General Mathematics I — ester Past Questions

University of Ilorin MTH101: 66 questions.

Topics: Sequence and Series, Trigonometry Function of Angle, Quadratic Equation, Binomial Theorem, Circular Measure, Complex Number, Set Theory and Venn Diagram, Number System

  1. If the nth term of a series is defined by $U_n = 2U_{n-1} - U_{n-2}$, and the first two terms are 1 and 2, find the sum of the next three terms.
    1. 3 + 4 + 5
    2. 4 + 5 + 6
    3. 5 + 6 + 7
    4. 6 + 7 + 8
  2. Expand $(2 - i)^3$.
    1. $8 - 12i + 6i^2 - i^3$
    2. $8 - 11i + 6i^2 - i^3$
    3. $8 - 10i + 6i^2 - i^3$
    4. $8 - 9i + 6i^2 - i^3$
  3. Evaluate $\\frac{a}{b} + \\frac{b}{a}$ if the sum of the roots is $\\frac{1}{2}$ and the product of the roots is 2.
    1. $-\\frac{15}{4}$
    2. $-\\frac{13}{4}$
    3. $-\\frac{11}{4}$
    4. $-\\frac{9}{4}$
  4. Insert four arithmetic means between 9 and 24.
    1. 12, 15, 18, 21
    2. 11, 14, 17, 20
    3. 10, 13, 16, 19
    4. 13, 16, 19, 22
  5. Expand $(1 - 2x)^4$.
    1. $1 - 8x + 24x^2 - 32x^3 + 16x^4$
    2. $1 - 6x + 24x^2 - 32x^3 + 16x^4$
    3. $1 - 8x + 20x^2 - 32x^3 + 16x^4$
    4. $1 - 8x + 24x^2 - 30x^3 + 16x^4$
  6. What is the root of the equation $x^2 + 3x - 7 = 0$?
    1. $\\frac{-3 \\pm \\sqrt{37}}{2}$
    2. $\\frac{-3 \\pm \\sqrt{29}}{2}$
    3. $\\frac{-3 \\pm \\sqrt{31}}{2}$
    4. $\\frac{-3 \\pm \\sqrt{35}}{2}$
  7. If the nth term of a series is defined by $U_n = 2U_{n-1} + U_{n-2}$, and the first two terms are 1 and 3, find the sum of the next five terms.
    1. 3 + 9 + 21 + 51 + 123
    2. 3 + 7 + 17 + 41 + 99
    3. 3 + 8 + 19 + 46 + 111
    4. 3 + 10 + 23 + 56 + 135
  8. Expand $(1 + x)^{1/2}$ up to the first three terms.
    1. $1 + \\frac{x}{2} - \\frac{x^2}{8}$
    2. $1 + \\frac{x}{2} - \\frac{x^2}{4}$
    3. $1 + \\frac{x}{2} + \\frac{x^2}{8}$
    4. $1 + \\frac{x}{2} + \\frac{x^2}{4}$
  9. If 400 students are in the mathematics department, 100 offer MAT111, 150 offer MAT113, and 75 offer both, how many students offer neither MAT111 nor MAT113?
