Elementary Mathematics III — 2025 Past Questions

University of Ilorin MTH103: 68 questions.

Topics: Scalar and Vector Products of Two Vectors, Geometric Representation of Vectors in 1-3 Dimensions, Addition, Scalar Multiplication, Differentiation of Vectors, Components and Direction Cosines

  1. Find the sum of vectors a, b, c, d, e in the diagram below
    1. 90
    2. 360
    3. 540
    4. 0
  2. The weight of a 7 kg mass and a north-easterly wind of 20 knots is:
    1. Vector quantity
    2. Direction and weight quantities
    3. Vector and physical quantity
    4. Physical quantities
  3. If P is a force of 30 N acting due North, what is the magnitude of the vector sum?
    1. 50 N
    2. 30 N
    3. 45 N
    4. 40 N
  4. Types of vectors include the following except:
    1. Free vector
    2. Position vector
    3. Static vector
    4. Line vector
  5. Suppose p = 2i + 4j + 3k and q = i + 5j - 2k. Find p × q:
    1. -23i + 7j - 6k
    2. -23i + 7j + 6k
    3. 23i - 7j - 6k
    4. 23i + 7j + 6k
  6. The modulus of vector x = 3i + 4j - 5k is:
    1. 5√2
    2. 3√2
    3. 4√2
    4. 2√5
  7. Given a = 2i - 2j + k, b = -i + j + 2k, and c = 2i + 2j - 4k, find a · (b × c):
    1. 16
    2. 20
    3. -16
    4. -20
  8. The position vector of a particle at time t is r = cos(t-2)i + sinh(t-2)j + at³k. Find the acceleration at time t = 2:
    1. -i + 12ak
    2. -i + 6ak
    3. i + 6ak
    4. i + 12ak
  9. Find the magnitude of vector TS, where S(1, 2, 3) and T(3, 2, 1):
    1. √2
    2. 2
    3. 3√2
    4. 2√2
  10. Find the sum of vectors AB + BC - DC - AD:
    1. 0
    2. AD
    3. DA
    4. -AD
  11. If x = 2i + 3j + k and y = i + 6j + k, find the direction cosine of x - y:
    1. (1/√10, -3/√10, 0)
    2. (1/√10, 3/√10, 0)
    3. (-1/√10, -3/√10, 0)
    4. (1/√10, -3/√10, 1/√10)
  12. Determine the vector product of x = 3i - j + 2k and y = i + 3j - 2k:
    1. 4i + 8j - 10k
    2. -4i - 8j - 10k
    3. 4i - 8j + 10k
    4. -4i + 8j + 10k
  13. Given a = 2i + 5j and b = 4i - 3j, find the unit vector in the direction of b:
    1. 1/√29 (4i - 3j)
    2. 1/5 (2i + 5j)
    3. 1/√29 (2i + 5j)
    4. 1/5 (4i - 3j)
  14. If vector A, B, and C are coplanar, which of the following is correct?
    1. A · (B × C) = 2
    2. A · (B × C) = 1
    3. A · (B × C) = 0
    4. A · (B × C) = -1
  15. If u = 3i + 5j and v = 2i - 4j, find 2u - v:
    1. 4i + 14j
    2. 4i - 14j
    3. -4i + 14j
    4. -4i - 14j
  16. A = (u + 3)i - (2 + u²)j + 2u³k. Determine dA/du at u = 3:
    1. i - 6j + 54k
    2. i + 6j + 54k
    3. i - 6j - 54k
    4. i + 6j - 54k
  17. If a = 4i - 2j and b = 2i - 2j, evaluate |3a - 2b|:
    1. 2√17
    2. √17
    3. 4√17
    4. 17
  18. If p = 6i - 2j and q = 4i + 2j, find p · q:
    1. 20
    2. 24
    3. -4
    4. 16
  19. Given: Ā = 2i - 3j + 4k, B = i - 2j - 3k, C = 2i + j + 2k. Find A · (B × C):
    1. 42
    2. 24
    3. 20
    4. 16
  20. Force of 30 N moving due North and a force of 40 N moving due East. Find the magnitude of the resultant force:
    1. 50 N
    2. 70 N
    3. 10 N
    4. 25 N
  21. A = (u + 3)i - (2 + u²)j + (2u³)k. Find |dA/du| at u = 3:
    1. √2953
    2. √2952
    3. √2954
    4. √2950
  22. P = 3i - 2j + 1k and Q = 2i + 3j - 4k. Find P · Q:
    1. -4
    2. 4
    3. 6
    4. -6
  23. r = cos(t - 2)i + sinh(t - 2)j + at³k. Find the acceleration at t = 2:
    1. i + 6ak
    2. -i + 6ak
    3. i + 12ak
    4. -i + 12ak
