(b) \( (-1,0) \)
(c) \( (0,3) \)

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Given that z=r(\cos heta+i \sin heta), express the absolute value and argument of the following complex numbers using r and heta. Assume r>0.
(1) 2 z
(2) -z
(3) ar{z}
(4) rac{1}{z}
(5) z^{2}
(6) -2 ar{z}

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Using the complex number $\sqrt{3}+i$, calculate the following.
(5)
Find \( (\sqrt{3}+i)^{10} + (\sqrt{3}-i)^{10} \).

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Let $\alpha$ and eta be complex numbers such that both the real and imaginary parts of $\alpha$ are positive, and |\alpha|=|eta|=1 . Given that the three points i \alpha, \frac{i}{\alpha}, eta form an equilateral triangle on the complex plane, find $\alpha$ and eta . [Shizuoka University]

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73 (1) 2( cos 5/3π + i sin 5/3π ) (2) √2/3 ( cos 3/4π + i sin 3/4π ) (3) 2√2( cos 4/3π + i sin 4/3π ) (4) 3( cos 3/2π + i sin 3/2π )