General Mathematics I — 2026-test Past Questions

University of Ilorin MTH101: 81 questions.

Topics: Real Sequences and Series, Theory of Quadratic Equations, Binomial Theorem, Mathematical Induction, Complements, Elementary Set Theory, Union, Intersection, Venn diagrams

  1. Which of the following is represented by the diagram above?
    1. A ⊂ B
    2. A ⊆ B
    3. A=B
    4. A≠B
  2. Insert 3 geometric means between 243 and 3.
    1. 81, 27, 9
    2. 90, 60, 30
    3. 100, 50, 25
    4. 72, 48, 24
  3. Find the measure of set A if A = {0, 1, 2}.
    1. 1
    2. 2
    3. 3
    4. 0
  4. If 2x² − 6x + 3 = 0, find α² + β² + α.
    1. 6 + α
    2. 9
    3. 6
    4. 15/2
  5. What is the arithmetic mean (A.M) between 2x + 7y and 4x + 5y?
    1. 3x + 6y
    2. 6x + 12y
    3. 3x + 4y
    4. 2x + 6y
  6. Find the coefficient of xⁿ in (1 − x)⁻¹.
    1. 1
    2. 0
    3. n
    4. −1
  7. The sum to infinity of a G.P. is possible when:
    1. |r| < 1
    2. r ≥ 1
    3. r > 1
    4. r = 1
  8. Find the sum to infinity of the sequence −2, 2/3, ...
    1. −3/2
    2. −3
    3. 3/2
    4. −5/2
  9. Insert 5 A.M between 3 and 12.
    1. 9/2, 6, 15/2, 9, 21/2
    2. 4, 5, 6, 7, 8
    3. 5, 7, 9, 11, 13
    4. 9/2, 13/2, 17/2, 21/2, 12
  10. If the fifth term of a G.P. is 162 and the 8th term is 4374, find the product of the first term and the common ratio.
    1. 6
    2. 12
    3. 18
    4. 2
  11. Find the geometric mean between (x − 1)/(x + 2) and (x − 1)(x + 2).
    1. x − 1
    2. x + 1
    3. (x − 1)²
    4. x + 2
  12. Find the middle term of (3a + 4b)⁸.
    1. 5th term
    2. 4th term
    3. 6th term
    4. 3rd term
  13. Find k in x² − kx − 10 = 0 when α² + β² = 29.
    1. ±7
    2. ±9
    3. ±3
    4. ±11
  14. Insert 2 arithmetic means between 4 and 22.
    1. 9, 13
    2. 10, 16
    3. 11, 15
    4. 12, 18
  15. If the fifth term of a G.P. is 162 and the 8th term is 4374, find the product of the first term and the common ratio.
    1. 6
    2. 12
    3. 18
    4. 2
  16. If 1/2 is a root of the equation x² + kx − 5/4 = 0, find k.
    1. 1
    2. 3
    3. 2
    4. 4
  17. Insert 3 arithmetic means between 4 and 16. Which of the following is not one of them?
    1. 7
    2. 10
    3. 9
    4. 13
  18. Find (α − β)² for x² − 4x + 3 = 0.
    1. ±8
    2. ±3
    3. ±2
    4. 4
  19. Find the sum to infinity of 2⁻¹ + 2⁻² + 2⁻³ + ...
    1. 2
    2. 1/2
    3. 1
    4. 3/2
  20. If P(n) = n(n + 1)(2n + 1)(3n² + 3n − 1)/30 is true, what is P(2)?
    1. 11
    2. 9
    3. 15
    4. 17
  21. Find (α − β)² for x² − 4x + 3 = 0.
    1. 8
    2. 6
    3. 4
    4. 2
  22. Find (α − β)² for the equation 3x² − 7x − 1 = 0.
    1. 61/9
    2. 37/9
    3. 49/9
    4. 55/9
  23. Find the 8th term of the sequence 1 + 3 + 9 + ...
    1. 6561
    2. 729
    3. 2187
    4. 19683
  24. Find the 4th term in the binomial expansion of (x − 2y)¹⁰.
    1. −960x⁷y³
    2. 960x⁷y³
    3. −120x⁷y³
    4. 45x⁸y²
  25. Find the sum to infinity of the G.P. 32, 16, 8, ...
    1. 64
    2. 32
    3. 48
    4. 16
  26. What is the coefficient of xⁿ in the expansion of (1 − x)⁻¹?
    1. 1
    2. −1
    3. 0
    4. n
  27. For the equation 9x² − 9x + 1 = 0, what is the nature of the roots?
    1. Real and equal
    2. Imaginary
    3. Real and distinct
    4. Rational
  28. For the equation x² − x − 6 = 0 with roots α and β, find the value of (α − β)².
    1. 24
    2. 20
    3. 25
    4. 15
  29. Find the 10th term of the sequence x, 3x, 5x, 7x, ...
    1. 19x
    2. 20x
    3. 21x
    4. 18x
  30. For the equation x² − x − 6 = 0 with roots α and β, find the value of 1/α² + 1/β².
    1. 13/36
    2. 11/36
    3. 5/18
    4. 7/18
  31. The third term of an A.P. is 10 and the seventh term is 34. Find the product of the first term and the common difference.