    1. 225
    2. 250
    3. 275
    4. 300
  10. For the sequence defined by $U_n = U_{n-1} + 2U_{n-2}$ with $U_1 = 1$ and $U_2 = 3$, find the 3rd and 4th terms.
    1. 3rd term: 5, 4th term: 11
    2. 3rd term: 4, 4th term: 10
    3. 3rd term: 6, 4th term: 12
    4. 3rd term: 7, 4th term: 13
  11. Evaluate $\\frac{1 - i}{2 - 3i}$.
    1. $\\frac{5}{13} + \\frac{1}{13}i$
    2. $\\frac{5}{13} - \\frac{1}{13}i$
    3. $\\frac{1}{13} + \\frac{5}{13}i$
    4. $\\frac{1}{13} - \\frac{5}{13}i$
  12. For the equation $6x^2 - 5x + 1 = 0$, evaluate $a^3 + b^3$ where $a$ and $b$ are the roots.
    1. $\\frac{215}{216}$
    2. $\\frac{205}{216}$
    3. $\\frac{195}{216}$
    4. $\\frac{185}{216}$
  13. Evaluate $(5 + 2i) - (7 - 3i) + (3 - 4i)$.
    1. $1 + i$
    2. $1 - i$
    3. $-1 + i$
    4. $-1 - i$
  14. Find the sum of the first 6 terms of a GP whose first term is 6 and the 5th term is 486.
    1. $2(3^6 - 1)$
    2. $3(3^6 - 1)$
    3. $4(3^6 - 1)$
    4. $5(3^6 - 1)$
  15. Find the sum of the 5th and 7th terms of an AP whose first term is 6 and the second term is 18.
    1. 84
    2. 96
    3. 108
    4. 120
  16. Evaluate $\\sin 15^\\circ$.
    1. $\\frac{\\sqrt{6} - \\sqrt{2}}{4}$
    2. $\\frac{\\sqrt{6} + \\sqrt{2}}{4}$
    3. $\\frac{\\sqrt{3} - \\sqrt{2}}{4}$
    4. $\\frac{\\sqrt{3} + \\sqrt{2}}{4}$
  17. Evaluate $\\sin 4x + \\sin 2x$.
    1. $2 \\sin 3x \\cos x$
    2. $2 \\sin 2x \\cos 2x$
    3. $2 \\sin x \\cos 3x$
    4. $2 \\sin 3x \\sin x$
  18. Express $30^\\circ$ in radians.
    1. $\\frac{\\pi}{6}$
    2. $\\frac{\\pi}{4}$
    3. $\\frac{\\pi}{3}$
    4. $\\frac{\\pi}{2}$
  19. Given $\\sin \\alpha = \\frac{2}{3}$ and $\\cos \\beta = \\frac{1}{2}$, find $\\sin(\\alpha + \\beta)$.
    1. $\\frac{2 + \\sqrt{15}}{6}$
    2. $\\frac{2 + \\sqrt{10}}{6}$
    3. $\\frac{2 + \\sqrt{5}}{6}$
    4. $\\frac{2 + \\sqrt{3}}{6}$
  20. Express $\\sin 75^\\circ$ in surd form.
    1. $\\frac{\\sqrt{6} + \\sqrt{2}}{4}$
    2. $\\frac{\\sqrt{6} - \\sqrt{2}}{4}$
    3. $\\frac{\\sqrt{3} + \\sqrt{2}}{4}$
    4. $\\frac{\\sqrt{3} - \\sqrt{2}}{4}$
  21. The 4th term of an AP is 15, and the 9th term is 35. Find the 15th term.
    1. 59
    2. 61
    3. 63
    4. 65
  22. Find the formula for $U_n$ in the sequence $\\frac{1}{2}, 2, 8, 32, 128, \\dots$.
    1. $U_n = \\frac{1}{2} \\cdot 4^{n-1}$
    2. $U_n = \\frac{1}{2} \\cdot 2^{n-1}$
    3. $U_n = \\frac{1}{2} \\cdot 8^{n-1}$
    4. $U_n = \\frac{1}{2} \\cdot 16^{n-1}$
