General Mathematics I — integral Past Questions

University of Ilorin MTH101: 35 questions.

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

1. Write $1 + \\sqrt{3}i$ in polar form
   1. $2[\\cos45^\\circ + i\\sin45^\\circ]$
   2. $2[\\cos60^\\circ - i\\sin60^\\circ]$
   3. $2[\\cos60^\\circ + i\\sin60^\\circ]$
   4. $\\sqrt{2}[\\cos60^\\circ + i\\sin60^\\circ]$
2. Simplify $\\sin60^\\circ\\cos45^\\circ - \\cos60^\\circ\\sin45^\\circ$
   1. $\\sin15^\\circ$
   2. $\\cos15^\\circ$
   3. $\\sin30^\\circ$
   4. $\\cos30^\\circ$
3. Find the 6th term in the binomial expansion of $(x - 2y)^{10}$
   1. $-252x^2y^5$
   2. $8064x^5y^5$
   3. $-8064x^5y^5$
   4. $252x^5y^5$
4. Find the value of $(\\cos\\theta - i\\sin\\theta)^2$
   1. $\\cos2\\theta - i\\sin2\\theta$
   2. $\\cos\\frac{\\theta}{2} - i\\sin\\frac{\\theta}{2}$
   3. $\\cos2\\theta + i\\sin2\\theta$
   4. $\\cos2\\theta - i\\sin2\\theta$
5. Find the value of $x$ and $y$ in the geometric sequence: $6, ar, ar^2, 162$
   1. 18, 54
   2. 3, 27
   3. 4, 64
   4. 18, -54
6. Find the sum of the series $7, \\frac{7}{4}, \\frac{7}{16}, \\dots$
   1. $-\\frac{28}{3}$
   2. $\\frac{3}{28}$
   3. $\\frac{28}{3}$
   4. $-\\frac{3}{28}$
7. Find the 5th term of the G.P $99, 33, 11, \\dots$
   1. $\\frac{81}{9}$
   2. $\\frac{9}{81}$
   3. $\\frac{99}{81}$
   4. $\\frac{11}{9}$
8. If $\\tan x = \\frac{12}{5}$, find the value of $\\cos x$
   1. $\\frac{14}{5}$
   2. $\\frac{5}{13}$
   3. $\\frac{12}{5}$
   4. $\\frac{12}{13}$
9. Find the value of $2.5\\pi$ in degrees
   1. 360
   2. 480
   3. 270
   4. 720
10. What is the value of 750° as an acute angle?
    1. 60°
    2. 30°
    3. 15°
    4. 10°
11. What is the length of an arc if the area of a sector from a circle of radius 10 cm is 50 cm²?
    1. 10 cm
    2. 15 cm
    3. 20 cm
    4. 25 cm
12. Find the length of an arc from a sector of radius 16 cm at angle 45°
    1. $\\frac{\\pi}{4}$ cm
    2. $4\\pi$ cm²
    3. $4\\pi$ cm
    4. $2\\pi$ cm
13. If 30 players play cricket, 45 play golf, 23 play hockey, and 11 play all, how many play cricket only?
    1. 20
    2. 22
    3. 30
    4. 14
14. For the quadratic equation $t^2 - 3t + 2 = 0$, find the value of $\\frac{\\alpha^2 + \\beta^2}{\\alpha\\beta}$
    1. $\\frac{7}{2}$
    2. $\\frac{2}{5}$
    3. $\\frac{5}{2}$
    4. $\\frac{2}{7}$
15. Find the 5th term of the expansion $(a + 2y)^{10}$
    1. 3630$a^6y^4$
    2. 3360$a^6y^4$
    3. 3306$a^5y^5$
    4. 3360$a^5y^5$
16. Express $\\cos240° - i\\sin240°$ as a polar form of an acute angle
    1. 30°
    2. 45°
    3. 60°
    4. 90°
