2024 VCAA Maths Methods Exam 2

Number of marks: 80

Reading time: 15 minutes

Writing time: 2 hours

Section A – Multiple-choice questions
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.

Question 1 [2023 Exam 2 Section A Q1]

The amplitude, ( A ), and the period, ( P ), of the function ( f(x) = -frac{1}{2} sin(3x + 2pi) ) are

A. ( A = -frac{1}{2}, quad P = frac{pi}{3} )
B. ( A = -frac{1}{2}, quad P = frac{2pi}{3} )
C. ( A = -frac{1}{2}, quad P = frac{3pi}{2} )
D. ( A = frac{1}{2}, quad P = frac{pi}{3} )
E. ( A = frac{1}{2}, quad P = frac{2pi}{3} )

Question 2 [2023 Exam 2 Section A Q2]

For the parabola with equation ( y = ax^2 + 2bx + c ), where ( a, b, c in mathbb{R} ), the equation of the axis of symmetry is

A. ( x = -frac{b}{a} )
B. ( x = -frac{b}{2a} )
C. ( y = c )
D. ( x = frac{b}{a} )
E. ( x = frac{b}{2a} )

Question 3 [2023 Exam 2 Section A Q3]

Two functions, ( p ) and ( q ), are continuous over their domains, which are [−2, 3) and (−1, 5], respectively. The domain of the sum function ( p + q ) is

A. [−2, 5]
B. [−2, −1) ∪ (3, 5]
C. [−2, −1) ∪ (−1, 3) ∪ (3, 5]
D. [−1, 3]
E. (−1, 3)

Question 4 [2023 Exam 2 Section A Q4]

Consider the system of simultaneous linear equations below containing the parameter ( k ):
( kx + 5y = k + 5 )
( 4x + (k + 1)y = 0 )
The value(s) of ( k ) for which the system of equations has infinite solutions are

A. ( k in {-5, 4} )
B. ( k in {-5} )
C. ( k in {4} )
D. ( k in mathbb{R} setminus {-5, 4} )
E. ( k in mathbb{R} setminus {-5} )

Question 5 [2023 Exam 2 Section A Q5]

Which one of the following functions has a horizontal tangent at (0, 0)?

A. ( y = x^{-1/3} )
B. ( y = x^{1/3} )
C. ( y = x^{2/3} )
D. ( y = x^{4/3} )
E. ( y = x^{3/4} )

Question 6 [2023 Exam 2 Section A Q6]

Suppose that ( int_3^{10} f(x), dx = C ) and ( int_7^{10} f(x), dx = D ). The value of ( int_3^7 f(x), dx ) is

A. ( C + D )
B. ( C + D - 3 )
C. ( C - D )
D. ( D - C )
E. ( CD - 3 )

Question 7 [2023 Exam 2 Section A Q7]

Let ( f(x) = log_e x ), where ( x > 0 ) and ( g(x) = sqrt{1 - x} ), where ( x < 1 ).
The domain of the derivative of ( (f circ g)(x) ) is

A. ( x in mathbb{R} )
B. ( x in (-infty, 1] )
C. ( x in (-infty, 1) )
D. ( x in (0, infty) )
E. ( x in (0, 1) )

Question 8 [2023 Exam 2 Section A Q8]

A box contains ( n ) green balls and ( m ) red balls. A ball is selected at random, and its colour is noted. The ball is then replaced in the box. In 8 such selections, where ( n ne m ), what is the probability that a green ball is selected at least once?

A. ( 8 left( frac{n}{n + m} right) left( frac{m}{n + m} right)^7 )
B. ( 1 - left( frac{n}{n + m} right)^8 )
C. ( 1 - left( frac{m}{n + m} right)^8 )
D. ( 1 - left( frac{n}{n + m} right) left( frac{m}{n + m} right)^7 )
E. ( 1 - 8 left( frac{n}{n + m} right) left( frac{m}{n + m} right)^7 )

Question 9 [2023 Exam 2 Section A Q9]

[ f(x) = begin{cases} tanleft(frac{x}{2}right) & 4 leq x < 2pi sin(ax) & 2pi leq x leq 8 end{cases} ]

The value of ( a ) for which ( f ) is continuous and smooth at ( x = 2pi ) is

A. −2
B. ( -sqrt{2} )
C. ( -frac{1}{2} )
D. ( frac{1}{2} )
E. 2

Question 10 [2023 Exam 2 Section A Q10]

A continuous random variable ( X ) has the following probability density function:

[ g(x) = begin{cases} frac{x - 1}{20} & 1 leq x < 6 frac{9 - x}{12} & 6 leq x leq 9 0 & text{elsewhere} end{cases} ]

