Math 2433 Online, Section 17180, Summer 2012 Multiple Choice Homework Week 1

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1. Find an equation of a sphere that has the line segment joining (0,4,2) and (6,0,2) as a

diameter. a. 2 2 2( 3) ( 2) ( 2) 13x y z + + = b. 2 2 2( 2) ( 3) ( 2) 13x y z + + = c. 2 2 2( 3) ( 1) ( 2) 12x y z+ + + = d. 2 2 2( 3) ( 1) ( 2) 12x y z + + = e. None of the above

2. Which of the following describes a sphere with radius of 2 and center at the origin together with its interior?

a. 2 2 2{( , , ) : 4}x y z x y z = + + b. 2 2 2{( , , ) : 2}x y z x y z = + + c. 2 2 2{( , , ) : 4}x y z x y z = + + d. 2 2 2{( , , ) : 2}x y z x y z = + + e. None of the above

3. Calculate the norm of the vector: 2 2i j k+ a. b. 3 c. 2 d. 5 e. None of the above

4. Find the unit vector with direction angles: 2, ,3 4 3

a. 1 2 12 2 2i j k+

b. 1 3 12 2 2i j k+

c. 3 2 32 2 2i j k+

d. 1 2 12 2 2i j k+ +

e. None of the above

5. Which of the following vectors are NOT parallel?

22 23 3 62 2 4

a i j kb i j kc i j kd i j k

= += += += +

a. a and d b. b and c c. a and c d. c and d e. None of the above

6. Taking 2a i j k= + and b i j k= + + , find the component of a on b.

a. 13

b. 23

c. 12

d. 23

e. None of the above

7. Taking 2a i j k= + and b i j k= + + , find the projection of a on b.

a. 2 2 13 3 3i j k+ +

b. 2 2 23 3 3i j k+ +

c. 2 2 13 3 3i j k+

d. 1 3 12 2 2i j k+ +

e. None of the above 8. The distance between (1, 3, -10) and (2, 5, 4) is

a. 3 14 b. 10 2 c. 201 d. 7 e. None of the above

9. The midpoint of the line segment containing (2, 4, -1) and (6, -2, 2) is:

a. (4, 3,2)

b. 53, 3,2

c. 3 1, 3,2 2

d. 14,1,2

e. None of the above 10. An equation for the sphere centered at (2, -1 ,3) and passing through the point (4, 3, -1) is:

a. 2 2 2( 4) ( 3) ( 1) 6x y z + + + = b. 2 2 2 4 2 6 22x y z x y z+ + + = c. 2 2 2 4 2 6 32 0x y z x y z+ + + = d. 2 2 2( 4) ( 3) ( 1) 36x y z + + + = e. None of the above

11. If 2 2 2 4 10 6 2 0x y z x y z+ + + + = is an equation for a sphere, then its center and radius are:

a. ( 2,4, 3); 2 10r = b. (2, 5,3); 4r = c. (2, 5,3); 6r =

d. (2,5,3); 4 2r = e. None of the above

12. The points :(2, 2,1), :(1,1,3), :(2,0,5)A B C are the vertices of a right triangle. The radius of the sphere with center at the midpoint of the hypotenuse and passing through the vertex opposite the hypotenuse is:

a. 2 3 b. 5 c. 6

d. 72

e. None of the above 13. The points :(2, 1,3)A and :( 4,5, 1)B are the endpoints of a diameter of a sphere. An

equation of the sphere is: a. 2 2 2( 2) ( 1) ( 3) 22x y z + + + = b. 2 2 2( 1) ( 2) ( 1) 22x y z+ + + = c. 2 2 2( 1) ( 2) ( 1) 16x y z + + + + = d. 2 2 2( 2) ( 1) ( 3) 16x y z + + + = e. None of the above

14. The set of all points :( , , )P x y z that satisfy the inequality 2 2 2 9x y z+ + < is a. A sphere centered at the origin with radius 3. b. The exterior of a sphere centered at the origin with radius 3. c. The interior of a sphere centered at the origin with radius 3. d. The straight line segment connecting (0,0,0) and (3,3,3). e. None of the above

