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1. Which vector is equal in magnitude but opposite in direction to KL? (LOOK AT IMAGE 1)
a) JL
b) LG
c) GL
d) GK
2. Express the Cartesian vector q = (3, 4) in polar form.
a) q = (7, 53°)
b) q = (5, 53°)
c) q = (7, 37°)
d) q = (5, 37°)
3. If p = (7, 2) q = (5, -1), find p - q.
a) p - q = (2, 1)
b) p - q(12, 1)
c) p - q = (12, 3)
d) p - q = (2, 3)
4. If A(3, 2, 9) and B(-4, 7, 2) are two points in 3-space, then find the vector AB.
a) AB= (-7, 5, -7)
b) AB = (-1, 9, 11)
c) AB = (-12, 14, 18)
d) AB = (7, 5, 7)
5. If p = (7, 2), then find -2p.
a) -2p = (5, 0)
b) -2p = (9, 4)
c) -2p =(-14, -4)
d) -2p = (14, 4)
6. If s = (3, 2, 9) and t = (-4, 7, 2), are two vectors in 3-space, then find the vector 3s - 2t.
a) 3s - 2t = (17, -8, 23)
b) 3s - 2t = (7, -5, 7)
c) 3s - 2t = (8, -14, -4)
d) 3s - 2t = (9, 6, 27)
7. If vector a = (3, 2, 9) and b = (-4, 7, 2), find a • b.
a) a • b =-9
b) a • b = 19
c) a • b = 20
d) a • b = 12
8. For which following condition, will the dot product of two vectors be zero?
a) If the angle between them is 90°.
b) If the angle between them is 0°.
c) If the angle between them is 180°
d) If the vectors have the same magnitude
9. If vector |a| = 12 cm, |b| = 13 cm, and the angle between them when placed tail to tail is 35°, find a • b to the nearest centimetre
a) a • b = 57 cm
b) a • b = 128 cm2
c) a • b = 11 cm2
d) a • b = 24 cm
10. The dot product represents:
a) The magnitude of the projection of one vector onto another.
b) A vector perpendicular to both given vectors.
c) A vector parallel to both given vectors.
d) The division of one vector by another.
11. For which following condition, will the cross product of two vectors be zero?
a) If the angle between them is 90°.
b) If the angle between them is 0°.
c) If the angle between them is 45°.
d) If the vectors have the same magnitude.
12. If vector |a| = 12 cm, |b| = 13 cm, and the angle between them when placed tail to tail is 35°, find |a × b| to the nearest centimetre.
a) |a × b| = 156 cm
b) |a × b| = 89 cm
c) |a × b| = 6 cm
d) |a × b| = 128 cm
13. Which statement is true about the cross product?
a) The magnitude of the projection of one vector onto another.
b) A vector perpendicular to both given vectors.
c) A vector parallel to both given vectors.
d) The division of one vector by another.
14. What is (0, 0, 1) × (0, 1, 0)?
a) (-1, 0, 0,)
b) (1, 0, 0)
c) (0, 0, 1)
d) (0, -1, 0)
15. Which of the following does not represent the direction shown above? (LOOK AT IMAGE 2)
a) N70°W
b) W70°N
c) W20°N
d) 290°
16. Which of the following represents the x – component of the vector in question 1 if the magnitude of the vector is 3 units?
a) 3 sin 20°
b) 3 cos 20°
c) 3 sin 70°
d) square root(3)
17. ABCD is the parallelogram shown below. (LOOK AT THE 3rd IMAGE)
a) AC = BC + CD
b) AC = AB + BE
c) AC = AD + DC
d) AC = AD + AB
18. If a = (5, 2) and b = (-3, 1), then which of the following vectors represent 3a - 2b?
a) (9, 4)
b) (9, 8)
c) (21, 8)
d) (21, 4)
19. Let a = (5, 2) and b = (-3, 1), find a • b.
a) 17
b) -13
c) 5
d) 7
20. Let a = (1, 2, 3) and b = (-2, 0, 3). Find 2a + 3b.
a) (-4, 4, 15)
b) (-2, 0, 9)
c) (8, 2, 6)
d) (-1, 2, 6)
21. Given the vectors a = (1, 3, 4) and a = (4, 5, -4), which of the following represent a × b?
a) (8, -20, 7)
b) (-32, 20, -7)
c) (4, 15, 16)
d) -3
22. In the previous question, what is the angle between the two given vectors to the nearest degree?
a) 94°
b) 86°
c) 36°
d) 24°
23. A math student is walking with a velocity of 5 m/s [N53°E]. Which of the following is the Eastern component of this velocity in m/s?
a) 2
b) 3
c) 4
d) 5
24. Which of the following is a vector equation of the line through the point (2,-5) with direction vector (3,2)?
a) (x,y) = (2,-5) +t(-2,3)
b) (x,y) = (3,2) + t(5,2)
c) (x,y) = (3,2) + t(2,-5)
d) (x,y) = (2,-5) + t(3,2)
25. Choose the scalar equation of the line through (-1,5) if (4,-1) is a normal vector to the line.
a) 4x – y – 9 = 0
b) x + 4y – 19 = 0
c) 4x – y + 9 = 0
d) x + 4y + 19 = 0
26. Which of the following is a vector equation for the line through (-1,2,-2) and parallel to x / -5 = y - 3 = z + 2/ 2
a) (x,y,z) = (0,3,-2) + t(-1,2,-2)
b) (x,y,z) = (-1,2,-2) + t(-5,1,2)
c) (x,y,z) = (-5,1,2) + t(-1,2,-2)
d) (x,y,z) = (-5,1,2) + t(0,3,-2)
27. A plane contains the point A(-1,2,5) and has direction vectors (2,-1,3) and (-3,1,5). Which of the following is a vector equation for the plane?
