Quèstion 6 Write z - 5- iz + 3i in a + bi form (x+y-5) + i(3-x +y) (x+y + 5)+ i(3 - x- y) (r- y- 5)+ i(3 –x- ) (x + y – 5) + i(3 –x -y)

Q: -1 -2 -3 2 3 5 6 1 Let A = 1 Compute A-1 using adjoint.

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Q: 4. Find all the roots for the equation, z5 = -3+ 3i in the form of a + bi. %3D

A: Concept introduction: De Moivre's formula The nth roots of a complex number a=rcosθ+isinθ are…

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: 5. Expressed in a + bi form, is equivalent to 3+1 1)- 2- 3)- i 4) 15- 5i -5i

A: Given : 53+i

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: [3-5i 2+ 4i] ++ 7i 1+8il find the conjugate of its transpose. Given A =

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Q: Show that 1/2 1+ 72 2 1. i%3D1 Xi n-1

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Q: Given that 2z-iz = 3(3-5/)where z denotes the complex conjugate of z a+ib. The value of a is and and…

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Q: Express 3-i/2+3i  in the form a + bi

A: Rationalise i^2=-1

Q: (b) Use Gaussian Elimination to solve: X1 - 2 x2 - X3 + 3x4 = 4 2x1 + X2 + X3- 4x4 3 3x1 - X2 2x3 +…

A: Apply gaussian Elimination to the augmented matrix and back substituting.

Q: Z1 = 3 + 4i Z2 = 5 – 31i Z3 = 8+ 2i Reduce to a + bi form 1. z1 + Z2 + Z3 2. Z1 * Z2 Z1 * Z2 Z3 3. –…

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Q: Exercise 1: Given that Z1 = 1– 2i, z, =-3+ 4i. Find Z1 and express it in a+bi Z2 form. 2: 1 Given z1…

A: we are given some complex numbers. as per our guidelines we can answer only first question or first…

Q: Exercise 6.4.1 Show that 1 +i,2+i are the only two roots to P(x) =x² – (3+2i).x+(1+3i) Hence complex…

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Q: 1- -1 -1 provethat K K_2I %3D

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Q: 1. If z = 1+ iy is a complex number. Find the principal argument and exponential form of z =

A: Given that z=x+iy is a complex number. where, Real Part = x and Imaginary Part= y Now,  z=i1+i…

Q: 6. Show that when ) < |z - 1| < 2, e - ( - 3) (2 - 1)(2 – 3) Complex Analysis n=0 (z-1)" 2+2 2(z-1)

A: We use Laurent series to prove given example.

Q: Find the conjugate of z if z = (1+ 3i) + (-2-4i) O -1+i O-1-i 01+i 01- i

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Q: 223+-5 -j Suppose z1 = 2+ j, 2 = 3- j2 and z3 = -+j Evaluate %3D 2-a+3-j

A: Substitute the given values of z1,  z2, z3 in the given expression and simplify.…

Q: 3 1] -1 2 If A = show that A² - 5A + 7I = 0. %3D %3D

A: Introduction :We have given a matrix A. And we have asked to show

Q: Let z1 = 2 – 3i , z2 = -1+ 5i , and z3 = -6- 2i. %3D 2123 Evaluate: z2 . Express final answer in…

A: The given complex numbers are z1=2-3i, z2=-1+5i and z3=-6-2i The addition  and subtraction of two…

Q: Express z =1/(2+5i) in the form a+ib.

A: Given:

Q: 6. Show that when 0 < |-1|< 2, (z-1)(z-3) Complex Analysis (z-1)" 2+2 =-36 z−3) ² A=0 2(z-1)

A: We use substitution and Laurent series expansion.

Q: 20. Express (-3 - 8i) (-7 + 2i) + (7 - 3i) in simplest a + bi form

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

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Q: If Z=1/√2+i/√2 then the value of conjugate of Z^5= ؟

A: We have z=12+i2 Now, a complex number z=a+ib can be written in polar form as: z=a+ib=rcosθ+i sinθ…

Q: Find the complex representation of H((-1,3),3)a, where a is the translation through i = (3, –4)

A: Given: The complex representation of H(-1,3),3), Where through

Q: (2+3i )( 3+2i) Express in the form of a + bi. 4-3i

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Q: Rewrite -5sin(15x)-3cos(15x)  in the form Asin(Bx+C) where 0<c<2pi

A: Given data is: -5sin15x-3cos15x To write -5sin15x-3cos15x in the form of AsinBx+C

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: Express the product (3 + 2i)(1 – 3i) in a + bi form.

A: Here we will use the product of two complex numbers

Q: = 2i-3j +3 and -3i +x +2î are orthogonal, then

A: If u and v are Orthogonal then determine x value

Q: 5) A = 2i+3j+5k and B = 3i+6k, Determine the direct cosine of (AXB). %3D %3D

A: Finding the cross product of vector A and vector B  i×i=j×j=k×k=0, i×j=k, j×k=i, k×i=j, j×i=-k,…

Q: 4) Let z= 1+ V 3 i. Compute all values of z". Exponential notation is sufficient. Sketch these…

A: Given: z=1+3i

Q: 2 The roots of the equation X + 2x+ 5 0 a 1) -3 and 1 2) -1, only 3)-1+ 2i and 1- 2i 4) -1+4i and…

A: given equation :

Q: 7. Derive the harmonic conjugate of u(x, y) = x² - y2. (2 2

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Q: Example 3: Find real numbers x, y and z such that xi + 2yj – zk + 3i – j = 4i + 3k

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Q: 6- The Conjugate of 4+3i =-(4+3i)

A: The objective is to find the conjugate of (4+3i).

Q: 6. Suppose that 1+2i is a complex zero of f(z)= z* – 2z³ +6z² – 2z +5, for all z eC then the other…

A: Given: The complex function is fz=z4-2z3+6z2-2z+5 for all z∈C. One of the root is (1+2i)

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

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Q: 10. Use i = (0, 1) and y = (y, 0) to verify that -(iy) = (-i)y. Thus show that the addi- tive…

A: i=0 ,1y=y ,0 verify :   -iy = -iy  To find additive inverse of z=x+iy

Q: Find all complex solutions of 3.x-2.r+ 1 = 0. %3D

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Q: 9. Express (3 – 2i)² in a + bi form.

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Q: Example:4 Verify Cayley-Hamilton theorem. Hence find 2 -1 A-1, A -1 2 -1 1 -1 2

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Q: Find harmonic conjugate of u=2xy+3y

A: The objective is to find the harmonic conjugate of u=2xy+3y If v is the harmonic conjugate of u then…

Q: 1 3-Find zeros and poles of x(z): %3D (1-금-7)(1-27-3)

A: We have to find Zeros and Poles to given function x(z).

Q: Question 3) Prove that if x and y are natural numbers then the equation 3x +2=y has no solution for…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…