15. Given the points :(4, 2, 3)P and :(6, 1,2)Q . The vector from P to Q is:

a. PQ

= (2,1,1)

b. PQ

= (2,3,5)

c. PQ

= (2,3,5)

d. PQ

= (10,1,1) e. None of the above

16. Given the points :(2, 1,5)P and :( 2,0,3)Q . The magnitude of the vector from P to Q is:

a. PQ

= 21

b. PQ

= 5

c. PQ

= 9

d. PQ

= 65

e. None of the above 17. A unit vector in the direction of 2 3 3a i j k= + + is:

a. 1 3 32 4 4a

u i j k=

b. 1 3 32 4 4a

u i j k= +

c. 1 3 32 4 4a

u i j k= + +

d. 1 3 32 4 4a

u i j k= +

e. None of the above 18. If 3 6a i j k= + and 2b i j k= + have the same magnitude, then =

a. 11 b. 11 c. 17 d. 21 e. None of the above

19. If (3,2, 1)a = and (6, , 2)b = are parallel, then = a. -4 b. -2 c. 2 d. 4 e. None of the above

20. The vector with norm 2, and with direction opposite to the direction of 3 4a i j k= + is:

a. 2 ( 3 4 )26

i j k +

b. 2( 3 4 )i j k +

c. 2 ( 3 4 )26

i j k+

d. 2( 3 4 )i j k+ e. None of the above

21. Let (1, 2,1), (2, 3, 2)a b= = , and (2,0,4)c = . Then (b i c)a = a. ( 4,8, 4) b. (12,0,24) c. 4 d. Cannot be determined e. None of the above

22. The angle between (2, 1,1)a = and ( 1,2,1)b = is:

a. 3

b. 56

c. 34

d. 23

e. None of the above 23. Calculate the length of line segment AB given ( 5, 2,0)A and (6,0,3)B

a. 134 b. 114 c. 12 d. 14 e. None of the above

24. Find the direction angles of the vector 2i j k +

a. , ,3 3 4 = = =

b. 2 3, ,3 3 4 = = =

c. 3, ,3 3 4 = = =

d. 2, ,3 3 4 = = =

e. None of the above 25. Find all numbers x for which 2 5 2 6 4i j k i j kx x+ + +

a. 0,4x = b. 4x = c. 4x = d. 1x = e. None of the above

26. A direction vector for the line x 42 =

y + 35 = z 1 is

a. d = (4, -3, 1) b. d = (-2, 5, 0) c. d = (2, -5, -1) d. d = (-2, 5, 1) e. None of the above

27. Find the distance from the point P(2, -1, 4) to the line : x = 1+ 3t, y = 4 2t, z = 2 + t

a. 1014

b. 514

c. 50714

d. 11014

e. None of the above 28. An equation for the plane that passes through the point P(2, -1, 5) and is parallel to the plane

2x 4y + z = 10 is a. 2x 4y z = 3 b. 2x 4y + z = 13 c. 2x 4y + z = 3 d. 2x 4y + z = 4 e. None of the above

29. Symmetric equations for the line that passes through P(2, -1, 4) and is parallel to the line r(t) = (3,7,2)+ t(3,4,6) are:

a. x 23 =y +14 =

z 46

b. x = 2 + 3t, y = 1+ 4t, z = 4 6t

c. x 33 =y 74 =

z 26

d. x + 23 =y 14 =

z + 46

e. None of the above 30. The lines 1 : x = 2 + 4t, y = 3 2t, z = 3+ 6t and 2 : x = 6 2s, y = 1+ s, z = 3 3s are

a. Coincident b. Parallel but not coincident c. Have a unique point of intersection d. Skew e. None of the above

