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4 Consider two vectors a and b in the Cartesian xy plane, both of which originate at ( 1,4) Suppose that vector a terminates at (0,5) and vector b terminates at (0,3) If we add these vectors and express the result in standard form, what do we get (a) a + b = (31/2,51/2) (b) a + b = ( 1,1) (c) a + b = ( 1,9) (d) a + b = (2,0) (e) a + b = (0,21/2) 5 Suppose that we have a complex number c in polar form, such that c = r cos q + j (r sin q) where r is the real-number polar vector magnitude and q is the real-number polar vector angle Also suppose that n is an integer DeMoivre s theorem tells us that the nth power of this complex number is equal to (a) rn cos (n q) + j [rn sin (n q)] (b) (r + n) cos (n q) + j [(r + n) sin (n q)] (c) r n cos (n q) + j [r n sin (n q)] (d) rn cos (q) + j [rn sin (q)] (e) r n cos (q) + j [r n sin (q)] 6 The point (x,y) = (0, 3) is (a) in the first quadrant of the Cartesian plane (b) in the second quadrant of the Cartesian plane (c) in the third quadrant of the Cartesian plane (d) in the fourth quadrant of the Cartesian plane (e) not in any quadrant of the Cartesian plane 7 Under what conditions is the cross product of two nonzero polar-plane vectors equal to the zero vector (a) When the two vectors point in the same direction (b) When the two vectors point in opposite directions (c) When the two vectors have equal magnitude (d) When the two vectors are mutually perpendicular (e) Under more than one of the above conditions (a), (b), (c), and (d) 8 In polar coordinates, which, if any, of the following equations can represent the dashed line graphed in Fig FE-1 (a) q = 0 (b) q = r (c) q = 2r p /3 (d) q = 2p /3 (e) We can t say without knowing the size of each radial increment
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Figure FE-1
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Illustration for Question 8
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9 The intersection of the sets of real and imaginary numbers is the set (a) { j } (b) {1} (c) { j } (d) { 1} (e) {0} 10 Which, if any, of these geometric figures would be the graph of a true function of x if drawn on the Cartesian xy plane, and the graph of a true function of q if drawn on the polar coordinate plane (a) A circle centered at the origin (b) A straight line passing through the origin (c) A straight line parallel to the Cartesian x axis, but not passing through the origin (d) A straight line parallel to the Cartesian y axis, but not passing through the origin (e) None of the above 11 When we multiply a polar vector by a negative scalar, what restrictions, if any, should we put on the direction angle of the product (a) It should be nonnegative, but less than p /2 (b) It should be nonnegative, but less than p (c) It should be nonnegative, but less than 2p (d) It should be at least p, but less than p (e) We don t have to take any precautions concerning the direction angle
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12 If we quadruple the value of each coordinate of a point in Cartesian three-space, then its distance from the origin increases by a factor of (a) 2 (b) 4 (c) 8 (d) 16 (e) 64 13 If we graph the unit circle in the Cartesian xy plane and then pick a point (x0,y0) on that circle such that x0 0, then 1/x0 is equal to (a) the cosine of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (b) the Arccosine of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (c) the tangent of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (d) the Arctangent of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) (e) the secant of the counterclockwise angle between the positive x axis and a ray going out from the origin through (x0,y0) 14 Suppose that we have a vector in Cartesian xyz space whose originating point is (1, 1,3) and whose terminating point is (7, 4, 3) What is the ordered triple representing the Cartesian standard form of this vector (a) We need more information to answer this question (b) (8, 5,0) (c) (6, 3, 6) (d) (7,4, 9) (e) (4, 5/2,0) 15 What are the Cartesian xy plane coordinates of the point P plotted in Fig FE-2 Assume that each concentric-circle radial division represents 2 units (a) (21/2,6) (b) (21/2, 6) (c) ( 21/2,6) (d) ( 21/2, 6) (e) None of the above 16 What is the range of values for x in the interval ( 4,0] (a) 4 < x < 0 (b) 4 x < 0 (c) 4 < x 0
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