Mathematical Methods Year 12 (QCE) External assessment — Paper 2

Queensland Curriculum and Assessment Authority · Mathematical Methods (Year 12 (QCE)) · 2024 AD · External assessment — Paper 2 — 31 questions with worked answers (solutions) — a free old question paper on Goodone.club’s exam-paper archive.

  1. A binomial random variable X has probability of success p = 0.72 for n = 8 trials. The mean of X is closest to: (A) 1.61 (B) 2.24 (C) 5.76 (D) 7.28 [1 marks]
  2. A continuous random variable X has probability density function p(x) = (1/4)x^2 for 0 <= x <= cube root of 12. The expected value E(X) is closest to: (A) 1.72 (B) 1.15 (C) 1.00 (D) 0.11 [1 marks]
  3. Given f'(x) = sin(2x) and f(pi/2) = 4, then f(x) = (A) -cos(2x) + 3 (B) cos(2x) + 5 (C) -0.5cos(2x) + 3.5 (D) 0.5cos(2x) + 4.5 [1 marks]
  4. A fair six-sided die is rolled once and success is rolling a four. The variance of this Bernoulli random variable is closest to: (A) 0.027 (B) 0.138 (C) 0.16 (D) 0.83 [1 marks]
  5. The mass of a kookaburra is normally distributed with mean 300 g and standard deviation 13 g. Which statement is correct? (A) 34% have a mass between 287 g and 313 g (B) 68% have a mass between 274 g and 326 g (C) 95% have a mass between 261 g and 326 g (D) 99.7% have a mass between 261 g and 339 g [1 marks]
  6. Determine the derivative of y = 2x*cos(3x). (A) 2cos(3x) - 6x*sin(3x) (B) 2cos(3x) + 6x*sin(3x) (C) -6sin(3x) (D) -2sin(3x) [1 marks]
  7. In the triangle shown (not to scale), one side is 8 cm, the base is 10 cm, and the angle opposite the 8 cm side is 40 degrees. Identify the possible values for the angle a in the triangle. (A) 13.5 or 126.5 (B) 53.5 or 126.5 (C) 53.5 or 86.5 (D) 13.5 or 86.5 [1 marks]
  8. Calculate the total enclosed area between the graph of y = x^2 - x - 6 and the x-axis from x = 1 to x = 5. (A) 5.33 (B) 7.33 (C) 12.67 (D) 20.00 [1 marks]
  9. It is known that f'(x) = 0 and f''(x) < 0 for one of the labelled points on the graph of f(x). Which point matches this description? (A) P (B) Q (C) R (D) S [1 marks]
  10. The velocity (m/s) at time t (s) of an object is given by v(t) = 0.4t^2 + 3t for t >= 0. The change in displacement (m) of the object from four to five seconds is: (A) 15.43 (B) 21.63 (C) 32.53 (D) 54.17 [1 marks]
  11. State the trapezoidal rule and use it with six strips to determine an approximate value of the definite integral for the curve of f(x) = 4(x - 3)^2 from x = 0 to x = 3. Show all substitutions made into the rule. [4 marks]
  12. The magnitude of an earthquake can be modelled by M_A = log10(I_A / I_0), where M_A is the magnitude at location A, I_A is the intensity at location A and I_0 is a constant. An earthquake at location P had a magnitude of 5.2. A different earthquake at location Q had a magnitude of 3.5.
  13. The number of termites in a particular nest can be modelled by N(t) = A / (2 + e^(-t)), where A is a constant and t is time (months) since the nest first became a visible mound above ground level. When the mound first became visible, the population was 3 x 10^5 termites.
  14. A football coach offered a 12-day intensive training clinic. One player's kick heights could be modelled by H(t) = log10(10t + 10) + 5, for 0 <= t <= 12, where H(t) is vertical height (m) and t is the time (days) spent in training.
  15. The term extremely tall is used to describe any person whose height is three standard deviations or more above the mean height of the population. A person who just qualifies as extremely tall in a country where heights are normally distributed with a mean of 180 cm and a standard deviation of 10 cm travels to another country, where they discover they are taller than exactly 90% of that country's population. Assuming the standard deviation of both countries is the same, determine the minimum height required to be considered extremely tall in the destination country. [4 marks]
  16. At council meetings in a particular town, new proposals are only discussed if more than 80% of the community are in favour of the proposal. To discover community opinion on a new bus route proposal, the council conducted several surveys, each with a sample size of 120. The distribution of the sample proportions from the surveys had a standard deviation of 0.04. Make a justified decision as to whether the new bus route proposal would be discussed at a council meeting. [4 marks]
  17. At a particular orchard, 3% of fruit is bruised during picking. After picking, the fruit is packed into boxes, each containing four pieces of fruit. A grocery shop orders 140 boxes of fruit to sell to their customers. Determine the expected number of boxes that will contain bruised fruit. [3 marks]
  18. An object experiencing straight-line motion along a path has an acceleration (m/s^2) defined by a(t) = 3sin(2t), where t is time (s) since the object begins moving, t >= 0. When t = 0, both displacement and velocity are zero. On the path is a motion sensor that is able to detect motion up to 2 metres away. The object passes directly by the motion sensor when t = 3. Determine the average velocity of the object while it moves through the range of the sensor. [5 marks]
  19. The normal distribution probability density function is p(x) = (1 / (sigma * sqrt(2*pi))) * e^(-0.5 * ((x - mu)/sigma)^2), with mean mu and standard deviation sigma. The speeds of electric scooter (e-scooter) riders on a particular section of a bike path are approximately normally distributed with a mean of 18 km/h. It is known that p(10) = 0.0135. The speed limit for e-scooters on this section of bike path is 23 km/h. A speed camera records the speeds of 75 e-scooter riders. Every rider travelling faster than the speed limit is given a fine of AUD 143. Evaluate the reasonableness of the sugge... [6 marks]