The value of ( k ) such that ( Pr(X < k) = 0.35 ) is

A. ( sqrt{14} - 1 )
B. ( sqrt{14} + 1 )
C. ( sqrt{15} - 1 )
D. ( sqrt{15} + 1 )
E. ( 1 - sqrt{15} )

Question 11 [2023 Exam 2 Section A Q11]

Two functions, ( f ) and ( g ), are continuous and differentiable for all ( x in mathbb{R} ). It is given that
( f(-2) = -7, quad g(-2) = 8, quad f'(-2) = 3, quad g'(-2) = 2 )
The gradient of the graph ( y = f(x) times g(x) ) at ( x = -2 ) is

A. −10
B. −6
C. 0
D. 6
E. 10

Question 12 [2023 Exam 2 Section A Q12]

The probability mass function for the discrete random variable ( X ) is shown below:

X	−1	0	1	2
Pr((X = x))	(k^2)	(3k)	(k)	(-k^2 - 4k + 1)

The maximum possible value for the mean of ( X ) is:

A. 0
B. ( frac{1}{3} )
C. ( frac{2}{3} )
D. 1
E. 2

Question 13 [2023 Exam 2 Section A Q13]

The following algorithm applies Newton’s method using a For loop with 3 iterations...

[ begin{array}{l} textbf{Inputs:} quad f(x), text{ a function of } x quadquadquadquad df(x), text{ the derivative of } f(x) quadquadquadquad x_0, text{ an initial estimate} textbf{Define } texttt{newton}(f(x), df(x), x_0) quad textbf{For } i text{ from } 1 text{ to } 3 quadquad textbf{If } df(x_0) = 0 textbf{ Then} quadquadquad textbf{Return } text{``Error: Division by zero''} quadquad textbf{Else} quadquadquad x_0 leftarrow x_0 - f(x_0) div df(x_0) quad textbf{EndFor} quad textbf{Return } x_0 end{array} ]

The return value of the function newton(x³ + 3x − 3, 3x² + 3, 1) is closest to

A. 0.83333
B. 0.81785
C. 0.81773
D. 1
E. 3

Question 14 [2023 Exam 2 Section A Q14]

A polynomial has the equation ( y = x(3x - 1)(x + 3)(x + 1) ).
The number of tangents to this curve that pass through the positive x-intercept is

A. 0
B. 1
C. 2
D. 3
E. 4

Question 15 [2023 Exam 2 Section A Q15]

Let ( X ) be a normal random variable with mean 100 and standard deviation 20. Let ( Y ) be a normal random variable with mean 80 and standard deviation 10.
Which of the diagrams below best represents the probability density functions for ( X ) and ( Y ), plotted on the same set of axes?

Question 16 [2023 Exam 2 Section A Q16]

Let ( f(x) = e^{x - 1} ).
Given that the product function ( f(x) times g(x) = e^{(x - 1)^2} ), the rule for the function ( g ) is:

A. ( g(x) = e^{x - 1} )
B. ( g(x) = e^{(x - 2)(x - 1)} )
C. ( g(x) = e^{(x + 2)(x - 1)} )
D. ( g(x) = e^{x(x - 2)} )
E. ( g(x) = e^{x(x - 3)} )

Question 17 [2023 Exam 2 Section A Q17]

A cylinder of height ( h ) and radius ( r ) is formed from a rectangular metal sheet of length ( x ) and width ( y ), by cutting along the dashed lines shown below.
Options A to E

The volume of the cylinder in terms of ( x ) and ( y ) is

A. ( pi x^2 y )
B. ( frac{pi x y^2 - 2 y^3}{4pi^2} )
C. ( frac{2 y^3 - pi x y^2}{4pi^2} )
D. ( frac{pi x y - 2 y^2}{2pi} )
E. ( frac{2 y^2 - pi x y}{2pi} )

Question 18 [2023 Exam 2 Section A Q18]

Consider the function ( f : [-api, api] to mathbb{R} ), ( f(x) = sin(ax) ), where ( a ) is a positive integer.
The number of local minima in the graph of ( y = f(x) ) is always equal to

A. 2
B. 4
C. ( a )
D. ( 2a )
E. ( a^2 )

Question 19 [2023 Exam 2 Section A Q19]

Find all values of ( k ), such that the equation
( x^2 + (4k + 3)x + 4k^2- frac{9}{4} = 0 )
has two real solutions for ( x ), one positive and one negative.

