Nationella prov (Sweden) · Mathematics

Probability, Statistics, Combinatorics and Calculus Fundamentals

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

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All 39 questions

  1. Evaluate the derivative of the function f(x) = 4x^3 - 5x^2 + 7x - 9 with respect to x.
  2. Find the equation of the tangent line to the curve f(x) = x^2 - 4x + 6 at the point where x = 3.
  3. How many distinct 4-person committees can be formed from a group of 9 eligible candidates?
  4. A bag contains 5 red balls, 4 blue balls, and 3 green balls. Two balls are drawn without replacement. What is the probability that both balls are red?
  5. Find all critical points of the function g(x) = 2x^3 - 9x^2 + 12x + 1.
  6. A discrete dataset has values: 4, 7, 7, 8, 9, 10, 11, 14. What is the interquartile range (IQR) of this data?
  7. Exam scores in a national cohort follow a normal distribution with mean mu = 68 and standard deviation sigma = 8. What percentage of students score between 60 and 76?
  8. Evaluate the definite integral: int_{1}^{3} (3x^2 - 2x + 1) dx.
  9. A fair six-sided die is rolled four times. What is the probability of obtaining at least one six?
  10. Determine the local maximum value of the function f(x) = -x^3 + 3x^2 + 9x - 5.
  11. In how many distinct ways can the letters of the word 'SWEDEN' be arranged?
  12. For events A and B, P(A) = 0.6, P(B) = 0.5, and P(A U B) = 0.8. Find the conditional probability P(A | B).
  13. Find the point of inflection of the cubic curve y = x^3 - 6x^2 + 9x + 2.
  14. Calculate the sample standard deviation for the dataset {3, 5, 7, 9} using sample formula s = sqrt(sum (x - bar{x})^2 / (n - 1)).
  15. Evaluate the limit: lim_{x -> 3} (x^2 - 9) / (x^2 - 2x - 3).
  16. What is the area of the region enclosed by the curve y = 4 - x^2 and the x-axis?
  17. A random variable X is uniformly distributed on the interval [10, 30]. What is the variance Var(X)?
  18. If f'(x) = 6x^2 - 4x + 1 and f(1) = 5, find the function f(x).
  19. How many 5-digit PIN codes can be formed using digits 0-9 if no digit may be repeated?
  20. Find the maximum rate of increase of the function y = -x^3 + 6x^2 - 5x + 3.
  21. In a class of 30 students, 18 study German, 15 study French, and 8 study both languages. How many students study neither language?
  22. Evaluate: lim_{x -> 0} (sin(4x)) / (2x).
  23. The mean of five test scores is 82. When a sixth score is added, the new mean becomes 85. What was the sixth test score?
  24. Find the second derivative f''(x) of the rational function f(x) = 1 / x for x != 0.
  25. A discrete probability distribution has P(X = 1) = 0.2, P(X = 2) = 0.5, and P(X = 3) = 0.3. What is the expected value E(X)?
  26. For what values of x is the curve y = x^4 - 6x^2 concave downward?
  27. A box contains 6 white balls and 4 black balls. Three balls are drawn at random without replacement. What is the probability that all three are white?
  28. If velocity is given by v(t) = 3t^2 - 12t + 9 for t >= 0, at what times is the particle at instantaneous rest?
  29. Find the maximum area of a rectangle that can be enclosed by 40 metres of fencing.
  30. Two fair coins are flipped 6 times. What is the probability of getting exactly 4 heads?
  31. Find the median of the data set 12, 7, 3, 15, 9, 10, 4.
  32. A fair six-sided die is rolled three times. What is the probability of getting exactly two sixes?
  33. Find the derivative of f(x) = 5e^(2x).
  34. Evaluate the definite integral: int_{0}^{pi} sin x dx.
  35. A ball is thrown straight up, and its height after t seconds is h(t) = 20t - 5t^2 metres. What is its greatest height?
  36. A and B are independent events with P(A) = 0.4 and P(B) = 0.5. What is the probability that exactly one of the two events occurs?
  37. In how many ways can a captain and a vice-captain be chosen from a football team of 11 players?
  38. Calculate the population standard deviation of the data 2, 4, 4, 4, 5, 5, 7, 9.
  39. One card is drawn at random from a standard deck of 52 cards. What is the probability that it is a heart or a king?

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