General Mathematics I — 2025A Past Questions

University of Ilorin MTH101: 81 questions.

Topics: Sequence and Series, Set Theory and Venn Diagram, Mathematical Induction, Quadratic Equation, Number System

  1. According to mathematical induction, which of these is a factor of \(2^{2n} - 1\)?
    1. 3
    2. 5
    3. 8
    4. 10
  2. A school went to a competition with a contingent of 11 football players and 8 sprinters. If 3 sprinters are also footballers, how many students are either footballers or sprinters?
    1. 16
    2. 17
    3. 18
    4. 19
  3. What is the sum to infinity of a geometric progression with first term 32 and common ratio \(\frac{1}{2}\)?
    1. 64
    2. 32
    3. 16
    4. 8
  4. Insert five arithmetic means between \(x\) and \(5x - 4\).
    1. \(x, x + \frac{5x - 4}{6}, x + \frac{2(5x - 4)}{6}, x + \frac{3(5x - 4)}{6}, x + \frac{4(5x - 4)}{6}, 5x - 4\)
    2. \(x, x + \frac{5x - 4 - x}{6}, x + \frac{2(5x - 4 - x)}{6}, x + \frac{3(5x - 4 - x)}{6}, x + \frac{4(5x - 4 - x)}{6}, 5x - 4\)
    3. \(x, x + \frac{4x - 4}{5}, x + \frac{2(4x - 4)}{5}, x + \frac{3(4x - 4)}{5}, x + \frac{4(4x - 4)}{5}, 5x - 4\)
    4. \(x, x + \frac{4x - 4}{6}, x + \frac{2(4x - 4)}{6}, x + \frac{3(4x - 4)}{6}, x + \frac{4(4x - 4)}{6}, 5x - 4\)
  5. The common difference of an arithmetic progression (AP) is:
    1. The difference between consecutive terms
    2. The difference between the first and last term
    3. The sum of consecutive terms
    4. The product of consecutive terms
  6. An \(A \cap (B \cup C')\) is:
    1. \(A - (B \cup C)\)
    2. \(A \cap B \cap C\)
    3. \((A \cap B)'\)
    4. \(A \cup B\)
  7. What is the sixth term of the sequence 1, 3, 9, …?
    1. 243
    2. 81
    3. 27
    4. 729
  8. Insert five arithmetic means between 3 and 12.
    1. 3, 4.5, 6, 7.5, 9, 10.5, 12
    2. 3, 5, 7, 9, 11, 13, 15
    3. 3, 4, 5, 6, 7, 8, 12
    4. 3, 6, 9, 12, 15, 18, 21
  9. The sum of the first 5 terms of an A.P where \(a = 2\) and \(d = 3\) is:
    1. 40
    2. 25
    3. 30
    4. 35
  10. If \(a = 10\), \(b = 20\), and \(c = 30\) in an A.P, the arithmetic mean of \(a\) and \(b\) is:
    1. 15
    2. 18
    3. 21
    4. 9
  11. Find the 8th term of a G.P whose first term is 3 and the common ratio is 2.
    1. 384
    2. 320
    3. 364
    4. 260
  12. Determine the sum of the first four terms of the series \(36 + 24 + 16 + \dots\)
    1. \(\frac{265}{3}\)
    2. \(\frac{65}{3}\)
    3. \(\frac{250}{3}\)
    4. \(\frac{260}{3}\)
  13. If A, B, and C are any three sets, what is \(A \cap (B \cup C)\)?
    1. \((A \cap B) \cup (A \cap C)\)
    2. \((A \cup B) \cap (A \cup C)\)
    3. \((A \cup B) \cup (A \cup C)\)
    4. \((A \cap B) \cap (A \cap C)\)
  14. Given that \(A = \emptyset\). Find the cardinality of \(P(A)\).
    1. 1
    2. 0
    3. 2
    4. No element
  15. Insert the arithmetic mean between 10 and 20.
    1. 15
    2. 12
    3. 18
    4. 14
  16. Find the sum to infinity: \(-2 + \frac{2}{3} - \frac{2}{9} + \dots\)
    1. -1.5
    2. 1.5
    3. 3
    4. -3
  17. The third term of a geometric progression is 63, and the fifth term is 567. Find the sum of the first 4 terms.
    1. 120
    2. 240
    3. 360
    4. 480
  18. Given the universal set \(U = \{5,6,7,8\}\), \(P = \{6,7\}\), and \(Q = \{6,8\}\), find \(P \cap Q'\).
    1. \{7\}
    2. \{6\}
    3. \{5,7\}
    4. \{6,7\}
  19. \(2^{2n} - 1\) is divisible by:
    1. 3
    2. 8
    3. 5
    4. 10
  20. Find the cardinality of natural numbers less than 11.
    1. 10
    2. 11
    3. 9
    4. 12
  21. Find the sum of the first five terms of the series: \(8 + 4 + 2 + \dots\)
    1. 15.5
    2. 16
    3. 15
    4. 14.5
  22. For every positive integer \(n\), \(7^{n} - 3^{n}\) is divisible by:
    1. 4
    2. 6
    3. 3
    4. 5
  23. What is the cardinality of the power set \(\{x : x \in \mathbb{N}, x < 11\}\)?
    1. 1024
    2. 1023
    3. 2048
    4. 10
  24. Insert the three arithmetic means between \(x - 2\) and \(x + 38\).
    1. \(x - 2, x + 7, x + 16, x + 25, x + 38\)
    2. \(x - 2, x + 10, x + 22, x + 34, x + 38\)
    3. \(x - 2, x + 8, x + 18, x + 28, x + 38\)
    4. \(x - 2, x + 12, x + 26, x + 30, x + 38\)
  25. According to the principle of induction, the propositional statement \(P(n)\) is true for \(n\) in the set of:
    1. Natural numbers
    2. Real integers
    3. Rational numbers
    4. Integers
  26. In a class of 60, 40 students speak Hausa, 19 students speak Igbo, and 7 students speak both Hausa and Igbo. How many students speak neither Hausa nor Igbo?