31. The lines 1 :

x + 24 =

y 32 =

z 36 and 2 : x = 1+ 3s, y = 1+ 2s, z = 4 + s are

a. Coincident b. Parallel but not coincident c. Have a unique point of intersection d. Skew e. None of the above

32. The lines 1 :r(t) = (3i + 4 j+ 7k)+ t(i + 3j+ 2k) and 2 :R(u) = (i j+ 4k)+ u(3i + 2 j+ k) are

a. Coincident b. Parallel but not coincident c. Have a unique point of intersection d. Skew e. None of the above

33. One of the problems 30-32 above have a unique point of intersection. That point of intersection is:

a. (1, -1, 0) b. (4, 1, 5) c. (4, -1, 9) d. (2, 1, 9) e. None of the above

34. The cosine of the angle between the lines 1 :

x + 23 =

y 22 =

z +12 and

2 : x = 2 3s, y = 4 + s, z = 1+ 4s is

a. 1926 17

b. 126 17

c. 1926 17

d. 1526 17

e. None of these 35. An equation for the plane which passes through the point P(-3, 2, -4) and is perpendicular to

the line x + 26 = y 3=z 52 is

a. 6(x + 3) 2(z + 4) = 0 b. 6x + y 2z = 0 c. 6(x 3)+ (y + 2) 2(z 4) = 0 d. 6(x + 3)+ (y 2) 2(z + 4) = 0 e. None of these

36. An equation for the plane which passes through the point P(2, -3, 1) and contains the line x 42 =

y 13 = z + 2 is

a. 13x + 8y 2z = 0 b. 5x 8y 14z = 4 c. 5x + 8y +14z = 0 d. 13x 8y + 2z = 4 e. None of these

37. The cosine of the angle between the planes 4x + y 2z = 3 and 2x + y + z = 1 is

a. 73 14

b. 1121 5

c. 719 6

d. 721 6

e. None of these

38. Given the point A(4,3, 2) and the plane P : 5x4y3z=24, let be the line through A and perpendicular to P. The point B at which intersects P is:

a. (9, -7, -1) b. (1, -1, -1) c. (-1, 1, 5) d. (1, 1, -1) e. None of these

39. Using your result from question 38, calculate the distance between A and B. a. 3 5 b. 22 c. 10 2 d. 5 2 e. None of these

40. The planes 2x 4y + 5z = 1 and x 2y 3z = 2 are a. Coincident b. Parallel c. Intersect in a straight line d. Skew e. None of the above

41. A direction vector for the line of intersection of the planes x y + 2z = 4 and 2x + 3y 4z = 6 is

a. d = i j+ 5k b. d = 2i + 8 j+ 5k c. d = 2i + 8 j+ k d. d = 10i 8 j 5k e. None of these

42. Continuing number 41, scalar parametric equations for the line of intersection are: a. x = 2t, y = 2 + 8t, z = 3+ 5t b. x = 10t, y = 2, z = 3+ 5t c. x = 2t, y = 2 + 8t, z = 3+ t d. x = 10t, y = 2 8t, z = 3 5t e. None of these

43. The dot product of two vectors is always orthogonal (perpendicular) to the plane through the two vectors.

a. True b. False

44. The vectors a and b lie flat on the page and point in the directions indicated below:

a b

Which of the following is true? a. a b = 0 b. a b = 0 c. a b < 0 d. a b > 0 e. None of these

45. Given points P(2, -1, 3), Q(4, 1, -1) and R(-3, 0, 5), find the area of PQR

a. 1392

b. 4 29 c. 2 29

d. 1492 e. None of these

46. What is the volume of the parallelepiped with sides: a = i 3j+ k, b = 2 j k, c = i + j 2k a. 2 b. 4 c. 6 d. 7 e. None of these

47. If a and b are vectors such that a b = 0 and a b = 0 then a. a = 0 and b = 0 b. only a = 0 c. only b = 0 d. at least one of a and b is 0

48. Which of the following DOESNT make sense? a. a (b c) b. a (b c) c. a (b c) d. a ((b c)d) e. all are okay