  23. Express $\\cos 300^\\circ$ in surd form.
    1. $\\frac{1}{2}$
    2. $\\frac{\\sqrt{3}}{2}$
    3. $-\\frac{1}{2}$
    4. $-\\frac{\\sqrt{3}}{2}$
  24. If $\\sin\\theta = \\sqrt{\\frac{2}{5}}$ and $\\theta$ is an obtuse angle, find $\\cos\\theta$.
    1. $-\\sqrt{\\frac{3}{5}}$
    2. $\\sqrt{\\frac{3}{5}}$
    3. $-\\frac{\\sqrt{3}}{5}$
    4. $\\frac{\\sqrt{3}}{5}$
  25. Evaluate $\\tan3A$.
    1. $\\frac{3\\tan A - \\tan^3 A}{1 - 3\\tan^2 A}$
    2. $\\frac{3\\tan A + \\tan^3 A}{1 - 3\\tan^2 A}$
    3. $\\frac{3\\tan A - \\tan^3 A}{1 + 3\\tan^2 A}$
    4. $\\frac{\\tan A - \\tan^3 A}{1 - 3\\tan^2 A}$
  26. Express $\\tan75^\\circ$ in surd form.
    1. $2 + \\sqrt{3}$
    2. $2 - \\sqrt{3}$
    3. $1 + \\sqrt{3}$
    4. $1 - \\sqrt{3}$
  27. A chord of length 3.1 cm subtends an angle of $60^\\circ$ at the centre of a circle. Calculate the length of the arc.
    1. 3.24 cm
    2. 3.14 cm
    3. 3.44 cm
    4. 3.34 cm
  28. Convert $75^\\circ$ to radians.
    1. $\\frac{5\\pi}{12}$
    2. $\\frac{\\pi}{12}$
    3. $\\frac{3\\pi}{12}$
    4. $\\frac{7\\pi}{12}$
  29. Express $\\frac{3\\pi}{4}$ in degrees.
    1. 135$^\\circ$
    2. 120$^\\circ$
    3. 150$^\\circ$
    4. 145$^\\circ$
  30. Evaluate $(1 - x)^6$.
    1. $1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6$
    2. $1 + 6x + 15x^2 + 20x^3 + 15x^4 + 6x^5 + x^6$
    3. $1 - 3x + 6x^2 - 10x^3 + 6x^4 - 3x^5 + x^6$
    4. $1 - 6x + 10x^2 - 15x^3 + 10x^4 - 6x^5 + x^6$
  31. What is the geometric mean between 3 and 243?
    1. 27
    2. 81
    3. 9
    4. 729
  32. Insert 5 arithmetic means between $\\frac{1}{4}$ and $-\\frac{39}{4}$.
    1. $-\\frac{7}{4}, -\\frac{15}{4}, -\\frac{23}{4}, -\\frac{31}{4}, -\\frac{39}{4}$
    2. $-\\frac{5}{4}, -\\frac{11}{4}, -\\frac{17}{4}, -\\frac{23}{4}, -\\frac{29}{4}$
    3. $-\\frac{9}{4}, -\\frac{19}{4}, -\\frac{29}{4}, -\\frac{39}{4}, -\\frac{49}{4}$
    4. $-\\frac{3}{4}, -\\frac{7}{4}, -\\frac{11}{4}, -\\frac{15}{4}, -\\frac{19}{4}$
  33. The sum of the 4th and 6th terms of an AP is 42. The sum of the 3rd and 9th terms is 52. Find the sum of the first 10 terms.
    1. 205
    2. 210
    3. 215
    4. 220
  34. If $\\alpha, \\beta$ are the roots of $2x^2 - x + 4 = 0$, find the value of $\\alpha^3 + \\beta^3$.
    1. $-\\frac{49}{4}$
    2. $\\frac{49}{4}$
    3. $-\\frac{47}{4}$
    4. $\\frac{47}{4}$
  35. If $\\alpha, \\beta$ are the roots of $2x^2 - 4x + 3 = 0$, find the value of $\\frac{1}{\\alpha^2} + \\frac{1}{\\beta^2}$.