  24. Calculate m × n if m = 3i - 2j + 3k and n = 5i + j - k:
    1. -i + 18j + 13k
    2. i + 18j + 13k
    3. -i - 18j + 13k
    4. i - 18j + 13k
  25. If x = 2i + 4j - 3k and y = i + 3j + 2k, determine the scalar product of the two vectors:
    1. 8
    2. 2
    3. 12
    4. 6
  26. Find the sum of vectors AB, -CB, CD, -ED:
    1. AE
    2. EA
    3. 0
    4. AD
  27. If P = 3i - 4j + 2k and Q = 2i + 5j - k, find -(P × Q):
    1. 6i - 7j - 23k
    2. -6i + 7j + 23k
    3. 6i + 7j - 23k
    4. -6i - 7j - 23k
  28. If A = 3i + 2j - k, B = 2i - j + 3k, and C = i - 2j + 2k, find A · (B × C):
    1. 13
    2. 12
    3. 3
    4. -3
  29. Find the area of the parallelogram determined by the vectors (2, 1, 0) and (-2, 1, 2):
    1. 9 units
    2. 6 units
    3. 12 units
    4. 18 units
  30. Given a = 2i + 5j and b = 4i - 3j, find the unit vector in the direction of b:
    1. 1/√29 (2i + 5j)
    2. 1/√29 (4i - 3j)
    3. 1/5 (4i - 3j)
    4. 1/5 (2i + 5j)
  31. If x = 2i + 9j + mk and y = 3i - 4j + 3k are perpendicular, find m:
    1. 9
    2. 6
    3. 3
    4. 10
  32. If Z₁ = 3i + j, Z₂ = -2i + j, and Z₃ = 4i - 2j, find Z₁ + Z₂ + Z₃:
    1. 5i
    2. 5j
    3. 5i + 5j
    4. 0
  33. Find the magnitude of the vector 2i - 5j + 4k:
    1. 6√8
    2. 3√5
    3. 2√3
    4. 4√2
  34. a = 2i + 3j + 5k and b = 4i + j + 6k. Find a · b:
    1. 41
    2. 31
    3. 21
    4. 11
  35. Find |(a + b) - (a - b)| when a = (6, 3) and b = (1, -2):
    1. 2√5
    2. √5
    3. 4√5
    4. 5
  36. Find the sum of the vectors AB, BC, -DC, -AD:
    1. 0
    2. AD
    3. DA
    4. -D
  37. Find the triple scalar product of x = 2i - 3j + k, y = 4i + j - 3k, z = -i + j - 3k:
    1. -40
    2. 40
    3. -45
    4. 45
  38. The weight of a 7 kg mass and a north-easterly wind of 20 knots is:
    1. Vector quantity
    2. Direction and weight quantities
    3. Vector and physical quantity
    4. Physical quantities
  39. If P is a force of 30 N acting due North, what is the magnitude of the vector sum?
    1. 50 N
    2. 30 N
    3. 45 N
    4. 40 N
  40. Types of vectors include the following except:
    1. Free vector
    2. Position vector
    3. Static vector
    4. Line vector
  41. Suppose p = 2i + 4j + 3k and q = i + 5j - 2k. Find p × q:
    1. -23i + 7j - 6k
    2. -23i + 7j + 6k
    3. 23i - 7j - 6k
    4. 23i + 7j + 6k
  42. The modulus of vector x = 3i + 4j - 5k is:
    1. 5√2
    2. 3√2
    3. 4√2
    4. 2√5
  43. Given a = 2i - 2j + k, b = -i + j + 2k, and c = 2i + 2j - 4k, find a · (b × c):
    1. 16
    2. 20
    3. -16
    4. -20
  44. The position vector of a particle at time t is r = cos(t-2)i + sinh(t-2)j + at³k. Find the acceleration at time t = 2:
    1. -i + 12ak
    2. -i + 6ak
    3. i + 6ak
    4. i + 12ak
  45. Find the magnitude of vector TS, where S(1, 2, 3) and T(3, 2, 1):
    1. √2
    2. 2
    3. 3√2
    4. 2√2
  46. Find the sum of vectors AB + BC - DC - AD:
    1. 0
    2. AD
    3. DA
    4. -AD
  47. If x = 2i + 3j + k and y = i + 6j + k, find the direction cosine of x - y:
    1. (1/√10, -3/√10, 0)
    2. (1/√10, 3/√10, 0)
    3. (-1/√10, -3/√10, 0)
    4. (1/√10, -3/√10, 1/√10)
  48. Determine the vector product of x = 3i - j + 2k and y = i + 3j - 2k:
    1. 4i + 8j - 10k
    2. -4i - 8j - 10k
    3. 4i - 8j + 10k
    4. -4i + 8j + 10k
  49. Given a = 2i + 5j and b = 4i - 3j, find the unit vector in the direction of b:
    1. 1/√29 (4i - 3j)
    2. 1/5 (2i + 5j)
    3. 1/√29 (2i + 5j)
    4. 1/5 (4i - 3j)
