General Secondary Certificate (Tawjihiya Kuwait) · Pure Mathematics

Vectors, Geometry, and Discrete Mathematics

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

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 31 questions

  1. Given vectors u = <3, -1, 4> and v = <2, 5, -1> in 3D space, what is their dot (scalar) product u . v?
  2. For what value of k are the vectors u = <2, k, -3> and v = <6, 4, 2> orthogonal (perpendicular)?
  3. Find the magnitude (norm) of the 3D vector v = <2, -3, 6>.
  4. What is the cross product u x v of vectors u = <1, 0, 0> and v = <0, 1, 0> in standard Cartesian coordinates?
  5. Find the standard equation of the plane passing through the point P(2, -1, 3) with normal vector n = <3, 4, -2>.
  6. Simplify the complex number expression: (3 + 4i) / (1 - 2i) into standard a + bi form.
  7. What is the modulus r and principal argument theta of the complex number z = -sqrt(3) + i?
  8. Using De Moivre's Theorem, evaluate (cos(pi/6) + i*sin(pi/6))^6.
  9. How many distinct 4-digit security codes can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if no digit can be repeated in a code?
  10. Find the coefficient of x^3 in the binomial expansion of (2x - 1)^5.
  11. A committee of 3 students is to be chosen from a group of 8 candidates. How many different committees can be selected?
  12. A box contains 5 red balls and 3 green balls. Two balls are drawn sequentially without replacement. What is the probability that both drawn balls are red?
  13. If A and B are independent events with P(A) = 0.4 and P(B) = 0.5, what is the probability of their union P(A or B)?
  14. A discrete random variable X has the probability distribution: P(X = 1) = 0.2, P(X = 2) = 0.5, and P(X = 3) = 0.3. What is the expected value E(X)?
  15. What is the distance from the point P(1, 2, 3) to the plane 2x - 2y + z + 5 = 0?
  16. Find the parametric equations of the line passing through point A(1, -2, 4) in the direction of vector v = <3, 2, -1>.
  17. In a standard normal distribution Z ~ N(0, 1), what is the probability P(-1 <= Z <= 1) given that Phi(1) approx 0.8413?
  18. What is the conjugate of the complex number z = (2 - 5i)^2?
  19. Determine the angle theta between vectors u = <1, 1, 0> and v = <0, 1, 1>.
  20. How many distinct arrangements of the letters in the word 'KUWAIT' can be formed?
  21. If a random variable X is normally distributed with mean mu = 70 and standard deviation sigma = 8, what is the standardized z-score corresponding to x = 86?
  22. What is the solution set of the trigonometric equation sin(2x) = cos(x) on the interval [0, 2*pi)?
  23. Find the area of the parallelogram determined by the 3D vectors u = <2, 3, 0> and v = <1, 4, 0>.
  24. A fair six-sided die is tossed 4 times. What is the probability of obtaining exactly two sixes?
  25. What is the eccentricity e of the ellipse given by the equation (x^2 / 25) + (y^2 / 16) = 1?
  26. Determine the sum of the infinite geometric series: 12 + 4 + 4/3 + 4/9 + ...
  27. What is the maximum value of on the closed interval ?
  28. Evaluate the definite integral: .
  29. In a geometric progression, the second term is 6 and the fifth term is 48. What is the sum of the first 6 terms?
  30. What is the equation of the normal line to the curve at the point ?
  31. How many distinct 4-letter arrangements can be formed using the letters of the word 'KUWAIT' if repetition is not allowed?

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