General Mathematics I — 2025B Past Questions

University of Ilorin MTH101: 50 questions.

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

  1. Insert three geometric means between \( X \) and \( 5X + 4 \).
    1. \( X, X+1, X+2 \)
    2. \( X, 2X, 4X \)
    3. \( X, Xr, Xr^2 \) where \( r \) is the common ratio
    4. \( X, X+4, X+8 \)
  2. Insert three geometric means between \( X + 7y \) and \( 5X + 3y \).
    1. \( X + 7y, X + 5y, X + 3y \)
    2. \( X + 7y, 2X + 5y, 4X + 3y \)
    3. \( X + 7y, (X + 7y)r, (X + 7y)r^2 \) where \( r \) is the common ratio
    4. \( X + 7y, X + 3y, X - y \)
  3. Consider the quadratic equation \( 2x^2 - 7x + 5 = 0 \). Calculate \( \alpha^3 + \beta^3 \).
    1. \( \frac{343}{8} \)
    2. \( \frac{203}{8} \)
    3. \( \frac{245}{8} \)
    4. \( \frac{147}{8} \)
  4. If the sum of 3 terms in an AP is 21, what is the arithmetic mean?
    1. 7
    2. 10
    3. 14
    4. 21
  5. For every possible integer \( n \), \( 7^n - 3^n \) is divisible by which \( n \)?
    1. 3
    2. 6
    3. 4
    4. 5
  6. Find the sum to infinity of a G.P with a first term of 81 and fourth term 3.
    1. 84.5
    2. 121.5
    3. 100
    4. 85
  7. In a school with 35 students, 22 play tennis and 18 play golf. If 4 play none of the sports, how many students play both?
    1. 9
    2. 10
    3. 11
    4. 12
  8. Find the sum to infinity of \( 32 + 16 + 8 + \dots \)
    1. 64
    2. 48
    3. 96
    4. 128
  9. Find the sum to infinity of \( -2 + \frac{2}{3} - \frac{2}{9} + \dots \)
    1. -3
    2. \( \frac{3}{2} \)
    3. 3
    4. \( -\frac{3}{2} \)
  10. The third term of a geometric progression is 63, the fifth term is 567. Find the sum of the first four terms.
    1. 120
    2. 210
    3. 105
    4. 315
  11. The fourth term of a G.P is 162 and the seventh term is 4374. What is the product of \( a \) and \( r \)?
    1. 9
    2. 6
    3. 12
    4. 18
  12. The first term of a G.P. is \( \frac{x(x+1)}{(x+1)} \), and the third term is \( \frac{x(x+1)^3}{(x+1)^3} \). What will be the second term?
    1. \(x^2(x+1)\)
    2. \(x(x^2+1)\)
    3. \(x(x+1)^2\)
    4. \(\frac{x(x+1)^2}{(x+1)^2}\)
  13. In a class of 60, 40 play football and 19 play basketball. If 7 play both sports, how many play none?
    1. 11
    2. 8
    3. 7
    4. 5
  14. The universal set \( \\xi \) is given as the set of all integers. \( P = \\{x : x < 3\\} \), \( Q = \\{x : -4 \\le x < 12\\} \). What is \( P \\cap (P \\cup Q)' \)?
    1. \( \\{x : x \\le -4\\} \)
    2. \( \\{-4 \\le x < 3\\} \)
    3. \( \\phi \)
    4. \( \\{x : x \\ge 12\\} \)
  15. If a quadratic equation is given as \( 2x^2 - 7x + 5 = 0 \), find \( \\frac{1}{\\alpha^2} + \\frac{1}{\\beta^2} \).
    1. \( \\frac{29}{25} \)
    2. \( \\frac{49}{4} \)
    3. \( -\\frac{49}{4} \)
    4. \( -\\frac{29}{25} \)
  16. Determine the sum of the first five terms in the series \( 1 + 3 + 9 + \\dots \)
    1. 321
    2. 121
    3. 242
    4. 221
  17. Find the cardinality of natural numbers less than 11.
    1. 9
    2. 10
    3. 11
    4. 12
  18. In the G.P \( 2, \\frac{2}{5}, \\frac{2}{25}, \\dots \), what is the fifth term?
    1. \( \\frac{2}{125} \)
    2. \( \\frac{2}{625} \)
    3. \( \\frac{2}{3125} \)
    4. \( \\frac{2}{15625} \)
  19. Insert three arithmetic means between \( x - 2 \) and \( x + 38 \).
    1. \( x - 2, x + 8, x + 18, x + 28 \)
    2. \( x - 2, x + 10, x + 22, x + 34 \)
    3. \( x - 2, x + 12, x + 26, x + 38 \)
    4. \( x - 2, x + 6, x + 14, x + 22 \)
  20. In the principle of induction, the propositional state \( P_n \) is true for \( n \) in the set of
    1. Rational Numbers
    2. Real Integers
    3. Natural numbers
    4. Integers
  21. In a class of 60 students, 40 Speak Hausa, 19 Speak Igbo and 7 Speak both languages. How many speak neither?
    1. 8
    2. 6
    3. 7
    4. 9
  22. For a quadratic equation \( ax^2 + bx + c = 0 \), where \( a \\neq 0 \) and \( D < 0 \), then the roots are?
    1. Real and distinct
    2. Real and equal
    3. Complex conjugates
    4. No roots
  23. If \( S_n = 2n^2 + 3n \), what is the nth term of the sequence?
    1. \( 4n + 1 \)
    2. \( 4n + 5 \)
    3. \( 4n - 1 \)
    4. \( 4n - 5 \)
