8. Let w cos(2π/5) + isin(2π/5). Here we describe how to express w in terms of...
detailed solution for this one ????? 11. (a) Gi) If w=z+z-' prove that (i) z2 + z 2 = w2 -2 ; 24 +2° + z²+z+1 = z2 (W2 + w+1) = (z? +[1+V5]+1)(22 +[1–V5]+1). (b) Show that the roots of 24 +2+z2+z+1=0 are the four non-real roots of z' =1. (c) Deduce that cos 72° = +(15 – 1) and cos 36° = (15+1).
4. (15 marks) Consider the following equation: where i denotes the complex number satisfying i2--1 (a) Rewrite the number -i in the exponential form and transform equation (5) into (b) Solve (6) to get the five solutions wo, ..., wa and draw them on the Argand diagramme (c) Show that wo··· , ua are the eigenvalues of the following real-valued matrix 0 0 0 0 0 cos(2m/5) A-10 -sin2(2π/5) 0 0 cos(2π/5) 0 0 0 2cos(4π/5) 2 Hint: compute the...
Please answer A, B, and C in full 2. Let f() € F[2] be a separable polynomial with roots {u1, ..., Un} contained in some splitting field K of f(x) over F. Define A= || (ui-u) = (ui - U2) (u - u3) ...(ui-un)(uz - u3) ..(un-1 - Un) EK. (a) (15 points) Consider GalpK < Sn by looking at its action on the set of roots for f(x). Show that if Te Galo K is a transposition then (A)...
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
1500 W 60 V 20 W 128 W 22 W Fuse Figure 2: Picture of the setup in problem 5. 4.) Which configuration of charge has more electric potential energy? (a.) an equilateral triangle (all sides the same length 1) with positive charge +q at each corner or (b.) a square (all sides have length l) where three corners are occupied by +q charges and the final is occupied by charge Show all work. (pun sort of intended "hint hint*)...
1 10 onvelge a636lutely, converges conditionally, or diverges. Justify your answer, including naming the convergence test you use. (1n(b) n7/3-4 (2k)! n-2 k-0 (-1)k 2k 4. (a) (10) Let* Find a power series for h(), and find the radius of convergence Ri for h'(x). Find the smallest reasonable positive integer n so that - (b) (10) Let A- differs from A by less than 0.01. Give reasons. 5. (a) (10) Let g(x) sin z. Write down the Taylor series for...
1500 W 128 W 822 W Fuse Figure 2: Picture of the setup in problem 5 4.) Which configuration of charge has more electric potential energy? (a.) an equilateral triangle (all sides the same length 1) with positive charge +g at each corner or (b.) a square (all sides have length 1) where three corners are occupied by +q charges and the final is occupied by charge -q? Show all work. (pun sort of intended "hint hint) 5.) A circuit...
Problem 25 please -Sesin(2x)-9ecos(2x). 21. W = Span(B), where Br(x2e-4x , xe®, e-4x); f(x)--5x2r" + 2e-4-1e 22. W= Span(B),where B= ({x25, x5*, 5x)); f(x)--4x2 5x+9s5x-2(5x). 3 W Span(B), where B (Exsin(2x), xcos(2x), sin(2x), cos(2x)y): f(x) = 4x sin(2x) + 9x cos(20-5 sin(2x) + 8 cos(2x). 24, In Exercise 21 of Section 3.6, we constructed the matrix [D, of the derivative operator D on W- Span(B), where B e sin(bx), e" cos(bx)): Dls a a. Find [D 1g and [D'lg: Observe...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
By 5. (a) Verify that y = {24 sin x is a solution to the differential equation dx2 dy + 5y = 0. dc [10 marks) (b) Differentiate the following functions with respect to c: (i) In(1 + sin? 2) (ii) * 2x3 - 4 - 8 dc. (c) Evaluate the integral / 272 * +432 – 4.7" [15 marks] [25 marks] 6. (a) let f: R+R be a function defined by f(x) 3 + 4 if : 51 ax+b...