1- Prove that (1- i)2 + (1 + i)2 =0. 2- Put in the form a+ ib .((1+i)?)2 - ((1- i)2)².

Q: Prove that Eo 2i = 2n+1 – 1 - i3D0

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Q: 1. Explain why Corollary 1 of Theorem 9-7 is true. That is, explain why L1 = L2.

A: Given that ∠1=∠2 is true statementsHint: Major arc; Minor arcand…

Q: Prove that (x - i) = > xi i=1

A: We will prove the following result.

Q: Prove that x4+4=(x+1+i)(x+1-i)(x-1-i)

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Q: 10 2. Use De Moivre's theorem to determine 1 in the form a + ib. 2

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Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: 2. prove that

A: I solved through matrix form.

Q: X sec10 can 1. Let u = 7 In х2 - 1

A: Given: ∫x sec107ln2x2 - 13tan7ln2x2 - 13dxx2 - 1.

Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: If z isa complex number satisfying|z-1=1. Then prove that z-2 = i tan {arg (z)} -itar

A: The given relation is: z-1=1 Here, let z = x + iy So, (x+iy)-1=1(x-1)+iy=1(x-1)2+y2=1(x-1)2+y2=1…

Q: Evaluate the integral. 2 (3u - 5)(u + 2) du -1

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Q: Prove that (3-2)(x+2) = 3+2i ifitues.

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Q: 2. Prove that if 5BA+ AC A - I = A, then A must be inver

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Q: Q4: 1- Prove that (1-0+(1+ () 0. 2. Put in the form a+ib ((1+i))- ((1-i)).

A: Here we have to prove that, 1) 1-i2+(1+i)2=0 2) Put in the form of a+ib, 1+i22-1-i22

Q: (2) Find all the numbers z such that elz = 10i.

A: Consider the given. eiz=10i

Q: ez = -2

A:

Q: i) Find an upper bound for the modulus 3z³ + 5z + 1, if |z|<1 where z is the complex number.

A: Now for the mod function: : a+b≤a+b. SO using the same method in the given complex function:…

Q: form of (3+3i)^5

A: Given,      (3+3i)5

Q: f for ƒ : R - {2} → R- { 5} indicated by f (x) =(5x +1)/(x -2a), a bijection can be made from [0, 1)…

A: Bijective function: The domain is set A and the co-domain in set B relates elements of two sets A…

Q: express z1z2 and (z1/z2) in the form a+ib z1=-1+3i, z2=2-i

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Q: Calculate x - z. (Image 24 JI 8х -2 2x +1 3z z + 22

A: As per the guidelines I am supposed to do the first question among multiple questions. So I am…

Q: Prove that x' +4=(x+1+i)(x+1-i)(x-1+i)(x-1-i).

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Q: %3D n-0o 2n

A: Given. We are given a limit and we need to prove that the limit of the given function at infinity…

Q: 6. Express in a + bi form: 2i(i + 3)

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Q: 4. Let z = -4 + 3i and zw = -7 - 2i. By showing clear steps, find w in the form of | | a+bi.

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Q: Ai How could you prove ZC AB ZCDB? Drag tiles to the empty boxes to correctly complete the proof.…

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Q: 34. Prove parts (a) and (c) of Theorem 3.2.3.

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Q: Prove that -(x+y)=(-x)+(-y).

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Q: 6. Determine the number of roots, counting multiplicities, of the equation in the annulus 1 ≤|z|< 2.…

A: We use Rouche's Theorem for this problem

Q: two positive iontages and bare by 2, them prove that. (2° +1) is most divisible by (2²-1) let a with

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Q: (i) find gcd(2,3+5i) in Z[i], (ii) show that 3+4i and 4-3i are associates in Z[i] , do all , dont…

A: Solution (i)

Q: #5 Let x, y, z be nonnegative real numbers such that y + z> 2. Prove that (x + y + z) >4x + 4yz.

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Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

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Q: 4x 1. Зу': y? %3D

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Q: 22. Given that Z= 2-3i, express Z/Z* and Z(cos-- isin -) in the form a + bi, where a and b are real…

A: Note: "Since you have asked two questions, we will solve the first question for you. If you want any…

Q: What is the midpoint between z1 = 9i and z2 = 4? (2, 4.5) (-4,-9) (-2,-4.5) (4, 9)

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Q: Question is attached, Thankyou

A: Let us take z = x + iy.

Q: Find a number z that satisfies: |z| - z = 2 + i

A: Complex number which satisfy |z|-z= 2+i

Q: 2.2.12 Prove that if x and y positive real numbers, then Vx+y # Vx+ Vy. Argue by contradiction. are

A: Let x and y are positive real numbers. Also let x+y=x+y Now squaring on both side,we get…

Q: 1+ 3k 1- 3k] 1 – 3k 1+3k 2 - prove that AK: 2 1 For A : -1 -

A: Given, A=2-1-12   ...i    ....(i)  We need to show that Ak=121+3k1+3k1+3k1+3k    ...ii     Now, A…

Q: 5 +1 Tsimplify denomina 7) (1+i) (1-1) = 4 the Following without using the de noctor Prove that

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Q: or complex number z and w, prove that zf w-|wfz=z-wif and onlyif z = w or, zw = 1. W

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Q: Prove that JI = J [1 0 0] I = |0 1 0 Lo 0 -1 2 -31 J = 4 5 -6 %3D –7 8 -9 1.

A: Considering the matrix multiplication of J and I

Q: (2.) Find another representation in which a. r is negative and 0 <0 < 2n. b. ris positive and -2n <0…

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Q: 5. Evaluate the integral JI. dA, R = [-2, 4] × [1, 3]. R

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Q: 1. (a) Let 5 -6 -6 B = |-1 4 2 3 -6 -4 Find oigon nigon un imil to D

A: To finding the eigenvalues and eigenvectors to the given matrix we use Gaussian elimination method.…

Q: 19. Express in a + bi form: (5 – 6yi)(4 + 8i) – [(2 – 3yi) – 3(1+ 6i)]

A: