Math 2412 - Final Exam Review Problems 36. Write 7 [cos (? ) + isin (3)] in rectangular form. 37. Given 21 = 3 [cos () + isin ()] and z2 = 8 [cos () + isin ()], find z1 . 22. Write your answer in polar form. 38. Given 21 = 72 [cos (150) + isin (15)] and z2 = 8 [cos (50) + i sin (50)], find 2. Write your answer in polar form. A 9 [cos(30) + i sin(39) ] B 9 [cos(150) sin(50) + isin(15) cos(59) ] 9[cos(10) + i sin(109)] D 9 [cos(30) - isin(39) ] 9 [cos(20) + i sin(209) ] None of A through E. 39. Use DeMoivre's Theorem to simplify (-1 - i)4. 40. Find all the complex roots of w = 16 [cos (*" ) + isin (4 )]; write your answers in rectangular form. 41. Solve x3 - 4+ 4iv3 =0. Write your answers in polar form. 8 [cos(100) + isin(1009)] B 2 [cos(300) + i sin(3009) ] 8 [cos(220) + i sin(2209)] 8 [cos(340) + i sin(3409) ] 8 [cos(300) + isin(3009)] D 2 [cos(100) + i sin(1009) ] 2 [cos(220) + i sin(2209) ] 2 [cos(340) + i sin(3409) ] 42. Solve x3 = 8i. 43. Use Polar Graph paper to graph r = 1 + 2 sin(0). 44. Let u = 2i + v2j and v = (1 + v2)i + (1 - V2)j. a) Find the exact value of lull. b) Find the exact value of lull. c) Find u . v. d) Find the angle between u and v. e) Find the unit vector in the direction of u. f) Find the projection of u onto v. g) Find 2u - 3v. 45. Which of the following expressions would be the correct form for the partial fraction decomposition of 1 + 15x -11, (x - 6)(1 + 4)2 A B A B C r -6 (x+4)2 B x - 6 (x+4) (x + 4)2 A Br + C A B Dr + E C - x - 6 (x +4)2 T-6 (1+4) (1+4)2 A Br +C Dr +E E - r - 6 (x +4) (x+4)2 None of A through B. 46. Complete the partial fraction decomposition of - 4x -5 12 - 5x - 24 - Created/Revised the Math Department 4Math 2412 - Final Exam Review Problems 47. Write the partial fraction decomposition of 8x -12, (z + 4)2 48. Write the partial fraction decomposition of 612 - 2+1 (2 + 1)(22+3) 19. Determine whether each statement is TRUE (T) or FALSE (F). T F The foci of an ellipse are located along the major axis. T F The foci of a hyperbola must satisfy the equation of the hyperbola. T F The vertices of a hyperbola are located along the transverse axis. T F The center of the circle r2 + y' - 6y + 4 =0 is on the y-axis. T F The major axis is the smaller of the two axes of symmetry of an ellipse. T F A curve in the ry-plane can have more than one set of parametric equations that describe it. T F The parametric equations x = 2 tan(t) y = 3sec(t) will describe a parabola. T F The parametric equations x = 12 y = t6 have the same graph as ly = 19 . T F If the parametric curve ex = f(t) ly = g(t) satisfies g(2) = 8, and f(2) = -3, then (-3, 8) is on its graph. 50. Consider the equation of the parabola: 2y = -y? - 8 + 11. (a) Complete the square to place the equation in standard form. (b) State the coordinates of the verter. (c) State the coordinates of the focus. (d) State the equations of the directrix and aris of symmetry. 51. Find the asymptotes of _ 25 9 = 1 . 52. Consider the ellipse with center at (3, -2), focus at (3, 1), and vertex at (3, 3). a) Plot the given points and find the coordinates of the remaining vertices and focus. b) Write the equation of the ellipse in standard form. 53. Identify the conic 25(x - 3)2 +9(y - 2)2 = 225. 54. Identify the center of the parabola (y + 5)? = 8x - 4. 55. Consider the equation of the hyperbola 4x2 - 8x - 68 = 9g2 + 36y. (a) Complete the square to place the equation in standard form. (b) Find the vertices. (c) Find the foci. (d) State the equation of the transverse aris (i.e., major aris) of the hyperbola. 56. Consider the conic: 5x2 - 4ry + 8y? = 36. Determine a positive angle of rotation (in radians) needed to eliminate the ry term. Do not perform the rotation. 57. Given x =1-43 y = 4-16 , find the rectangular equation. Sketch the curve and indicate the orientation. 58. Identify ry = 2 without applying a rotation of axes