Let z be a complex number such that z, z3 and z5 form an equilateral triangle on the complex plane. Find all possible values of z.

Q: The complex numbers 0, z+ 1 and z2 are distinct and collinear on the complex plane. Draw the locus…

A: As per our guideline we are supposed to answer only first question of a multiple type questions.…

Q: Let z is a complex number and Z is the conjugate number of z then () equals O a. z1 Z2 b. 21 Z2 C. O…

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Q: et P(x)=2x^4+9x^3+13x^2+81x-45 Factor p(x) over the complex numbers

A: We have fx=2x4+9x3+13x2+81x−45Put…

Q: 10 2) z = (-j)()

A: We have to use De Moivre's theorem cosθ+jsinθn=cosnθ+jsinnθ and Eular form of complex number…

Q: Let a = 1+bi be a complex number, b>0, such that a is a real number. Evaluate 1+a+a² + . + a". ....

A: Given a complex number : a=1+bi where b>0 It is given that  a3 is a real number. We need to find…

Q: Describe and plot the set of all complex numbers that satisfy the equation Iz - 2il = Im(z) %3D -2

A: We have to plot the complex graph satisfying the given equation.

Q: Plot each of the given complex numbers in an Argand diagram and label all the necessary parts. Also,…

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Q: Let a, b, and c be any three nonzero complex numbers. If |z| = 1 and 'z' satisfies the equation az +…

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Q: The graph of the complex number a + bi is a ________ that extends from the ______to the point ______…

A: The graph of the complex number a+bi is a ________ that extends from the ______ to the point ______…

Q: Let , ² be the complex cube roots of unity. Let z₁, Z₂, Z3 be complex numbers such that Z₁ + Z₂ +Z₁…

A: Given that, Let ω,ω2 be the complex cube roots of unity. Let z1,z2,z3 be complex numbers such that…

Q: Let a be a complex number that is algebraic over Q. Show that√a is algebraic over Q.

A: Consider the polynomial: Clearly,  Hence, we have following: Let f(x) be the minimal polynomial…

Q: Let C be the set of complex numbers. Define additionon C by (a+bi) + (c+di) = (a+c) + (b+d)i And…

A: The solution is given below:

Q: The conjugate of complex number z satisf y: Z-Z=2ilmz O z+z=10 z+z=2Rez O

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Q: use the picture to find at least one complex number c such that z^7 + 3z^4 + c has exactly 3 zeroes…

A: consider the equation

Q: The distance between two points in the complex plane is the ________ of the difference of the…

A: Let (a,b) and (m,n) are two points in the complex plane. Then the difference of the complex numbers…

Q: Find the complex conjugate z¡ of z1 = –7i + 6. Z1 = %3D

A: conjugate of a function means change the sign of i term

Q: 10. Find a complex number z such that (a) z – z+7 – 6i = 0 - | (b) = 3 + 4i %3D 1+ z

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Q: Consider a complex number z such that Arg (z) = - 120º. Then, the image that represents all the…

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Q: Find all real numbers a and b such that the complex numbers 21 = a² + 4b – bi and 22 = 4+b- 2 a²i…

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Q: Plot each of the given complex numbers in an Argand diagram and label all the necessary parts. Also,…

A: Since there are multiple questions the first question is answered as per our guidelines. Kindly…

Q: If z is a complex number then the sum of z and its conjugate is an imaginary number. O true false

A: Let z be a complex number given by, z=a+ib where a is the real part and b is the complex part of z.

Q: Let A, B, C be the set of complex numbers defined as: A = {z: | z+2|-|z- 2 || = 2} 6. %3D z-1 TC B =…

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Q: The set of all complex numbers is denoted by C={x+iy:x,y€R}

A: A complex number is the combination of real and imaginary number in the form of x+iy , x, y are real…

Q: The conjugate of complex number z satisfy: z-z=2ilmz O z+z=10 - z+z=2Rez O - z-z=00

A: The complex number, z is usually represented as z=x+iy . Here, x is the real part and is represented…

Q: If z is a complex number, show that the product of z and its conjugate is a realnumber.

A: Let z=a+ib be any complex number where a and b are real number. Then conjugate of z is z¯=a-ib

Q: Find the complex conjugate z, of z1 = -4i + 4. ž =

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Q: Write down the definition of the complex conjugate, z, of z = a + ib where a, b are real numbers.…

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Q: Give all the elements of Ug and write them in rectangular form. Draw the regular octagon in the…

A: The given question is related with group theory. We have to write all the elements of U8 and we need…

Q: The complex numbers z, and z2 are such that z, = -2 + 2i and zz = -2– 2i. Find (a) arg z1 (b) |z21".

A: we have to find the value of the following

Q: If zi and z2 are two distinct non-zero complex numbers such. that |z|| = |z2), then 2 +Z, is always…

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Q: Find the fourth roots of 2 + v2 i, namely all complex numbers z such that z* = v2 + V2 i

A: The representation of a complex number can be done in two forms, rectangular and polar forms. The…

Q: Given two complex numbers z and w such that z = (1+ i)w + (3 –- i)w, where w is the conjugate of w.…

A: z and w are two complex numbers. Such that

Q: Find the location of z = -3-4i on the complex plane and find |z|.

A: The given complex number is z=-3-4i Compare with,

Q: If z is a complex number that satisfies the condition: |z − 2 − 3i| = 2, then show that the…

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Q: Let z₁,z, and z, be complex numbers such that |z₁|=|z₂|=|z₁|=|z₁+z₂+z₁|= 2. If and z₂ #z,. Then find…

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Q: Let z1 , Z2 and z3 be the complex numbers given in the following argand diagram. Find the complex…

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Q: C. Find all complex numbers z of the form 2=x+y such that that zz = 25 and *+y = 7.

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Q: Find the graph of the complex number z = -3 + 4i in the complex plane Imaginary Real A Imaginary…

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Q: The complex number z is given by Z = z²+1 a²-b² 2z a² + b² a+bi a-bi show clearly that

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Q: Find all the complex numbers z where z^4 = -16i. Write both a + ib form and polar form, and sketch…

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Q: Given the complex numbers z1 = -3+i and z2 = 1+ 4i. Show that |z1 z2 = |z1||22| and org (z1 2) = arg…

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Q: Find all complex numbers z satisfying = -i

A: i = -1(i)2 = -1

Q: Find all complex numbers z of the form 2 =x+iy such that that ZZ = = 25 and *+ y = 7.

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Q: 5. Find all complex numbers of the form z = a + bi , where a and b are real numbers such that zz' =…

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Q: Let z₁ = 5+2i, Z₂=1+3i, z3=2-3i, Z4--4-7i. (a) Plot the complex numbers z₁, Z2, Z3, Z4 on an Argand…

A: Solution

Q: Find a complex number z=a+bi such that z*z_1=1. Also, convert z into its three (3) other forms.…

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Q: (c) Let z be a complex number such that z, z³ and z5 form an equilateral triangle on the complex…

A: Let z be a complex number such that z, z3, z5 form an equilateral triangle on the complex plane.We…

Q: Let z be a complex number such that Iz| = 4 and arg(z) = 5T/6, then z = (a) -2v3 + 2i (b) 2v3 + 2i…

A: Recall:    Let z=x+iy be an complex number then             z=r(cosθ +i sinθ)…

Q: Let z2 = -V2 + i/3, w = -3+ 3i and p= -3– 2i. 1. Find all the complex numbers given by In z2.

A: It is given that

Q: If the complex number z and its conjugate z' satisfy zz' + 2(z - z') = 12 + 8i, then the values of z…

A: Explanation of the answer is as follows