Qatar Senior Secondary Certificate (Thanaweya) · Advanced Mathematics

Algebra, Trigonometry and Probability

103 practice questions, each with the answer and an explanation.

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All 103 questions

  1. What is the modulus and principal argument of the complex number z = -1 + i * sqrt(3)?
  2. Using De Moivre's Theorem, evaluate (cos(pi/6) + i sin(pi/6))^6.
  3. In the binomial expansion of (2x - 1/x)^8, what is the term independent of x (the constant term)?
  4. Solve the trigonometric equation 2 cos^2(x) - cos(x) - 1 = 0 on the interval [0, 2pi).
  5. What is the sum of all three-digit positive integers that are multiples of 7?
  6. A committee of 4 students is to be selected at random from 6 boys and 5 girls. What is the probability that the committee contains exactly 2 boys and 2 girls?
  7. Solve the system of linear equations using Cramer's Rule:
    2x + 3y = 8
    5x - y = 3
  8. If the roots of the quadratic equation x^2 - 6x + k = 0 are in the ratio 1:2, what is the value of k?
  9. What is the exact value of tan(75°)?
  10. In a normal distribution with mean mu = 70 and standard deviation sigma = 8, what percentage of the population lies between 54 and 86 according to the Empirical Rule (68-95-99.7)?
  11. Find the value of x that satisfies the logarithmic equation: log_3(x + 6) - log_3(x - 2) = 2.
  12. What is the inverse matrix A^(-1) of the 2x2 matrix A = [3, 5; 1, 2]?
  13. A biased coin is tossed 5 times. The probability of obtaining heads on any single toss is p = 0.6. What is the probability of getting exactly 3 heads?
  14. Simplify the trigonometric expression: [sin(2x)] / [1 + cos(2x)].
  15. What is the dot product of vectors u = 3i - 2j + 4k and v = 2i + 5j - k in three-dimensional space?
  16. Solve the exponential equation 4^x - 3 * 2^x - 4 = 0 for real values of x.
  17. What is the mathematical expectation E(X) of a discrete random variable X with probability distribution: P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3?
  18. Find the acute angle between the two lines y = 2x + 1 and y = 3x - 4 using the tangent angle formula.
  19. How many terms of the geometric sequence 2, 6, 18, 54, ... must be added to obtain a sum of 2186?
  20. If events A and B are independent events with P(A) = 0.4 and P(B) = 0.5, what is the probability of the union P(A union B)?
  21. What is the cross product u x v of vectors u = (1, 0, 2) and v = (0, 3, -1)?
  22. Solve the logarithmic inequality: log_2(x - 3) < 3.
  23. What is the coefficient of x^3 in the polynomial expansion of (1 - 2x)^5?
  24. If sin(alpha) = 3/5 with alpha in Quadrant I, and cos(beta) = -12/13 with beta in Quadrant II, what is the value of sin(alpha + beta)?
  25. What is the distance between the two parallel planes 2x - y + 2z = 4 and 2x - y + 2z = 13?
  26. In how many ways can 5 distinct books be arranged on a shelf such that two specific mathematics books are always placed next to each other?
  27. A bag holds 3 red and 2 blue marbles. Two are drawn without replacement. Given that the second marble is red, what is the probability that the first was also red?
  28. How many different arrangements of the letters of the word BALLOON have the two L's next to each other?
  29. How many solutions does the equation 2 sin^2 x + 3 cos x - 3 = 0 have in the interval [0, 2π)?
  30. What is the coefficient of x^3 in the expansion of (2x - 1/x)^9?
  31. An arithmetic sequence has first term 3 and common difference 4. Find the sum of its first 10 terms.
  32. An arithmetic sequence has first term 5 and common difference 2. Find the sum of its first 12 terms.
  33. An arithmetic sequence has first term -4 and common difference 3. Find the sum of its first 8 terms.
  34. An arithmetic sequence has first term 2 and common difference 5. Find the sum of its first 15 terms.
  35. An arithmetic sequence has first term 10 and common difference -2. Find the sum of its first 9 terms.
  36. An arithmetic sequence has first term 7 and common difference 3. Find the sum of its first 14 terms.
  37. A geometric sequence has first term 2 and common ratio 3. Find the sum of its first 5 terms.
  38. A geometric sequence has first term 1 and common ratio 4. Find the sum of its first 4 terms.
  39. A geometric sequence has first term 4 and common ratio 3. Find the sum of its first 4 terms.