  24. If \( P(n) = n^2 (n+3) (n^2 - 1) \), find \( P(k+1) \).
    1. \( (k+1)^2 (k+4) (k^2 + 2k) \)
    2. \( (k+1)^2 (k+3) (k^2 - 1) \)
    3. \( (k+1)^2 (k+2) (k+4) \)
    4. \( (k+1) (k+2) (k+4) \)
  25. If \( x - x^2 + x^3 - x^4 + \\dots \) is a geometric series. Find the sum to infinity.
    1. \( \\frac{x}{1 + x} \)
    2. \( \\frac{x}{1 - x} \)
    3. \( \\frac{1}{1 + x} \)
    4. \( \\frac{1}{1 - x} \)
  26. Suppose \( x \\neq 1 \), for what values of \( x \) will \( x, \\frac{x(x-2)}{(x+1)}, \\frac{x(x-2)^2}{(x+1)^2} \) have a Sum to Infinity?
    1. \( \\left| \\frac{x-2}{x+1} \\right| \\le 1 \)
    2. \( x < 1 \)
    3. \( \\left| \\frac{x-2}{x+1} \\right| < 1 \)
    4. \( \\left| \\frac{x(x-2)}{x+1} \\right| < 1 \)
  27. Find the Sum of the first 6 terms of a Series \( 18, 6, 2, \\dots \)
    1. \( \\frac{728}{27} \)
    2. \( \\frac{27}{729} \)
    3. \( \\frac{729}{27} \)
    4. \( \\frac{27}{728} \)
  28. The sum of the nth term of an AP is \( 5n^2 - 3n \). Find the 10th term.
    1. 92
    2. 94
    3. 96
    4. 102
  29. The nature of the root of an equation \( x^2 - 2x + 1 = 0 \) is
    1. Repeated and imaginary
    2. Repeated and real
    3. Real and distinct
    4. Imaginary and distinct
  30. For sets \( A, B \) and \( C \), \( A \\cap (B \\cup C) \) is
    1. \( (A \\cap B) \\cap (A \\cap C) \)
    2. \( (A \\cup B) \\cup (A \\cup C) \)
    3. \( (A \\cap B) \\cup (A \\cap C) \)
    4. \( (A \\cup B) \\cap (A \\cup C) \)
  31. For which integer values of \( n \) does \( n^2 \\le 2^n \) hold?
    1. \( n = 0, 1, 2, 4, 5 \)
    2. \( n = -1, 0, 1, 2, 4 \)
    3. \( n = 2, 3, 4 \)
    4. \( n = 0, 1, 2, 3, 4 \)
  32. Given the universal set \( U \) has subsets \( P = \\{x: x \\le 3\\} \) and \( Q = \\{x: -5 < x < 12\\} \). Find \( P \\cap Q \).
    1. \( \\{x: -5 < x \\le 3\\} \)
    2. \( \\{x: x \\le 3\\} \)
    3. \( \\{x: -5 < x < 12\\} \)
    4. \( \\{x: x < 12\\} \)
  33. If the nth term of a Sequence is \( (-1)^n \), it is
    1. Divergent
    2. Alternating
    3. Increasing
    4. Decreasing
  34. If the second term of a GP is 24 and the 5th term is 81, find the first term.
    1. 16
    2. 18
    3. 20
    4. 22
  35. Insert three geometric means between -243 and -1.
    1. -81, -27, -9
    2. -81, 27, -9
    3. 81, -27, 9
    4. -81, -27, 9
  36. What is the sum to infinity of \( 1 + \\sqrt{3} + 3 + \\dots \)?
    1. Does not converge
    2. \( \\frac{1}{1 - \\sqrt{3}} \)
    3. \( \\frac{1}{1 + \\sqrt{3}} \)
    4. 0
  37. For which values of \( n \) will \( 2^{4n-1} - 2^n \) be divisible by 8?
    1. 0
    2. 1
    3. 3
    4. 5
  38. Given that \( A = \\emptyset \), find the cardinality of \( P(A) \).
    1. 0
    2. 2
    3. 1
    4. No element
  39. The 6th term of an AP is 20 and the 10th term is 32. Find the sum of the first 10 terms.
    1. 185
    2. 200
    3. 195
    4. 225
  40. Insert 2 geometric means between \( \\frac{x}{p^2} \) and \( xp^2 \).
    1. \( \\frac{x}{p}, x \)
    2. \( x, xp \)
    3. \( \\frac{x}{p}, xp \)
    4. \( xp, xp^2 \)
  41. In a school with 166 students, 32 study only Math, 7 study only Physics, 27 study only Chemistry, and others study combinations. How many study at least two subjects?
    1. 100
    2. 95
    3. 90
    4. 85
  42. From the Venn diagram, find \( n(R \\cap Q \\cap N) \).
    1. 2
    2. 3
    3. 4
    4. 7
  43. From the Venn diagram, find \( n(N) \).
    1. 37
    2. 44
    3. 51
    4. 58
  44. From the Venn diagram, find \( n(E - R) \), where \( E \) is the universal set.
    1. 41
    2. 58
    3. 63
    4. 72
  45. 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
  46. According to mathematical induction, which of these is a factor of \( 2^{4n-1} \)?
    1. 3
    2. 5
    3. 8
    4. 10
  47. A school has 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
  48. \( A \\cap (B \\cup C') \) is
    1. \( A - (B \\cup C) \)
    2. \( A \\cap B \\cap C \)
    3. \( (A \\cap B)' \)
    4. \( A \\cup B \)
  49. If \( a = 10 \), \( b = 20 \), and \( c = 30 \) in an A.P, the arithmetic mean of \( a \) and \( b \) is?
    1. 21
    2. 15
    3. 18
    4. 9
  50. 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

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