QCAA Unit 4 Maths Methods IA3 – Short Answer Practice Questions

Question 6 (4 marks)

An object travels in a straight line. Its displacement from the origin is xm/s, at time t seconds is given by the following function:

x(t) = 0.5e^x + 2x – 0.5, 0 ≤ t ≤ 5

a. Determine an expression for the velocity of the particle as a function of time

b. Determine an expression for the acceleration of the particle as a function of time

c. Calculate the initial speed of the particle 

d. Determine when and at what position the particle is at rest

Question 7 (2 marks)

Solve 2 sin sin (2x) – 2 = 0, 0 ≤ x ≤2 π

Question 8 (2 marks)

Solve the following:

2 cos cos (x) – √3 = 0, 0 ≤ x ≤ 3π

Question 9 (3 marks)

A new medication has been developed for the treatment of hay fever. In lab trials, the medication was found to be effective in 81% of the cases. Kim has been prescribed the medication. Letting X be the effectiveness of the medication:

a. Construct a probability distribution table for X 

b. Determine E(X)

c. Determine the variance and standard deviation of X, correct to four decimal places

Question 10 (5 marks)

Sketch the graph of f(x) = xe^x + 2. Calculate and label the x and/or y-intercepts, stationary points and points of inflection. Giver your answers to two decimal places.

Question 11 (2 marks)

The probability of Jim hitting the centre of the dartboard is 0.65. What is the smallest number of darts he needs to shoot to ensure that the probability of him hitting the centre at least once is more than 0.8?

Question 12 (2 marks)

Amy has a math test that consists of 40 multiple choice questions which have 4 answer options each. She chooses her answers all at random. Let X be the discrete random variable that describes the number of answers she gets correct. 

a. Determine the expected number of questions Amy has gotten correct.

b. Determine the standard deviation of the number of questions answered correctly. Give your answer to two decimal places. 

Question 13 (3 marks)

Z, a binomial random variable, has a mean of 6.3 and a variance of 2.3. 

a. Determine the probability of success (p). Give your answer to two decimal places.

b. Determine the number of trials (n). Give your answer to two decimal places.

Question 14 (4 marks)

A recent study has shown that 14% of the population has asthma. A sample of 20 people were tested for asthma. Let X be the random variable that represents the number of people who have asthma. 

a. Determine P(X≤5). Give your answer to four decimal places.

b. Determine E(X) and SD(X). Give your answer to four decimal places.

Question 15 (6 marks)

Sketch the graphs of the following functions and state whether each function is a probability density function. 

a. f(x) = ∫ 2(x + 2) – 6,  1 ≤ x ≤ 4, 0 elsewhere 

b. f(x) = ∫ 0.5x – 2,  4 ≤ x ≤6, 0 elsewhere

Question 16 (2 marks)

Z is a normally distributed random variable, determine:

a. x given that P(Z > x) = 0.45, Z ~ N (14.5, 1.3²)

b. n given that P(Z < n) = 0.67, Z ~ N (19, 2.1²)

Question 17 (4 marks)

The amount of strawberries packed into one carton for a particular brand is normally distributed with a mean of μg and a standard deviation of 5g. The advertised weight of each strawberry carton is 300g. 

a. Calculate the number of packets that will be under the advertised weight when μ = 310g. Give your answer to four decimal places. 

b. Calculate the value of μ required to ensure that only 2% of packets are below the advertised weight. Give your answer to four decimal places.

Question 18 (4 marks)

James surveys 200 people and finds that 30 of them like mint ice-cream. Estimate the proportion of the population that likes mint ice-cream. Determine a 95% confidence interval. Give your answers to four decimal places.

Question 19 (2 marks)

On average, 30% of people spend at least 2 hours a day on social media. Using the normal approximation, determine the approximate probability that, in a sample size of 500, more than 45% of people have spent at least 2 hours a day on social media. Give your answer to three decimal places. 

Question 20 (2 marks)

Find the mean of the following probability density function:

f(x) = ∫ 3x² – 6x, 2 ≤ x ≤ 4, 0 elsewhere

Question 21 (3 marks)

Find the median of the following function

f(x) = ∫ 2x – 3, 4 ≤ x ≤ 6, 0 elsewhere

Question 22 (2 marks)

Question 23 (2 marks)

The probability of the train being late is 0.25. What is the probability that the train is on time at least once in the next four days?

Question 24 (2 marks)

Find the length of the side labelled p. Give your answer to two decimal places.

Image sourced from Centre for Innovation in Mathematics Teaching

Question 25 (2 marks)

Find the length of the unknown side.

Image sourced from Centre for Innovation in Mathematics Teaching

BONUS QUESTION (2 marks) 

If N = 450, n = 50 and p = 0.4, calculate:

a. the mean of the distribution. 

b. the standard deviation of the distribution. Give your answer to two decimal places.

QCAA Unit 4 Maths Methods Short Answer Practice Solutions

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