  40. A geometric sequence has first term 1 and common ratio 3. Find the sum of its first 7 terms.
  41. Find the sum to infinity of the geometric series 6 + 3 + 3/2 + ...
  42. Find the sum to infinity of the geometric series 8 + 8/3 + 8/9 + ...
  43. Find the sum to infinity of the geometric series 10 + 20/3 + 40/9 + ...
  44. Find the sum to infinity of the geometric series 12 + 9 + 27/4 + ...
  45. Find the sum to infinity of the geometric series 9 + 9/4 + 9/16 + ...
  46. Find the sum to infinity of the geometric series 5 + 1 + 1/5 + ...
  47. Solve log_2(x) = 5.
  48. Solve log_3(x) = 3.
  49. Solve log_2(x) = 7.
  50. Solve log_5(x) = 2.
  51. Solve log_3(x) = 4.
  52. Solve log_4(x) = 3.
  53. Find the product of the roots of x^2 - 5x + 6 = 0.
  54. Find the product of the roots of x^2 - 5x + 4 = 0.
  55. Find the product of the roots of x^2 - 8x + 15 = 0.
  56. Find the product of the roots of x^2 - 8x + 12 = 0.
  57. Find the product of the roots of x^2 - 8x + 7 = 0.
  58. Find the product of the roots of x^2 - 9x + 20 = 0.
  59. Write (1 + 2i)(3 - i) in the form x + yi.
  60. Write (2 + i)(1 + 3i) in the form x + yi.
  61. Write (3 - 2i)(2 + i) in the form x + yi.
  62. Write (2 - 3i)(4 + i) in the form x + yi.
  63. Write (5 + 2i)(1 - 2i) in the form x + yi.
  64. Write (1 + 4i)(2 + 3i) in the form x + yi.
  65. In how many ways can a committee of 3 be chosen from 6 people?
  66. In how many ways can a committee of 2 be chosen from 7 people?
  67. In how many ways can a committee of 3 be chosen from 8 people?
  68. In how many ways can a committee of 2 be chosen from 6 people?
  69. In how many ways can a committee of 2 be chosen from 9 people?
  70. Events A and B satisfy P(B) = 1/3 and P(A and B) = 1/6. Find P(A | B).
  71. Events A and B satisfy P(B) = 1/2 and P(A and B) = 3/8. Find P(A | B).
  72. Events A and B satisfy P(B) = 3/5 and P(A and B) = 2/5. Find P(A | B).
  73. Events A and B satisfy P(B) = 2/5 and P(A and B) = 1/5. Find P(A | B).
  74. Events A and B satisfy P(B) = 1/2 and P(A and B) = 3/10. Find P(A | B).
  75. A basketball player scores with probability 1/2 on each independent shot. Find the probability of exactly 2 scores in 3 shots.
  76. A basketball player scores with probability 2/3 on each independent shot. Find the probability of exactly 2 scores in 3 shots.
  77. A basketball player scores with probability 1/3 on each independent shot. Find the probability of exactly 2 scores in 3 shots.
  78. A basketball player scores with probability 1/2 on each independent shot. Find the probability of exactly 2 scores in 4 shots.
  79. A basketball player scores with probability 3/4 on each independent shot. Find the probability of exactly 2 scores in 3 shots.
  80. A basketball player scores with probability 1/2 on each independent shot. Find the probability of exactly 3 scores in 4 shots.
  81. Given that theta is acute and sin(theta) = 3/5, find sin(2 theta).
  82. Given that theta is acute and sin(theta) = 5/13, find sin(2 theta).
  83. Given that theta is acute and sin(theta) = 8/17, find sin(2 theta).
  84. Given that theta is acute and sin(theta) = 7/25, find sin(2 theta).
  85. Given that theta is acute and sin(theta) = 20/29, find sin(2 theta).
  86. Given that theta is acute and sin(theta) = 9/41, find sin(2 theta).
  87. Find the dot product of the vectors a = (1, 2) and b = (3, -1).
  88. Find the dot product of the vectors a = (2, 3) and b = (-1, 4).
  89. Find the dot product of the vectors a = (3, 1) and b = (2, 5).
  90. Find the dot product of the vectors a = (-2, 4) and b = (1, 3).
  91. Find the dot product of the vectors a = (1, -3) and b = (5, 2).
  92. Find the determinant of the matrix with rows (2, 3) and (1, 4).
  93. Find the determinant of the matrix with rows (3, 5) and (2, 1).
  94. Find the determinant of the matrix with rows (4, 1) and (3, 2).
  95. Find the determinant of the matrix with rows (5, 2) and (1, 3).
  96. Find the determinant of the matrix with rows (6, 4) and (2, 1).
  97. Find the determinant of the matrix with rows (7, 2) and (3, 1).
  98. A culture of 100 bacteria doubles every 3 hours. How many bacteria are there after 12 hours?
  99. A culture of 50 bacteria doubles every 4 hours. How many bacteria are there after 12 hours?
  100. A culture of 200 bacteria doubles every 2 hours. How many bacteria are there after 10 hours?

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