Name                                     MATH 107 PRACTICE MIDTERM II Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.   Printable Key PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true explain why. A.     (15 Points)  If A, B, and C are points, v is the vector from A to B, w is the vector from B to C, and v x w  =  0 , then A, B and C are collinear. Solution B.      (15 Points) If x  =  x(t),  y  =  y(t) are parametric equations of a line then dx/dt  is a constant.   Solution   PROBLEM 2 (21 Points) Consider the surface x2 + z2 - e2y  =  0 .  This surface is formed by revolving a generating curve about an axis.  Find an equation of this generating curve and state the axis of revolution. Solution   PROBLEM 3  (21 Points)  Use vectors to find the equation of the line that passes through the point (2,3,4) and is perpendicular to the plane 5x - 4y + 2z = 7. Solution     PROBLEM 4  (21 Points)  Find all points (if any) of horizontal and vertical tangency.  Make sure to present your answer by listing the points not just the values of q.         x  =  cos q        y  =  2sin(2q) PROBLEM 5 (21 Points)  Determine the area of the first quadrant loop of  r  =  3sin(2q) PROBLEM 6  (21 Points)  Show that the polar equation for the hyperbola            x2          y2                    -            =  1                      a2          b2      is                         -b2             r2  =                                             1 - e2 cos2 q  given that                                b2             e2  =  1 +                                            a2  PROBLEM 7  (21 Points)  Use vectors to determine if the triangle with vertices (1,0,1), (2,1,0), (0,0,4) is a right triangle.   PROBLEM 8  (21 Points)  Find parametric equations for the a particle moves along the line through (1,4,2) and (3,5,7) such that it is at the point (1,4,2) when t = 0 is at the point (3,5,7) when t = 2 and is speeding up as time progresses