    1. 8
    2. 6
    3. 7
    4. 9
  27. If $\\alpha$ and $\\beta$ are the roots of the quadratic equation $2x^2 - 7x + 5 = 0$, find $\\alpha \\beta$.
    1. $\\frac{5}{1}$
    2. $-\\frac{5}{2}$
    3. $\\frac{7}{2}$
    4. $-\\frac{7}{5}$
  28. Given the sequence $-\\frac{1}{3}, \\frac{1}{2}, ...$, find the product of $a$ and $r$.
    1. $-\\frac{1}{6}$
    2. $\\frac{1}{6}$
    3. $-\\frac{2}{3}$
    4. $\\frac{2}{3}$
  29. For a quadratic equation $ax^2 + bx + c = 0$, where $a \\neq 0$, when $D < 0$, then the roots are:
    1. Real and equal
    2. Real and distinct
    3. Complex and distinct
    4. Rational and distinct
  30. If $P(A) = \\emptyset$, find the cardinality of $P(A)$.
    1. 0
    2. 1
    3. 2
    4. Infinite
  31. By the principle of mathematical induction, $2^{4n-1}$ is divisible by:
    1. 8
    2. 5
    3. 3
    4. 10
  32. Find the sum of the first four terms of the series $1 + \\sqrt{3} + 3 + ...$.
    1. $4(\\sqrt{3} - 1)$
    2. $2(\\sqrt{3} + 1)$
    3. $2(\\sqrt{3} - 1)$
    4. $4(\\sqrt{3} + 1)$
  33. Let $A = \\emptyset$. Find $P(A)$.
    1. 0
    2. 1
    3. 2
    4. Infinite
  34. Find the 6th term of the geometric progression: $1 + 3 + 9 + ...$.
    1. 243
    2. 729
    3. 81
    4. 162
  35. Given the quadratic equation $2x^2 - 7x + 5$, find $\\alpha^3 + \\beta^3$.
    1. 125
    2. 343
    3. 216
    4. 512
  36. What is the formula for the sum of terms in an arithmetic progression (AP)?
    1. $S_n = \\frac{n}{2} (2a + (n-1)d)$
    2. $S_n = \\frac{n(a + l)}{2}$
    3. $S_n = \\frac{a(1 - r^n)}{1 - r}$
    4. $S_n = ar^{n-1}$
  37. In the principle of mathematical induction, the propositional statement $P(n)$ is true for $n$ in the set of:
    1. Real numbers
    2. Natural numbers
    3. Integer numbers
    4. Rational numbers
  38. Find the 4th term of a geometric progression whose first term is 6 and common ratio is 3.
    1. 486
    2. 192
    3. 120
    4. 162
  39. Find the sum of the first 6 terms of an arithmetic progression where $a = 2$ and the last term $l = 20$.
    1. 66
    2. 54
    3. 84
    4. 72
  40. Find the sum of the first 5 terms of a sequence whose first and last terms are 3 and 3 respectively.
    1. 360
    2. 243
    3. 242
    4. 363
  41. The sum of the first $n$ terms of an arithmetic progression is given by $S_n = 5n^2 - 3n$. Find the 10th term.
    1. 92
    2. 96
    3. 80
    4. 94
  42. Insert three geometric means between $\\frac{x}{p^2}$ and $xp^2$.
    1. $\\frac{1}{p}, x, p$
    2. $\\frac{x}{p}, p, xp$
    3. $\\frac{x}{p}, x, p$
    4. $\\frac{x}{p}, x, xp$
  43. Determine the nature of the roots of the equation $x^2 - 2x + 1 = 0$.
    1. Real and equal
    2. Real and distinct
    3. Imaginary
    4. None of the above
  44. If the quadratic equation $2x^2 - 7x + 5 = 0$ has $\\alpha$ and $\\beta$ as its roots, find $\\alpha^2 + \\beta^2$.
    1. 17
    2. 25
    3. 29
    4. 35
  45. What will be $P(k+1)$ for $P(n) = n^3(n+1)$?
    1. $(k+1)^3(k+2)$
    2. $(k+1)^3(k+1)$
    3. $k^3(k+2)$
    4. $k^3(k+1)$
  46. Insert three arithmetic means between $x + 7y$ and $5x + 3y$.
    1. $2x + 6y, 3x + 5y, 4x + 4y$
    2. $2x + 8y, 3x + 6y, 4x + 5y$
    3. $2x + 7y, 3x + 5y, 4x + 3y$
    4. $2x + 6y, 3x + 4y, 4x + 2y$
  47. What are the five numbers between 3 and 12 in an arithmetic progression?
    1. 5, 7, 9, 11, 13
    2. 5, 7, 9, 11, 12
    3. 4, 6, 8, 10, 12
    4. 4, 5, 7, 10, 12
  48. For $P(n)$, $n$ is true for $n$ in the set of:
    1. Natural numbers
    2. Integers
    3. Real numbers
    4. Whole numbers
  49. What is the nature of the roots of the equation $2x^2 - 3x + 5 = 0$?
    1. Real and equal
    2. Real and distinct
    3. Imaginary
    4. Cannot be determined
  50. For the quadratic equation $ax^2 + bx + c = 0$, $a \\neq 0$, where $D > 0$, what is the nature of the roots?
    1. Real and equal
    2. Real and distinct
    3. Imaginary
    4. None of the above
  51. $P = \\{x : x \\leq 3\\}$, $Q = \\{x : 5 < x < 12\\}$, $R = \\{x : -2 \\leq x < 5\\}$. What is $P \\cap (Q \\cup R)$?
    1. $-4 \\leq x \\leq 3$
    2. $x < 0$
    3. $x < -4$
    4. $\\emptyset$
  52. What is the sum of the first four terms of the series $1 + \\sqrt{3} + 3 + \\dots$?
    1. $2(\\sqrt{3} + 1)$
    2. $4(\\sqrt{3} + 1)$
    3. $4(\\sqrt{3} - 1)$
    4. $2(\\sqrt{3} - 1)$
  53. The first term of an arithmetic progression is 2, the last term is 50, and the common difference is 4. What is the number of terms in the AP?
