Question 1 [2024 Exam 2 Section B Q1]

Consider the function \( f: \mathbb{R} \rightarrow \mathbb{R},\ f(x) = (x + 1)(x + a)(x - 2)(x - 2a) \) where \( a \in \mathbb{R} \).

a. State, in terms of \( a \) where required, the values of \( x \) for which \( f(x) = 0 \). 1 mark

b. Find the values of \( a \) for which the graph of \( y = f(x) \) has

i. exactly three x-intercepts. 2 marks

ii. exactly four x-intercepts. 1 mark

c. Let \( g \) be the function \( g: \mathbb{R} \rightarrow \mathbb{R},\ g(x) = (x + 1)^2(x - 2)^2 \), which is the function \( f \) where \( a = 1 \).

i. Find \( g'(x) \). 1 mark

ii. Find the coordinates of the local maximum of \( g \). 1 mark

iii. Find the values of \( x \) for which \( g'(x) > 0 \). 1 mark

iv. Consider the two tangent lines to the graph of \( y = g(x) \) at the points where \( x =\frac{-\sqrt{3} + 1}{2} \) and \( x = \frac{\sqrt{3} + 1}{2} \). Determine the coordinates of the point of intersection of these two tangent lines. 2 marks

d. Let \( g \) remain as the function \( g: \mathbb{R} \rightarrow \mathbb{R},\ g(x) = (x + 1)^2(x - 2)^2 \), which is the function \( f \) where \( a = 1 \).

Let \( h \) be the function \( h: \mathbb{R} \rightarrow \mathbb{R},h(x) = (x + 1)(x - 1)(x + 2)(x - 2) \), which is the function \( f \) where \( a = -1 \).

i. Using translations only, describe a sequence of transformations of \( h \), for which its image would have a local maximum at the same coordinates as that of \( g \). 1 mark

ii. Using a dilation and translations, describe a different sequence of transformations of \( h \), for which its image would have both local minimums at the same coordinates as that of \( g \). 2 marks

Question 1 [2024 Exam 2 Section B Q2]

A model for the temperature in a room, in degrees Celsius, is given by

a. Express the derivative \( f'(t) \) as a hybrid function. 2 marks

b. Find the average rate of change in temperature predicted by the model between \( t = 0 \) and \( t = \frac{1}{2} \). Give your answer in degrees Celsius per hour. 1 mark

c. Another model for the temperature in the room is given by \( g(t) = 22 - 10e^{-6t},\ t \geq 0 \).

i. Find the derivative \( g'(t) \). 1 mark

ii. Find the value of \( t \) for which \( g'(t) = 10 \). Give your answer correct to three decimal places. 1 mark

d. Find the time \( t \in (0, 1) \) when the temperatures predicted by the models \( f \) and \( g \) are equal. Give your answer correct to two decimal places. 1 mark

e. Find the time \( t \in (0, 1) \) when the difference between the temperatures predicted by the two models is the greatest. Give your answer correct to two decimal places. 1 mark

f. The amount of power, in kilowatts, used by the heater \( t \) hours after it is switched on, can be modelled by the continuous function \( p \), whose graph is shown below.

\\[ p(t) = \begin{cases} 1.5 & 0 \leq t \leq 0.4 \\ 0.3 + Ae^{-10t} & t > 0.4 \end{cases} \]

The amount of energy used by the heater, in kilowatt hours, can be estimated by evaluating the area between the graph of \( y = p(t) \) and the \( t \)-axis.

i. Given that \( p(t) \) is continuous for \( t \geq 0 \), show that \( A = 1.2e^4 \). 1 mark

ii. Find how long it takes, after the heater is switched on, until the heater has used 0.5 kilowatt hours of energy. Give your answer in hours. 1 mark

iii. Find how long it takes, after the heater is switched on, until the heater has used 1 kilowatt hour of energy. Give your answer in hours, correct to two decimal places. 2 marks

Question 1 [2024 Exam 2 Section B Q3]

The points shown on the chart below represent monthly online sales in Australia.

The variable \( y \) represents sales in millions of dollars.

The variable \( t \) represents the month when the sales were made, where \( t = 1 \) corresponds to January 2021, \( t = 2 \) corresponds to February 2021 and so on.

a. A cubic polynomial \( p : (0, 12] \rightarrow \mathbb{R}, \, p(t) = at^3 + bt^2 + ct + d \) can be used to model monthly online sales in 2021.