    1. −12
    2. 12
    3. 10
    4. −10
  32. Find the sum to infinity of the geometric progression 32, 16, 8, ...
    1. 64
    2. 48
    3. 32
    4. 16
  33. For the equation x² − x − 6 = 0 with roots α and β, find the value of α² + β².
    1. 11
    2. 13
    3. 12
    4. 10
  34. Suppose that P(n): Σi² (from i = 1 to n) = n(n + 1)(2n + 1)/6 is true for n ∈ ℕ. Which of these is NOT a factor of P(k + 1)?
    1. 2k + 1
    2. k + 2
    3. 2k + 3
    4. k + 1
  35. Find the sum of the first eight terms of the sequence 1, 1 + √2, 1 + 2√2, ...
    1. 10 + 28√2
    2. 4 + 14√2
    3. 8 + 28√2
    4. 6 + 16√2
  36. Suppose that for n ∈ ℕ, the statement P(n): Σj⁴ (from j = 1 to n) = n(n + 1)(2n + 1)(3n² + 3n − 1)/30 is true. What is P(2)?
    1. 16
    2. 20
    3. 17
    4. 24
  37. The second term of a geometric progression is 24 and the fifth term is 81. Find the first term.
    1. 12
    2. 18
    3. 16
    4. 14
  38. Find the sum of the first twelve terms of the sequence 3, 2½, 2, 1½, ...
    1. 5
    2. 9
    3. 3
    4. 7
  39. If α and β are roots of the equation 2x² − x + 4 = 0, find the value of 2α − 1/β + 2β − 1/α.
    1. 1
    2. 1/4
    3. 3/4
    4. 1/2
  40. Given the universal set U = {x ∈ ℕ: 1 ≤ x ≤ 10}, A = {all odd numbers in U} and B = {all even numbers less than 10}, find (A ∪ B)′.
    1. {2, 4, 6, 8}
    2. {10}
    3. ∅
    4. {1, 3, 5, 7, 9}
  41. Given the universal set U = {x ∈ ℕ: 1 ≤ x ≤ 10}, A = {all odd numbers in U} and B = {all even numbers less than 10}, find A ∩ B′.
    1. {2, 4, 6, 8, 10}
    2. {1, 3, 5, 7}
    3. {1, 3, 5, 7, 9}
    4. {9, 10}
  42. U = {x : 1 ≤ x ≤ 10} and B = {x : x is an even number less than 10}. Find B′ (the complement of B).
    1. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
    2. 1, 3, 5, 7, 9
    3. 2, 4, 6, 8
    4. 1, 3, 5, 7, 9, 10
  43. Find the value of n if the sum of the first n terms of the series 1, 5, 9, … is equal to the sum of the first n terms of the series 19, 17, 15, 13, …
    1. 4
    2. 5
    3. 6
    4. 7
  44. What is the sum of the coefficients of the expansion of (1 − 2x)⁷?
    1. −1
    2. 0
    3. 1
    4. 128
  45. If α and β are the roots of the equation ax² + bx + c = 0, find the value of (α + 1)(β + 1).
    1. (a + b − c)/a
    2. (a − b + c)/a
    3. (c − b + a)/a
    4. (a + b + c)/a
  46. What are the roots of the equation 3k² − 2k − 1 = 0?
    1. k = 1, k = −1/3
    2. k = −1, k = 1/3
    3. k = 1, k = 1
    4. k = −1, k = −1/3
  47. U = {x : 1 ≤ x ≤ 10}, A = {x : x is an odd number less than 10} and B = {x : x is an even number less than 10}. Find A′ ∪ B′.
    1. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
    2. 2, 4, 6, 8, 10
    3. 1, 3, 5, 7, 9
    4. Ø (the empty set)
  48. Find 4 geometric means between 243 and −1.
    1. −81, 27, −9, 3
    2. 81, −27, 9, −3
    3. 81, 27, 9, 3
    4. −81, −27, −9, −3
  49. If P(n) = n²(n + 1)(n + 2), find P(k + 1).
    1. (k + 1)²(k + 2)(k + 3)
    2. (k + 1)²(k + 1)(k + 2)
    3. k²(k + 1)(k + 2)
    4. (k + 1)(k + 2)(k + 3)
  50. If α and β are the roots of the equation ax² + bx + c = 0, find the value of (α + 1)(β + 1).
    1. (a + b − c)/a
    2. (a − b + c)/a
    3. (c − b + a)/a
    4. (a + b + c)/a
  51. What is the sum of the coefficients of the expansion of (1 + x)⁸?
    1. 512
    2. 128
    3. 65,536
    4. 256
  52. Find the first three terms of the expansion of (1 + 3x)¹⁰.
    1. 1 + 30x + 405x²
    2. 1 + 30x + 90x²
    3. 1 + 10x + 45x²
    4. 1 + 3x + 9x²
  53. Find the geometric mean between (x − 2)/(x + 1) and (x − 1)(x + 2).
    1. √[(x − 2)(x − 1)(x + 2)/(x + 1)]
    2. √[(x + 2)(x − 1)/((x − 2)(x + 1))]
    3. (x − 2)(x − 1)/[(x + 1)(x + 2)]
    4. √[(x − 2)(x + 1)(x − 1)(x + 2)]
  54. The set of numbers is given as {1 ≤ x ≤ 10}. A = {all odd numbers less than 10} and B = {all even numbers less than 10}. What is the measure of (A′ ∪ B′)?