    1. $\\frac{4}{9}$
    2. $\\frac{5}{9}$
    3. $\\frac{7}{9}$
    4. $\\frac{8}{9}$
  36. Express $-1 + \\sqrt{3}i$ in polar form.
    1. $2(\\cos120^\\circ + i\\sin120^\\circ)$
    2. $2(\\cos60^\\circ + i\\sin60^\\circ)$
    3. $\\sqrt{2}(\\cos135^\\circ + i\\sin135^\\circ)$
    4. $\\sqrt{2}(\\cos45^\\circ + i\\sin45^\\circ)$
  37. Evaluate $\\sin4x + \\sin2x$.
    1. $2\\sin3x\\cosx$
    2. $2\\cos3x\\sinx$
    3. $2\\sin3x\\sinx$
    4. $2\\cos3x\\cosx$
  38. Evaluate $\\cos4x - \\cos2x$.
    1. $-2\\sin3x\\sinx$
    2. $2\\sin3x\\sinx$
    3. $-2\\cos3x\\cosx$
    4. $2\\cos3x\\cosx$
  39. Express $30^\\circ$ to radians.
    1. $\\frac{\\pi}{6}$
    2. $\\frac{\\pi}{3}$
    3. $\\frac{\\pi}{4}$
    4. $\\frac{\\pi}{2}$
  40. Given that $\\sin\\alpha = \\frac{2}{3}$ and $\\cos\\beta = \\frac{1}{2}$, find $\\sin(\\alpha + \\beta)$.
    1. $\\frac{2 + \\sqrt{15}}{6}$
    2. $\\frac{2 - \\sqrt{15}}{6}$
    3. $\\frac{1 + \\sqrt{5}}{3}$
    4. $\\frac{1 - \\sqrt{5}}{3}$
  41. If $\\alpha$ and $\\beta$ are the roots of $3x^2 - 7x - 1 = 0$, find $\\frac{1}{\\alpha^3} + \\frac{1}{\\beta^3}$.
    1. $\\frac{343}{27}$
    2. $\\frac{343}{9}$
    3. $\\frac{343}{81}$
    4. $\\frac{343}{243}$
  42. The 4th term of an AP is 15, and the 9th term is 35. Find the 15th term.
    1. 55
    2. 50
    3. 45
    4. 60
  43. Express $2\\sin3x\\cosx$ as a sum of two trigonometric functions.
    1. $\\sin4x + \\sin2x$
    2. $\\cos4x + \\cos2x$
    3. $\\sin4x - \\sin2x$
    4. $\\cos4x - \\cos2x$
  44. Express $2\\sin5x\\sin4x$ as a sum of two trigonometric functions.
    1. $\\cosx - \\cos9x$
    2. $\\cos9x - \\cosx$
    3. $\\sin9x - \\sinx$
    4. $\\sinx - \\sin9x$
  45. Expand $(x + 2y)^6$.
    1. $x^6 + 12x^5y + 60x^4y^2 + 160x^3y^3 + 240x^2y^4 + 192xy^5 + 64y^6$
    2. $x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6$
    3. $x^6 + 12x^5y + 30x^4y^2 + 40x^3y^3 + 30x^2y^4 + 12xy^5 + y^6$
    4. $x^6 + 6x^5y + 30x^4y^2 + 80x^3y^3 + 120x^2y^4 + 96xy^5 + 32y^6$
  46. Given that $A = \\{x: x \\text{ are integers greater than 11}\\}$ and $B = \\{y: y \\text{ are odd numbers less than 10}\\}$, find $A \\cap B$.
    1. $\\{\\}$
    2. $\\{11, 13, 15, \\dots\\}$
    3. $\\{1, 3, 5, 7, 9\\}$
    4. $\\{11\\}$
  47. Find the value of $x$ given that $x+1, x+2, 3x+5$ are consecutive terms of an AP.
    1. 3
    2. 2
    3. 4
    4. 1
  48. Find the sum and product of the roots of the equation $5x^2 + 9x = -8$.
    1. Sum: $-\\frac{9}{5}$, Product: $\\frac{8}{5}$
    2. Sum: $\\frac{9}{5}$, Product: $-\\frac{8}{5}$
    3. Sum: $-\\frac{9}{5}$, Product: $-\\frac{8}{5}$
    4. Sum: $\\frac{9}{5}$, Product: $\\frac{8}{5}$
  49. Find the arithmetic mean of 4 and 12.
    1. 8
    2. 6
    3. 10
    4. 9
  50. In a class of 20 students, 10 like mathematics, 14 like English. If 6 students like both, how many students like neither mathematics nor English?