The graph of \( y = p(t) \) is shown as a dashed curve on the set of axes above.

It has a local minimum at (2, 2500) and a local maximum at (11, 4400).

i. Find, correct to two decimal places, the values of \( a \), \( b \), \( c \) and \( d \). 3 marks

ii. Let \( q : (12, 24] \rightarrow \mathbb{R}, \, q(t) = p(t - h) + k \) be a cubic function obtained by translating \( p \), which can be used to model monthly online sales in 2022.

Find the values of \( h \) and \( k \) such that the graph of \( y = q(t) \) has a local maximum at (23, 4750). 2 marks

b. Another function \( f \) can be used to model monthly online sales, where

\( f : (0, 36] \rightarrow \mathbb{R}, \quad f(t) = 3000 + 30t + 700\cos\left(\frac{\pi t}{6}\right) + 400\cos\left(\frac{\pi t}{3}\right) \)

Part of the graph of \( f \) is shown on the axes below.

i. Complete the graph of \( f \) on the set of axes above until December 2023, that is, for \( t \in (24, 36] \).
Label the endpoint at \( t = 36 \) with its coordinates. 2 marks

ii. The function \( f \) predicts that every 12 months, monthly online sales increase by \( n \) million dollars.
Find the value of \( n \). 1 mark

iii. Find the derivative \( f'(t) \). 1 mark

iv. Hence, find the maximum instantaneous rate of change for the function \( f \), correct to the nearest million dollars per month, and the values of \( t \) in the interval \( (0, 36] \) when this maximum rate occurs, correct to one decimal place. 2 marks

Question 1 [2024 Exam 2 Section B Q4]

At an airport, luggage is weighed before it is checked in.

The mass of each piece of luggage, in kilograms, is modelled by a continuous random variable \( X \), whose probability density function is

\[ f(x) = \begin{cases} \frac{1}{67500} x^2(30 - x), & 0 \leq x \leq 30 \\ 0, & \text{elsewhere} \end{cases} \]

A piece of luggage is labelled as heavy if its mass exceeds 23 kg.

a. Write a definite integral which gives the probability that a piece of luggage is labelled as heavy. 1 mark

b.

i. Find the mean of \( X \). 1 mark

ii. Find the standard deviation of \( X \). 2 marks

iii. Given that the mass of a piece of luggage is more than the mean, find the probability that it is labelled as heavy, correct to three decimal places. 2 marks

Use the following information to answer parts c and d of this question.

Of the travellers flying from the airport

• • 10% do not check in any luggage
• • 40% check in exactly one piece of luggage
• • 50% check in exactly two pieces of luggage

Assume that the mass of each piece of luggage is independent of the number of pieces checked in by each traveller.

Use the value of 0.234 for the probability that a piece of luggage is labelled as heavy.

c. Let \( W \) be the discrete random variable that represents the number of pieces of luggage labelled as heavy checked in by each traveller.

i. Show that \( \Pr(W = 2) = 0.027 \), correct to three decimal places. 1 mark

ii. Complete the table below for the probability distribution \( W \), correct to three decimal places. 2 marks

\[ \begin{array}{|c|c|c|c|} \hline w & 0 & 1 & 2 \\ \hline \Pr(W = w) & & & 0.027 \\ \hline \end{array} \]

d. On a particular day, a random sample of 35 pieces of luggage was selected at the airport.

Let \( \hat{P} \) be the random variable that represents the proportion of luggage labelled as heavy in random samples of 35.

i. Find \( \Pr(\hat{P} > 0.2) \), correct to three decimal places. 2 marks

ii. Determine the probability that \( \hat{P} \) lies within one standard deviation of its mean, correct to three decimal places. Do not use a normal approximation. 2 marks

e.

i. In one random sample of 50 pieces of luggage, 10 are labelled as heavy. Use this sample to find an approximate 90% confidence interval for \( p \), the population proportion of luggage labelled as heavy, correct to three decimal places. 1 mark

ii. A second random sample of 50 pieces of luggage is selected. Using this sample, the approximate 90% confidence interval for \( p \), the population proportion of luggage labelled as heavy, is wider than the one obtained above in part e.i.

State the minimum and maximum possible number of pieces of luggage labelled as heavy in the second sample. 1 mark