17. Find the sum of the first 4 terms of the geometric series $6, 18, 54, \\dots$
    1. 3
    2. 63
    3. 240
    4. 218
18. Find the 11th term of the AP $2, 5, 8, 11, \\dots$
    1. 30
    2. 31
    3. 32
    4. 33
19. Find the value of $\\frac{-9 + 9i}{i - 2i}$
    1. 9 - 9i
    2. -9 + 9i
    3. 9 + 9i
    4. -9 - 9i
20. Find the angle form of the complex number $-5 - 5i$
    1. 30°
    2. 45°
    3. 60°
    4. 90°
21. If $\\tan\\theta = \\frac{-5}{-12}$, find $\\sin2\\theta$
    1. $\\frac{60}{169}$
    2. $\\frac{120}{169}$
    3. $-\\frac{120}{169}$
    4. $\\frac{60}{169}$
22. Find the 4th term in $(1 + x)^9$
    1. $84x^3$
    2. $81x^3$
    3. $80x^3$
    4. $50x^3$
23. Write $4[\\cos30^\\circ - i\\sin30^\\circ]$ in normal form
    1. $2\\sqrt{3} - 2i$
    2. $2\\sqrt{3} + 2i$
    3. $2 + 2\\sqrt{3}i$
    4. $2 + 3\\sqrt{2}i$
24. Find the 3rd term in $(x - y)^7$
    1. $37x^5y^2$
    2. $35x^3y^2$
    3. $21x^2y^2$
    4. $21x^5y^2$
25. Find the eleventh term of the series $8, 14, 20, 26, \\dots$
    1. 68
    2. 69
    3. 70
    4. 71
26. Find the tenth term of the series $5, 10, 20, 40, \\dots$
    1. 400
    2. 2560
    3. 2600
    4. 2550
27. Given $Z_1 = 2 + 4i$ and $Z_2 = 3 + 5i$, find $Z_1Z_2$
    1. $14 + 22i$
    2. $14 - 22i$
    3. $12 + 22i$
    4. $-14 + 22i$
28. Given $Z_1 = 2 + 4i$ and $Z_2 = 3 + 5i$, find $Z_1 + Z_2$
    1. $5 + 9i$
    2. $5 - 9i$
    3. $-5 + 9i$
    4. $-9 + 5i$
29. Find the value of $\\cos105^\\circ$
    1. $\\frac{1 + \\sqrt{3}}{2\\sqrt{2}}$
    2. $\\frac{1 - \\sqrt{3}}{2\\sqrt{2}}$
    3. $\\frac{3 - \\sqrt{2}}{2\\sqrt{2}}$
    4. $\\frac{2\\sqrt{2}}{1 - \\sqrt{3}}$
30. Find the value of $\\tan750^\\circ$
    1. $\\frac{2}{4 + 2\\sqrt{3}}$
    2. $4 + 2\\sqrt{3}$
    3. $4 + 2\\sqrt{3}$
    4. $\\frac{2}{4 - 2\\sqrt{3}}$
31. Convert $\\frac{\\pi}{2}$ to degrees
    1. 45°
    2. 60°
    3. 90°
    4. 180°
32. Find the equivalent of $290^\\circ$
    1. $-\\sin70^\\circ$
    2. $-\\sin76^\\circ$
    3. $-\\sin90^\\circ$
    4. $-\\sin50^\\circ$
33. Find the equivalent of $22^\\circ$
    1. 62°
    2. 42°
    3. 52°
    4. 40°
34. Given $Z_1 = 5[\\cos50^\\circ + i\\sin50^\\circ]$ and $Z_2 = 2[\\cos65^\\circ + i\\sin65^\\circ]$, find $Z_1Z_2$
    1. $25[\\sin115^\\circ + i\\cos115^\\circ]$
    2. $\\frac{5}{2}[\\cos115^\\circ + i\\sin115^\\circ]$
    3. $10[\\sin115^\\circ + i\\cos115^\\circ]$
    4. $25[\\sin15^\\circ + i\\cos15^\\circ]$
35. Find the sum to infinity of the series $2 + \\frac{3}{2} + \\frac{9}{8} + \\frac{27}{32} + \\dots$
    1. 4
    2. 8
    3. 10
    4. 14

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