Homework Answers
1) For nth roots, we will get n roots
2) Magnitude of each root is
3) Angle between each root is 
4) Angle of the first root is 
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6. Sketch the roots. (Approximate) yi To find the nth roots of z rcise: 1. We will getroots 2. The magnitude of the roots is 3. The angle between the roots on the complex plane is 4. The angle of the...
fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n...
fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n for which 2. The roots off(x) = x"-1 are the n complex numbers Cn and are ged(n, k) 1. It is easy to see that Q(C) is the splitting field of zn - 1. (a) For each n 3,... ,8, sketch the nth roots of unity in the complex plane. Use a different set of axes for each n. Next to each root, write...
1. Sketch the region in the complex plane that contains the elements of {Z – 3+i:ze...
1. Sketch the region in the complex plane that contains the elements of {Z – 3+i:ze C,1<\2-11 <2} n {z EC: Im(2) >0}. Justify your answer.
Problem 3 (30 points) Given the following unity feedback system we wish to sketch the root...
Problem 3 (30 points) Given the following unity feedback system we wish to sketch the root locus of KG() = -16+-10) for 0 < K<0. (a) Indicate the following on the above s-plane (show all your works): 1) (2 points) Finite poles and zeros of G(3) ii) (2 points) real axis section of root locus i.e. real axis roots) m) (4 points) departure angles and amival angles if any iv) (4 points) Approximate breakaway and break-in points if any. v)...
1(a) Find the square roots of the complex number z -3 + j4, expressing your answer in the form a + jb. Hence find the roots for the quadratic equation: x2-x(1- 0 giving your answer in the form p+ q w...
1(a) Find the square roots of the complex number z -3 + j4, expressing your answer in the form a + jb. Hence find the roots for the quadratic equation: x2-x(1- 0 giving your answer in the form p+ q where p is a real number and q is a complex number. I7 marks] (b) Express: 3 + in the form ω-reje (r> 0, 0 which o is real and positive. θ < 2π). Hence find the smallest value of...
Consider the function xtan x -1 defined over all x. Sketch the function to get an idea of the roots 1 find the first couple of roots using bisection to a precision of machine epsilon 2 after straddli...
Consider the function xtan x -1 defined over all x. Sketch the function to get an idea of the roots 1 find the first couple of roots using bisection to a precision of machine epsilon 2 after straddling a root, find its value using the Newton-Raphson method. 3 after straddling a root, find its value using the secant method 4 after straddling a root, find its value using the false position method. Determine the order of the methods and comment...
1) Using Matlab, find all real and complex roots of the following polynomial equation: (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)=8 2)...
1) Using Matlab, find all real and complex roots of the following polynomial equation: (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)=8 2) Using Matlab, find the root for the following system of equations. Both x and y are positive. a: (x^2)cos(y)=1 b: e^(-4x)+1
3. Find the polar form of each given complex number and sketch its position in the...
3. Find the polar form of each given complex number and sketch its position in the complex plane. a) 2-4 b) z 4i c) 2-1-i
Ks+8-0. For this system. 6. A negative feedback system has characteristic equation 1+ s2 +2s +2 (a) Sketch the root-locus, marking all important points, numerical values, incl. the angle of departure...
Ks+8-0. For this system. 6. A negative feedback system has characteristic equation 1+ s2 +2s +2 (a) Sketch the root-locus, marking all important points, numerical values, incl. the angle of departure (possibly in terms of tan(x (b) Find the gain when the roots are both equal and find these 2 equal roots. 6 pts) 4 pts)
Ks+8-0. For this system. 6. A negative feedback system has characteristic equation 1+ s2 +2s +2 (a) Sketch the root-locus, marking all important points,...
1) find all value of i^i, and show that they are all real 2) Find all values of log(-1-i) 3) find a) the cube roots of -1 b) the sixth root of i c) the cube roots of 1-i 4) Find (d/dz) i^z
1) find all value of i^i, and show that they are all real 2) Find all values of log(-1-i) 3) find a) the cube roots of -1 b) the sixth root of i c) the cube roots of 1-i 4) Find (d/dz) i^z
(2 points) Here are several points on the complex plane: The red point represents the complex...
(2 points) Here are several points on the complex plane: The red point represents the complex number zı = and the blue point represents the complex number Z2 = The "modulus" of a complex number z = x+iy, written [z], is the distance of that number from the origin: z) = x2 + y2. Find the modulus of zi. |zıl = 61^(1/2) We can also write a complex number z in polar coordinates (r, 6). The angle is sometimes called...
fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n for which 2. The roots off(x) = x"-1 are the n complex numbers Cn and are ged(n, k) 1. It is easy to see that Q(C) is the splitting field of zn - 1. (a) For each n 3,... ,8, sketch the nth roots of unity in the complex plane. Use a different set of axes for each n. Next to each root, write...
1. Sketch the region in the complex plane that contains the elements of {Z – 3+i:ze C,1<\2-11 <2} n {z EC: Im(2) >0}. Justify your answer.
Problem 3 (30 points) Given the following unity feedback system we wish to sketch the root locus of KG() = -16+-10) for 0 < K<0. (a) Indicate the following on the above s-plane (show all your works): 1) (2 points) Finite poles and zeros of G(3) ii) (2 points) real axis section of root locus i.e. real axis roots) m) (4 points) departure angles and amival angles if any iv) (4 points) Approximate breakaway and break-in points if any. v)...
1(a) Find the square roots of the complex number z -3 + j4, expressing your answer in the form a + jb. Hence find the roots for the quadratic equation: x2-x(1- 0 giving your answer in the form p+ q where p is a real number and q is a complex number. I7 marks] (b) Express: 3 + in the form ω-reje (r> 0, 0 which o is real and positive. θ < 2π). Hence find the smallest value of...
Consider the function xtan x -1 defined over all x. Sketch the function to get an idea of the roots 1 find the first couple of roots using bisection to a precision of machine epsilon 2 after straddling a root, find its value using the Newton-Raphson method. 3 after straddling a root, find its value using the secant method 4 after straddling a root, find its value using the false position method. Determine the order of the methods and comment...
3. Find the polar form of each given complex number and sketch its position in the complex plane. a) 2-4 b) z 4i c) 2-1-i
Ks+8-0. For this system. 6. A negative feedback system has characteristic equation 1+ s2 +2s +2 (a) Sketch the root-locus, marking all important points, numerical values, incl. the angle of departure (possibly in terms of tan(x (b) Find the gain when the roots are both equal and find these 2 equal roots. 6 pts) 4 pts)
Ks+8-0. For this system. 6. A negative feedback system has characteristic equation 1+ s2 +2s +2 (a) Sketch the root-locus, marking all important points,...
(2 points) Here are several points on the complex plane: The red point represents the complex number zı = and the blue point represents the complex number Z2 = The "modulus" of a complex number z = x+iy, written [z], is the distance of that number from the origin: z) = x2 + y2. Find the modulus of zi. |zıl = 61^(1/2) We can also write a complex number z in polar coordinates (r, 6). The angle is sometimes called...
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