    1. 10
    2. 12
    3. 13
    4. 15
  54. The mean of two terms is 18. If the first term is 12, what is the second term?
    1. 18
    2. 24
    3. 26
    4. 30
  55. In an A.P, the sum of the first $n$ terms is $S_n = 2n^2 + 3n$. What is the $n$th term ($a_n$)?
    1. $4n + 1$
    2. $4n - 1$
    3. $2n + 3$
    4. $3n^2$
  56. Find the sum to infinity of $32 + 16 + 8 + \\dots$
    1. 64
    2. 60
    3. 20
    4. 128
  57. Find the sum to infinity of $-2 + \\frac{2}{3} - \\frac{2}{9} + \\dots$
    1. $-2$
    2. $-3$
    3. $-\\frac{6}{7}$
    4. $-\\frac{3}{2}$
  58. Determine the sum of the first five terms of the series $1 + 3 + 9 + \\dots$
    1. 121
    2. 221
    3. 242
    4. 321
  59. Find the 5th term of the sequence $2, \\frac{2}{5}, \\frac{2}{25}, \\dots$
    1. $\\frac{2}{125}$
    2. $\\frac{2}{3125}$
    3. $\\frac{2}{625}$
    4. $\\frac{2}{15}$
  60. If $a = 10$, $b = 20$, $c = 30$, what is the arithmetic mean of $a$ and $b$?
    1. 15
    2. 16
    3. 19
    4. 20
  61. In a class of 60 students, 40 speak Hausa, 19 speak Igbo, and 7 speak both. How many speak neither?
    1. 11
    2. 6
    3. 8
    4. 7
  62. The 2nd term of a G.P is 9, and the 4th term is 81. Find the first term.
    1. 3
    2. 9
    3. 2
    4. 6
  63. Find the sum of the first six terms of an A.P where the first term $a = 2$ and the last term $l = 20$.
    1. 84
    2. 66
    3. 72
    4. 80
  64. For a quadratic equation $ax^2 + bx + c = 0$, if the determinant is less than zero, describe its roots.
    1. Real and distinct
    2. Real and repeated
    3. Not real roots
    4. None
  65. Insert four arithmetic means between 243 and -1.
    1. 81, 27, 9, 3
    2. 128, 64, 32, 16
    3. 100, 75, 50, 25
    4. 90, 60, 30, 15
  66. Insert a geometric mean between $\\frac{x(x + 1)}{x - 2}$ and $\\frac{x(x + 1)^3}{(x - 2)^3}$.
    1. $\\frac{x(x + 1)^2}{(x - 2)^2}$
    2. $\\frac{x(x + 1)}{(x - 2)^2}$
    3. $\\frac{(x + 1)^2}{(x - 2)^2}$
    4. $\\frac{x^2(x + 1)^2}{(x - 2)^4}$
  67. Given \( P(n) = n^2(n+3)(n^2-1) \), which of the following correctly represents \( P(k+1) \)?
    1. \( (k+1)^2(k+4)(k^2 + 2k) \)
    2. \( (k+1)^2(k+1)(k^2-1) \)
    3. \( (k+1)^2(k+3)(k^2-1) \)
    4. \( (k+1)(k+2)(k+4) \)
  68. Given the universal set has subsets \( P = \\{ x \\mid x \\leq 3 \\} \) and \( Q = \\{ x \\mid -5 \\leq x < 12 \\} \), find \( P \\cap Q \).
    1. \( -5 \\leq x \\leq 3 \)
    2. \( -5 < x \\leq 3 \)
    3. \( -5 \\leq x < 3 \)
    4. \( -5 < x < 3 \)
  69. Find the sum to infinity of the series $-2 + \\frac{2}{3} - \\frac{2}{9} + \\dots$
    1. -3
    2. -2
    3. -4
    4. -1
  70. Determine the sum of the first five terms of the series $1 + 3 + 9 + \\dots$
    1. 221
    2. 242
    3. 321
    4. 121
  71. In a class of 60 students, 40 speak Hausa, 19 speak Igbo, and 7 speak both. How many speak neither?
    1. 11
    2. 6
    3. 8
    4. 7
  72. The 2nd term of a G.P is 35, and the 4th term is 875. Find the 3rd term.
    1. 175
    2. 245
    3. 105
    4. 125
  73. If the 8th term of a G.P is 4374 and the 5th term is 162, find the sum of the first four terms.
    1. 80
    2. 40
    3. 160
    4. 120
  74. Find the 8th term of a G.P whose first term is 3 and the common ratio is 2.
    1. 384
    2. 260
    3. 320
    4. 364
  75. In an A.P, the first term is 2, the last term is 50, and the common difference is 4. How many terms are there in the series?
    1. 12
    2. 13
    3. 14
    4. 15
  76. Find the sum of the first six terms of an A.P where the first term $a = 2$ and the last term $l = 20$.
    1. 66
    2. 72
    3. 78
    4. 84
  77. For what value of $n$ is $2^n > n^2$ true?
    1. $n \\geq 5$
    2. $n \\geq 2$
    3. $n \\geq 1$
    4. $n \\geq 4$
  78. Find the number that can divide $7^n - 3^n$ when $n$ is a positive integer.
    1. 4
    2. 5
    3. 6
    4. 7
  79. If $n(P \\cup Q) = 50$, $n(P) = 40$, and $n(Q) = 18$, find $n(P \\cap Q)$.
    1. 8
    2. 10
    3. 12
    4. 14
  80. Given the quadratic equation \( x^2 + 2x + 1 = 0 \), where \( \alpha \) and \( \beta \) are its roots, what is the value of \( \alpha^2 + \beta^2 \)?
    1. 2
    2. 0
    3. 1
    4. -2
  81. Suppose \( \alpha \) and \( \beta \) are the roots of the quadratic equation associated with the infinite series \( x, \frac{x(x-2)}{x+1}, \frac{x(x-2)^2}{(x+1)^2}, \dots \). For what values of \( x \) will this series have a sum to infinity?
    1. a. \( \left| \frac{x-2}{x+1} \right| \leq 1 \)
    2. b. \( x < 1 \)
    3. c. \( \left| \frac{x-2}{x+1} \right| < 1 \)
    4. d. \( \left| \frac{x(x-2)}{x+1} \right| < 1 \)

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