8.3) Solve the equation z – 8 = 0, using De Moivre's theorem, giving the results as complex numbers. -

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: Consider the complex number 2 + i. Working in exponential form, investigate the effect of…

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Q: 4. Consider the complex equation - -32i. (a) Solve for all values of z.

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Q: 4. Find the modulus, Iz], and argument, 0 , of the complex number. -1 + i

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Q: Prove the Euler's formula: e" = cos0+ jsin 0 using the definition of the complex number

A: We have to prove that ejθ=cosθ+jsinθ. We know that ex=1+x+x22!+x33!+x44!+.......... Now,…

Q: Q2\ If z = 1 + i, what is z3 + z? + z + 1 as a single complex number?

A: Use the following formulae, to simplify the given expression.…

Q: Given that x = 1-4i is one root of the quartic equation 2-82' + 54z - 152z +425 = 0, find the other…

A: We will use long division to factor the four degree polynomial into the product of two quadratic…

Q: Plot the point corresponding to, z =2(cos30ο +isin30ο ) in the complex plane, and write an…

A: z = 2(cos 30° + i sin 30°)

Q: Use De Moivre's Formula to compute the three complex cubic roots of v3 + i. List your answers in the…

A: To find: complex cubic roots of given complex number.

Q: 5. Determine the roots of the complex numbers studied in class. z³ = 2 – 3i

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Q: (a) Use the formula zk = Vr(co e+ 27tk cos +i sin 0+277K to find the indicated roots of the complex…

A: Given Z=-322(-1+i) Zk=rncosθ+2πkn+sinθ+2πkn, 0≤θ<2π The trigonometric form of the complex number…

Q: 2. Solve the eight eighth roots of z = -2 – j2y3. (express roots in polar form)

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Q: Determine the four fourth roots of the complex number Z 3-4j, using De Moivre's theorem and express…

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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: Express the following complex number in the form x + iy respectively 3em/2

A: The given complex number is 3eiπ2. Write the given complex number in the standard form.…

Q: iv) in complex number z3. (7)? = |z|?. z

A: Complex number: A complex number has the form z=x+ιy, where x is called the real part and y is the…

Q: Find two complex numbers a + bi in which a 0 V17 and b 0 with a modulus of

A: topic - complex numbers

Q: 4. Let a be a complex number. Show that equation |z – al + |z +a| = 2 %3D admits a complex solution…

A: Given equation is |z-a|+|z+a|=2    (1) since a is complex number. let  z=x+iya=c+id

Q: Determine all complex solutions of the equation cos(z) = 2.

A: First represent cosz in exponential form then make quadratic equations to find solutions as in next…

Q: The complex conjugate of z is denoted as z̄. Solve z in equation: z-12=j(9-2z̄) giving the answer…

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Q: How may the following complex numbers be obtained from z = x + iy geometrically? Sketch. a. (-z) b.…

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Q: 3) The complex conjugate of z is denoted as z. Solve for z in the equation: 27(1 – j2), j 2- j3 1 +…

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Q: 2) The complex conjugate of z is denoted as z. Solve for z in the equation: Z – 12 = j(9 – 27) -…

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Q: 5. Determine the roots of the complex numbers studied in class. z3 = 2 – 3i %3|

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Q: Find the modulus (r) for the given complex number z = -6.3 + 8.2i.

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Q: How may the following complex numbers be obtained fromz = x + iy geometrically? Sketch. -z

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Q: B. Express in the form of a complex number x+iy 1. (2+34)+(-4+5i) 2. (3+ 2i)(3 – 31) 8+4i 3. 1-i

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Q: Solve the following  complex numbers in their rectangular form.  Where in Z1=2+4i, Z2 = 5e2i and Z3…

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Q: V3 1 (- 2

A: given(-32-12i)20 =(-1)20(32+12i)20 =(cos30+isin30)20 The formula of Demoivre's theorem is cos θ+isin…

Q: 4. This question has two parts. a) Compute the value of eln 2020+3xi/4 b) Find all complex numbers z…

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Q: 2. Find the Cartesian (x+iy) form of this complex number 3π 3π Z=4(cos + 2 isin 377)

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Q: 3) The complex conjugate of z is denoted as z. Solve for z in the equation: 27(1- j2) j 2-j3 1+ j2…

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Q: 2) The complex conjugate of z is denoted as z. Solve for z in the equation: z – 12 = j(9 – 2z)…

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Q: - Find the roots of the following complex nos. 3. %3D 3. Z =\ Z-27-0 %3D 3. |

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Q: Use DeMoivre's Theorem to find the 5th power of the complex number z = 2(cos 24° + i sin 24°).…

A: Solution :  Given that  Z = 2 (cos 24° + i sin 24°)

Q: find all complex values z satisfying the given equation cosh z = 2+3i

A: We know that - cosh x= (ex-e-x)/2 using it, the explanation is given below-

Q: Given that the complex number z is such that 22. Iz| = 4 and arg z = - then z =

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Q: :If z is a complex number satisfying the equation (-1)" = =", where n e N. Show that the locus of z…

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Q: (a) Express the complex number 2 + (2/3)i in the form r(cos 0 +i sin 0), -n < 0 <T. (b) Solve the…

A: Complex numbers are those numbers which is represented in the form of a + ib where is a real part…

Q: logz, (z3":) \0gz;

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Q: 2) The complex conjugate of z is denoted as z. Solve for z in the equation: z – 12 = j(9 – 2z) %3D…

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Q: 5. Determine all roots of the equation z4 – 9 = 0, and sketch them on the complex plane.

A: Given equation:                            z4-9=0

Q: Let z be a complex number .then e^(3z+i) is bounded over the set of complex numbers?

A: Let z is a complex number. Let f(z) = e(3z+i)  Let us assume that f(z) is bounded. Hence f(z) is an…

Q: ) The complex conjugate of z is denoted as z. Solve for z in the equation: 27(1 – j2) 2 - j3 5z 1+j2…

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Q: a) Express the complex number Z 10 (cos 150° +i sin 150° ) in rectangular and exponential form. V3…

A: (a)z=10(cos 150°+i sin 150)where r=10and θ=150*π180=56πExponential form is z=reiθz=10ei5π/6In…

Q: Q2\ If z = 1 + i, what is z + z? + z + 1 as a single complex number?

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Q: 4 9) Use Euler's formula to compute (--i. Express as a 2 2 complex number of the form a + bi.

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Q: 2(2) A complex number z = v2(cos+i sin ), find In ( Z.Z' )

A: z = √2(cosπ/4 + i sinπ/4)  z = √2(1/√2 +i/√2 )  z = (1+i)  z* = 1-i

Q: Exer. 21–46: Express the complex number in trigonometric form with 0 so<27. 25 2V3 + 2i 4 cis 37 29…

A: Given: The complex number 25) 23+2i. 29) -20i 34) -5.

Q: If a is a complex number such th equation az²+z+1=0 has one p

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.