(6 points) Determine if the following is possible for some real number z. Explain why using...
C1= 5
C2= 6
A1 Rewrite the following sentence using variables and logical or mathematical symbols. Limit yourself to as few English words as possible, but it must be an equivalent statement. "e to the power of some integer times the square root of minus 1 is a complex number that is not real”. A2 Let S := {kt, ..., kg;} be a set of containing certain possibly equal complex numbers, and let T be the set of integers lying...
Exercise 12: Residues and real integrals (a) [6+4 points) Compute the residues for all isolated singularities of the following functions (i) f(2)== (2-) tan(2), (i) 9(2):= z2 sin () (b) (4+6+5 points) Compute (using the Residue theorem) (1) cos(72) ( d, A3 := {z € C:<3), 243 := {Z EC: | = 3}, 34, (2-1)(2 + 2)2(2-4) : 43 € C:21 <3}, po 12 To (x2 + 4)2 da, 24 2 + 4 cosat. J 5 + 4 sin(t)
Let z = t(1 + i) be a complex number, where t is some real parameter. Using the polar form of a complex number, find Z6. teri/4 (V2t) erila (2t) 26. 6i/4 teori/4 (V2)%e6ni14 (12)etail4
(10 points) First, determine the quadrant for 2; then find x, y, and r; and finally, give all six trigonometric ratios for a given the following information: csc(0) = 1 and cos(0) < 0 e lives in quadrant • X= • y = 1. sin(O) = 2. cos(0) = 3. tan(O) = 4. sec(0) = 5. csc(0) = 6. cot(0) =
4) 3s points 11. Given the unity feedback system of Figure P9.1 with G(s) K (s + 6) do the following: [Section: 9.3 a. Sketch the root locus. b) Using the operating point of -3.2+j2.38 find the gain K. c) if the system is to be cascade-compensated so that T, -1 sec, find the compensator compensator zero is at -45. pole if the d) Sketch the root locus for the new compensated system.
4) 3s points 11. Given the unity...
mints) Give the following examples if possible. If it is not possible, explain why. You do not have ove the properties hold 2 points) A list of 4 vectors in Rồ that is linearly independent. (3 points) A subspace W of R* that is complementary to V = | 22 23=0, 1334=0 =) (4 points) A linear map T:R?R? with image(T) - ker(T) = span ()
14. Let so(z) = 1+c(x+1)3 for-1 < z < 0 for some real constant c. Determine s1() for 0 S 1 such that so(x) s1() for 0S for-1 < x < 0; s(x) = 1 is a natural cubic spline on [-1, 1] with nodes at 1,0,1. How must c be chosen if one wants s(1) =-1?
(15 points) 6. For the following structure, please determine the number of infrared and Raman bands for their CO groups. (Please note: Modes with translational symmetry (x, y, z) will be infrared active while modes with xy, xz, yz, xº, ya or zº symmetry are Raman active.) (Please show detailed steps for full credit!) O=C—M-CEO
Economics: 1) Why is it possible to change real economic factors in the short run simply by printing and distributing more money? 2) Explain why a stable 5% inflation rate can be preferable to one that averages 4% but varies between 1-7% regularly. 3) Explain the difference between active and passive monetary policy. 4) Suppose the economy is in long-run equilibrium, with real GDP at $16 trillion and the unemployment rate at 5%, Now assume that the central bank unexpectedly...
6. For a positive real number z, the difference 1.-z- is called the fractional part of r. Given arbitrary positive real numbers a and b, state a condition in terms of the fractional parts of and that is necessary and sufficient for la + bl = lal + Ibl Prove that this equation is true if and only if your condition holds. 7. Evaluate 10002+4 23k+5 k- 2 algebraically and simplify as much as possible. You ust show all steps....