QCAA Unit 3 Maths Methods IA2 – Short Answer Practice Questions

Question 6 (4 marks)

The graph of the function f(x) = 0.3log_e(x – 2) is shown below.

a. Determine the value of a given that it is the x-intercept
b. State the domain and range of the function
c. Determine the gradient of the curve at (a,0)
d. Determine the equation of the tangent at (a,0)

Question 7 (3 marks)

Solve 4 sin sin (2x) – 2 = 0, 0 ≤ x ≤2 π

Question 8 (4 marks)

a. Calculate the depth of the water at low tide and when it will next occur

b. Determine the rate of change of the depth of water with respect to time at 3:00 pm

Question 9 (2 marks)

Differentiate the following:

a. log₃2 + 2log₃x = log₃(7x – 3)

b. 3lnx = ln125

Question 10 (3 marks)

A deposit of $5000 is invested at Bank A, and a deposit of $7000 is invested at bank B. Bank A offers compound interest continuously at a rate of 6% per annum, whereas Bank B offers compound interest continuously at a rate of 5% per annum. In how many years will the two investments be the same?

Give your answer to the nearest appropriate year. Use the compound interest equation below: 

A = Pe^rt

where:

A = accumulated dollars
P = initial deposit
r = rate
t = years

Question 11 (2 marks)

Derive the following functions:

Question 12 (3 marks)

Given that f(x) = x(5 – 4x)², evaluate f'(x) and determine the equation of the tangent to the curve at point (1,1)

Question 13 (4 marks)

The formula below represents the profit (P), in dollars, that a business makes by selling its products each day.

a. Calculate the number of items that should be sold to maximise the profit on each item.

b. Calculate the profit per item by selling this number of items

c. Calculate the total profit per day made by selling this number of items

Question 14 (4 marks)

The displacement of a particle, in x metres, from a fixed origin at t seconds can be modelled by the following function:

a. Calculate the initial position of the particle

b.Determine the velocity function for the particle and calculate its initial velocity

c. Calculate the time and position at which the particle is at rest

Calculator Questions

Question 15 (2 marks)

Solve the following. Give your answer to two decimal places.

Question 16 (1 mark)

The area of the region enclosed by the graphs of y = x(x²– 3) and y = e^-x to two decimal places is…

Question 17 (1 mark)

The graph of y = e(x³– 4x) is concave up for which x values?

Question 18 (4 marks)

The temperature (T) in Brisbane in °C on a particular day can be modelled by the following function: 

T = -3cos cos (0.5t) + 20, 0 ≤ x ≤ 24

where t is the number of hours after 12am (start of that day)

a. Calculate the temperature at 6am. Give your answer to the nearest degree. 

b. Calculate the rate of change at any time 

c. What is the rate of change of temperature at noon? Give you answer correct to two decimal places.

Question 19 (2 marks)

Determine f(x) of the following:

Question 20 (3 marks)

The population of tadpoles in a pond changes at a rate modelled by the following function:

where:

• N is the number of tadpoles in the pond
• t is the number of days after the first frogs arrived at the pond 

a. Determine the rule for the population of tadpoles at a given time if initially there were 0 tadpoles 

b. What is the tadpole population after 4 days? Round answer to nearest whole number

Question 21 (5 marks)

a. Domain and Range 

b. Any x-intercepts and y-intercepts

c. Horizontal and vertical asymptotes 

d. Local maximum and/or minimum

e. Intervals for which the function is increasing and decreasing

Question 22 (3 marks)

Calculate the values of x for which the following functions have the same gradient

Question 23 (3 marks)

The velocity of a particle (ms^-1) is given by the following function:

a. Calculate the velocity of the particle after 3 seconds (correct to two decimal places)

b. Calculate the acceleration of the particle after 4 seconds (correct to two decimal places)

Question 24 (5 marks)

The number of magpies, K, in a forest on a particular day can be modelled by the following equation:

where t is the number of weeks after the start of the spring season (September). Find:

a. The number of magpies at the start of September 

b. The number of magpies at the start of November (round to nearest whole number)

c. The rate of change at any time 

d. The rate of change when t = 6 (round to two decimal places)

Question 25 (5 marks)

Consider the function f(x) = (-4 + e^-x).

a. Determine the coordinates of the x-intercepts and y-intercepts 

b. Sketch and label the function with all important features

c. State the transformations required to map f(x) = e^x onto f(x) = (-4 + e^-x)

QCAA Unit 3 Maths Methods Short Answer Practice Solutions

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