    1. 10
    2. 9
    3. 1
    4. 5
  55. The third term of an AP is 10 and the seventh term is 34. Find the product of the common difference and the first term.
    1. −12
    2. 10
    3. 12
    4. −10
  56. The sum to infinity of a geometric series is 12 and the common ratio is 2/3. Find the first term.
    1. 36
    2. 8
    3. 4
    4. 6
  57. Given sets A = {0, 2, 4, 6} and B = {1, 2, 3, 4}, find m(A ∪ B) where m denotes the measure (number of elements).
    1. 8
    2. 4
    3. 6
    4. 7
  58. Find the sum of the coefficients of the expansion of (x + y)⁸.
    1. 256
    2. 128
    3. 512
    4. 64
  59. The set of numbers is given as {1 ≤ x ≤ 10}. A = {all odd numbers less than 10} and B = {all even numbers less than 10}. What is the measure of (A ∪ B)′?
    1. 9
    2. 5
    3. 8
    4. 1
  60. If P(n) = n(n + 1)(2n + 1)/6, find the value of P(3).
    1. 10
    2. 20
    3. 14
    4. 6
  61. Find the limiting sum (sum to infinity) of the geometric series 6, −2, 2/3, −2/9, …
    1. 2
    2. 18
    3. 4.5
    4. 9
  62. In the expansion of (2x − 1/x)⁴, find the term independent of x.
    1. 24
    2. −24
    3. 16
    4. 6
  63. The first term of a geometric progression is 81 and its 4th term is 3. Find the sum to infinity.
    1. 121.5
    2. 243
    3. 108
    4. 120
  64. In the equation 2x² − x + c = 0, find the value of c if one root is double the other.
    1. 1/9
    2. 1/2
    3. 2/9
    4. 1/18
  65. In a G.P., the 1st term is 6 and the ratio is 3. Find the 4th term.
    1. 54
    2. 18
    3. 162
    4. 486
  66. In the expansion of (2x − y)¹⁰, find the coefficient of the 4th term.
    1. 15,360
    2. −15,360
    3. −1,920
    4. 960
  67. In a G.P., the 4th term is 24 and the 8th term is 8/27. Find its common ratio.
    1. 1/3
    2. 3
    3. 1/9
    4. 2/3
  68. If α + β = 5 and αβ = −2, find the value of 1/α² + 1/β².
    1. 29/4
    2. 21/4
    3. 25/4
    4. 33/4
  69. In a Geometric Progression (G.P.), the 4th term is 162 and the 7th term is 4374. Find the product of the first term and the common ratio.
    1. 18
    2. 54
    3. 6
    4. 12
  70. Find the 6th term of the series 1 + 3 + 9 + …
    1. 81
    2. 729
    3. 243
    4. 486
  71. Find the total number of distinct terms in the expansion of (1 + 2x + x²)¹⁰.
    1. 11
    2. 22
    3. 21
    4. 10
  72. Find the 7th term of a GP whose common ratio is 4 and 6th term is 42.
    1. 168
    2. 126
    3. 10.5
    4. 210
  73. Find the value of p for which (x + p)² = 2 − 3p has equal roots.
    1. p = 2/3
    2. p = 1
    3. p = −1
    4. p = −2
  74. Given 27x² + bx + 8 = 0, find b if one of the roots is the square of the other.
    1. −20
    2. 20
    3. −30
    4. 30
  75. For the quadratic equation x² − x + 2 = 0 with roots a and b, find 2a − 1/a + 2b − 1/b.
    1. 1.5
    2. 2.0
    3. 2.5
    4. 3
  76. Find the sum of the first 12 terms in the series 3, 2.5, 2, 1.5, …
    1. 3
    2. 1.5
    3. 6
    4. 0
  77. Find the coefficient of the middle term in the expansion of (2x + 1/x)⁸.
    1. 1120
    2. 560
    3. 70
    4. 2240
  78. For the quadratic equation 2x² − 6x + 3 = 0 with roots a and b, find a² + ab + b².
    1. 7.5
    2. 5.5
    3. 6
    4. 4.5
  79. Insert a geometric mean between (x − 1)/(x + 2) and (x − 1)(x + 2).
    1. x + 2
    2. x + 1
    3. x − 1
    4. (x − 1)²
  80. What is the nature of the roots for a quadratic equation ax² + bx + c = 0 if b² − 4ac = 0?
    1. Real and distinct
    2. Imaginary
    3. Real and equal
    4. Complex
  81. Given 2x² − x + c = 0, find c if one of the roots is double the other.
    1. 1/6
    2. 1/2
    3. 1/3
    4. 1/9

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