  50. If vector A, B, and C are coplanar, which of the following is correct?
    1. A · (B × C) = 2
    2. A · (B × C) = 1
    3. A · (B × C) = 0
    4. A · (B × C) = -1
  51. If u = 3i + 5j and v = 2i - 4j, find 2u - v:
    1. 4i + 14j
    2. 4i - 14j
    3. -4i + 14j
    4. -4i - 14j
  52. A = (u + 3)i - (2 + u²)j + 2u³k. Determine dA/du at u = 3:
    1. i - 6j + 54k
    2. i + 6j + 54k
    3. i - 6j - 54k
    4. i + 6j - 54k
  53. If a = 4i - 2j and b = 2i - 2j, evaluate |3a - 2b|:
    1. 2√17
    2. √17
    3. 4√17
    4. 17
  54. If p = 6i - 2j and q = 4i + 2j, find p · q:
    1. 20
    2. 24
    3. -4
    4. 16
  55. Given: Ā = 2i - 3j + 4k, B = i - 2j - 3k, C = 2i + j + 2k. Find A · (B × C):
    1. 42
    2. 24
    3. 20
    4. 16
  56. Force of 30 N moving due North and a force of 40 N moving due East. Find the magnitude of the resultant force:
    1. 50 N
    2. 70 N
    3. 10 N
    4. 25 N
  57. A = (u + 3)i - (2 + u²)j + (2u³)k. Find |dA/du| at u = 3:
    1. √2953
    2. √2952
    3. √2954
    4. √2950
  58. P = 3i - 2j + 1k and Q = 2i + 3j - 4k. Find P · Q:
    1. -4
    2. 4
    3. 6
    4. -6
  59. r = cos(t - 2)i + sinh(t - 2)j + at³k. Find the acceleration at t = 2:
    1. i + 6ak
    2. -i + 6ak
    3. i + 12ak
    4. -i + 12ak
  60. Calculate m × n if m = 3i - 2j + 3k and n = 5i + j - k:
    1. -i + 18j + 13k
    2. i + 18j + 13k
    3. -i - 18j + 13k
    4. i - 18j + 13k
  61. If x = 2i + 4j - 3k and y = i + 3j + 2k, determine the scalar product of the two vectors:
    1. 8
    2. 2
    3. 12
    4. 6
  62. Find the sum of vectors AB, -CB, CD, -ED:
    1. AE
    2. EA
    3. 0
    4. AD
  63. If P = 3i - 4j + 2k and Q = 2i + 5j - k, find -(P × Q):
    1. 6i - 7j - 23k
    2. -6i + 7j + 23k
    3. 6i + 7j - 23k
    4. -6i - 7j - 23k
  64. If A = 3i + 2j - k, B = 2i - j + 3k, and C = i - 2j + 2k, find A · (B × C):
    1. 13
    2. 12
    3. 3
    4. -3
  65. Find the area of the parallelogram determined by the vectors (2, 1, 0) and (-2, 1, 2):
    1. 9 units
    2. 6 units
    3. 12 units
    4. 18 units
  66. Given a = 2i + 5j and b = 4i - 3j, find the unit vector in the direction of b:
    1. 1/√29 (2i + 5j)
    2. 1/√29 (4i - 3j)
    3. 1/5 (4i - 3j)
    4. 1/5 (2i + 5j)
  67. If x = 2i + 9j + mk and y = 3i - 4j + 3k are perpendicular, find m:
    1. 9
    2. 6
    3. 3
    4. 10
  68. If Z₁ = 3i + j, Z₂ = -2i + j, and Z₃ = 4i - 2j, find Z₁ + Z₂ + Z₃:
    1. 5i
    2. 5j
    3. 5i + 5j
    4. 0

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