a) (x,y,z) = (2,-1,3) + s(-1,2,5) + t(-3,1,5)
b) (x,y,z) = (-3,1,5) + s(-1,2,5) + t(2,-1,3)
c) (x,y,z) = (-1,2,5) + s(2,-1,3) + t(-3,1,5)
d) (x,y,z) = (-1,2,5) + s(2,-1,3) + t(-8,-19,-1)
28. Which of the following are parametric equations for the plane through the point (2,-3,4) and parallel to the yz-plane?
a) x = 2, y = -3 + s, z = 4 + t
b) x = 2 + t, y = -3, z = 4 + s
c) x = 2 + t, y = -3 + s, z = 4
d) x = 2, y = -3, z = 4
29. If ƒ(x) = -2sin(x) then ƒ?(x)?
a) 2sin x
b) 2cos x
c) -2cos x
d) -2sin x
e) 2tan x
30. If ƒ(x) = (15)x then ƒ?(x) = ?
a) ln(15)x
b) ln(15)(15)x
c) (15)x ln (15)x
d) 15(15)x
e) 1.176(15)x
31. If ƒ(x) = sin(x) then ƒ?(x) = ?
a) sin x
b) cos x
c) -cos x
d) -sin x
e) tan x
32. If ƒ(x) = (2.5)x then ƒ?(x) = ?
a) ln(2.5)x
b) 2.5(2.5)x
c) (2.5)x ln(2.5)x
d) ln(2.5)(2.5)x
e) 0.916(2.5)x
33. If ƒ?(x) = 0 then a possible function is
a) ƒ(x) = cos(x)
b) ƒ(x) = sin(x)
c) ƒ(x) = 2x
d) ƒ(x) = ex
34. If ƒ?(x) = sin(x) then ƒ(x) = ?
a) sin x
b) cos x
c) -cos x
d) -sin x
e) tan x
35. if ƒ(x) = 1/ex then ƒ?(x) = ?
a) 1/ex
b) -1/ex
c) 1/ex ln ex
d) -ex
36. Find the derivative of s(x) = (1 + x^2)^5
a) s?(x) = 5(1 + x^2)^4
b) s?(x) = 5(1 + x^2)^4 (1 + x)^2
c) s?(x) = 10x(1 + x^2)^4
d) s?(x) = 2x(1 + x^2)^5
37. Find the derivative of ƒ(x) = sin (1 + x^2)
a) ƒ?(x) = cos(1 + x^2)
b) ƒ?(x) = cos(2x)
c) ƒ?(x) = -2x cos (1 + x^2)
d) ƒ?(x) = 2x cos (1 + x^2)
38. Find the derivative of y = e^3x+2
a) y? = e^3x+2
b) y? = 3e^3x+2
c) y? = (3x + 2)e^3x+2
d) y? = (3x)e^3x+2
39. If ƒ?(x) < 0 when x < c then ƒ(x) is decreasing when x < c.
True
False
40. The function ƒ(x) = x3 – 3x + 2 is increasing on the interval -1 < x < 1.
True
False
41. If ƒ'(c) < 0 then ƒ(x) is decreasing and the graph of ƒ(x) is concave down when x = c.
True
False
42. A local extreme point of a polynomial function ƒ(x) can only occur when ƒ?(x) = 0.
True
False
43. If ƒ?(x) > 0 when x < c and ƒ?(x) < 0 when x > c, then ƒ(x) has a maximum value when x = c.
True
False
44. If ƒ?(x) has a minimum value at x = c, then the graph of ƒ(x) has a point of inflection at x = c.
True
False
45. If ƒ?(c) > 0 and ƒ?(c), then ƒ(x) is increasing and the graph is concave up when x = c.
True
False
46. If ƒ?(c) = 0 then ƒ(x) must have a local extreme point at x = c.
True
False
47. The graph of ƒ(x) has an inflection point at x = c so ƒ?(x) has a maximum or minimum value at x = c.
True
False
48. ƒ?(x) is increasing when x < c and decreasing when x > c so the graph of ƒ(x) has an inflection point at x = c.
True
False
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Short Answers:
49. Consider A(3, -2, 6) and B(2, 1, 0). What is the magnitude of AB?
50. What is the dot product of a = (3, -2, 6) and b = (2, 1, 0)?
51. What is the cross product of a = (3, -2, 6) and b = (2, 1, 0)?
52. What is the average rate of change between x = 3 and x = 4 for the function ƒ(x) = -x2 + 4x + 1?
53. What is the simplified average rate of change between x = 3 and x = 3 + h for the function ƒ(x) = -x2 + 5? Use the definition of a limit: lim h-->0 f(x+h)-f(x) / h
54. What is the graphical representation of the instantaneous rate of change?
55. What is the instantaneous rate of change for ƒ(x) = 2x + 5 at x = -7
56. How is the average rate of change and the instantaneous rate of change related for ƒ(x) = 2x + 5
57. If the population growth (P) in a community is projected to follow the function P = 7t2 + 5t + 350 (t is time in years), then find the average rate of growth from the 2nd to the 3rd year.
58. If the population growth (P) in a community is projected to follow the function P= 7t2 + 5t + 350 (t is time in years), then find the instantaneous rate of growth in the 2nd year.
59. How many maximum or minimum points does the equation ƒ(x) = -(x2 + 4x - 4)(x + 2)(x – 1) have?
60. What is the inverse of the function ƒ(x) = ex
61. What is the inverse of the function ƒ(x) = ln x
62. if ƒ?(x) = ex then ƒ(x) = ?
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