    1. 2
    2. 4
    3. 6
    4. 8
  51. Find the 4th term of the sequence $1 + 3 + 9 + \\dots$.
    1. 27
    2. 9
    3. 81
    4. 243
  52. Find the sum of the 12th term of the arithmetic series \(3 + 2\frac{1}{2} + 2 + 1\frac{1}{2} + \dots\)
    1. 18
    2. 20
    3. 22
    4. 24
  53. Find the geometric mean between 2 and 8
    1. 4
    2. 5
    3. 6
    4. 7
  54. Find the sum to infinity of \(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \dots\)
    1. 1
    2. \(\frac{1}{2}\)
    3. \(\frac{2}{3}\)
    4. \(\frac{3}{4}\)
  55. The sum of three terms of a G.P is 38 and their product is 1728. Find the terms.
    1. 6, 12, 20
    2. 8, 12, 18
    3. 6, 12, 24
    4. 9, 12, 17
  56. Evaluate \(\sin 3A\)
    1. \(3\sin A - 4\sin^3 A\)
    2. \(4\sin^3 A - 3\sin A\)
    3. \(3\sin A + 4\sin^3 A\)
    4. \(4\sin^3 A + 3\sin A\)
  57. Evaluate \(\cos B - \cos 13B\)
    1. \(2\sin 7B \sin 6B\)
    2. \(-2\sin 7B \sin 6B\)
    3. \(2\sin 7B \cos 6B\)
    4. \(-2\sin 7B \cos 6B\)
  58. Rational and irrational numbers belong to which set?
    1. Natural numbers
    2. Integers
    3. Real numbers
    4. Whole numbers
  59. Evaluate \(\tan 520^\circ\)
    1. \(\tan 20^\circ\)
    2. \(-\tan 20^\circ\)
    3. \(\cot 20^\circ\)
    4. \(-\cot 20^\circ\)
  60. Evaluate \(\sin 160^\circ\)
    1. \(\sin 20^\circ\)
    2. \(-\sin 20^\circ\)
    3. \(\cos 20^\circ\)
    4. \(-\cos 20^\circ\)
  61. If \(P = \{a, b, c, d, e\}\), \(Q = \{b, d, e\}\), and \(U = \{a, b, c, d, e, f, \dots\}\), find \(P' \cap Q'\)
    1. \(\{a, c\}\)
    2. \(\{f, g, \dots\}\)
    3. \(\{b, d, e\}\)
    4. \(\{a, c, f, g, \dots\}\)
  62. Express \(\sqrt{2} + i\sqrt{2}\) in polar form
    1. \((2, 30^\circ)\)
    2. \((2, 45^\circ)\)
    3. \((2, 60^\circ)\)
    4. \((2, 90^\circ)\)
  63. Given \(M = \{a, b, c, d, e, f\}\), \(P = \{b, c, k, f, i, j\}\), and \(N = \{e, f, h, i, j, g\}\), find \(M \cap (N \cap P)\)
    1. \(\{f\}\)
    2. \(\{e, f\}\)
    3. \(\{b, c\}\)
    4. \(\{i, j\}\)
  64. Given \(Y = (3u + 2x)^4\), what is the coefficient of \(u^3\) in the expansion?
    1. 96
    2. 108
    3. 216
    4. 324
  65. Find three consecutive terms of an A.P. whose sum is 3 and product is -15
    1. -1, 1, 3
    2. 3, 1, -1
    3. 5, 3, 1
    4. -3, 1, 5
  66. Find the geometric progression of the series \(-\frac{1}{3}, \frac{1}{2}, \frac{3}{4}, \frac{9}{8}\)
    1. First term \(a = -\frac{1}{3}\), common ratio \(r = -\frac{3}{2}\)
    2. First term \(a = -\frac{1}{3}\), common ratio \(r = \frac{3}{2}\)
    3. First term \(a = \frac{1}{2}\), common ratio \(r = \frac{2}{3}\)
    4. First term \(a = -\frac{1}{3}\), common ratio \(r = -\frac{2}{3}\)

Register free to unlock all 66 questions with answers and explanations

SmartDriller logo

SmartDriller

Loading your past questions, mock exams, notes, and video tutorials...

Starting up…