A. ( k > -frac{3}{4} )
B. ( k geq -frac{3}{4} )
C. ( k > frac{3}{4} )
D. ( -frac{3}{4} < k < frac{3}{4} )
E. ( k < -frac{3}{4} text{ or } k > frac{3}{4} )

Question 20 [2023 Exam 2 Section A Q20]

Let ( f(x) = log_eleft(x + frac{1}{sqrt{2}}right) ).
Let ( g(x) = sin(x) ) where ( x in (-infty, 5) ).
The largest interval of ( x ) values for which ( (f circ g)(x) ) and ( (g circ f)(x) ) both exist is

A. ( left( -frac{1}{sqrt{2}}, frac{5pi}{4} right) )
B. ( left[ -frac{1}{sqrt{2}}, frac{5pi}{4} right) )
C. ( left( -frac{pi}{4}, frac{5pi}{4} right) )
D. ( left[ -frac{pi}{4}, frac{5pi}{4} right) )
E. ( left[ -frac{pi}{4}, -frac{1}{sqrt{2}} right] )

Question 1 [2024 Exam 2 Section A Q1]

The asymptote(s) of the graph of ( y = log_e(x + 1) - 3 ) are

A. ( x = -1) only
B. ( x = 1) only
C. ( y = -3) only
D. ( x = -1) and ( y = -3 )

Question 2 [2024 Exam 2 Section A Q2]

A function ( g: mathbb{R} to mathbb{R} ) has the derivative ( g'(x) = x^3 - x ).
Given that ( g(0) = 5 ), the value of ( g(2) ) is

A. 2
B. 3
C. 5
D. 7

Question 3 [2024 Exam 2 Section A Q3]

A discrete random variable ( X ) is defined using the probability distribution below, where ( k ) is a positive real number.

x	0	1	2	3	4
Pr(X = x)	2k	3k	5k	3k	2k

Find ( Pr(X < 4 mid X > 1) ).

A. (frac{13}{15})
B. (frac{11}{13})
C. (frac{4}{5})
D. (frac{8}{15})

Question 4 [2024 Exam 2 Section A Q4]

If ( int_a^b f(x),dx = -5 ) and ( int_a^c f(x),dx = 3 ), where ( a < b < c ), then ( int_b^c 2f(x),dx ) is equal to

A. -16
B. 16
C. -4
D. 4

Question 5 [2024 Exam 2 Section A Q5]

Consider the functions ( f : (1, infty) to mathbb{R}, f(x) = x^2 - 4x ) and ( g: mathbb{R} to mathbb{R}, g(x) = e^{-x} )
The range of the composite function ( g(f(x)) ) is

A. ( (0, e^3) )
B. ( (0, e^3] )
C. ( (0, e^4) )
D. ( (0, e^4] )

Question 6 [2024 Exam 2 Section A Q6]

Consider the function ( f(x) = frac{2x+1}{3-x} ), with domain ( x in mathbb{R} setminus {3} ).
The inverse of ( f ) is

A. ( f^{-1}(x) = frac{3x - 1}{x + 2} ) with domain ( x in mathbb{R} setminus {3} )
B. ( f^{-1}(x) = 3 - frac{7}{x + 2} ) with domain ( x in mathbb{R} setminus {-2} )
C. ( f^{-1}(x) = 3 + frac{5}{x + 2} ) with domain ( x in mathbb{R} setminus {-2} )
D. ( f^{-1}(x) = frac{1 - 3x}{x + 2} ) with domain ( x in mathbb{R} setminus {-2} )

Question 7 [2024 Exam 2 Section A Q7]

A fair six-sided die is repeatedly rolled. What is the minimum number of rolls required so that the probability of rolling a six at least once is greater than 0.95?

A. 15
B. 16
C. 17
D. 18

Question 8 [2024 Exam 2 Section A Q8]

Some values of the functions ( f: mathbb{R} to mathbb{R} ) and ( g: mathbb{R} to mathbb{R} ) are shown below.

x	1	2	3
f(x)	0	4	5
g(x)	3	4	-5

The graph of the function ( h(x) = f(x) - g(x) ) must have an x-intercept at

A. (2, 0)
B. (3, 0)
C. (4, 0)
D. (5, 0)

Question 9 [2024 Exam 2 Section A Q9]

At a Year 12 formal, 45% of the students travelled to the event in a hired limousine, while the remaining 55% were driven by a parent.
Of the students who travelled in a hired limousine, 30% had a professional photo taken.
Of the students who were driven by a parent, 60% had a professional photo taken.
Given that a student had a professional photo taken, what is the probability that the student travelled in a hired limousine?

A. (frac{1}{8})
B. (frac{27}{200})
C. (frac{9}{31})
D. (frac{22}{31})

Question 10 [2024 Exam 2 Section A Q10]

Suppose a function ( f: [0, 5] to mathbb{R} ) and its derivative ( f': [0, 5] to mathbb{R} ) are defined and continuous on their domains.
If ( f'(2) < 0 ) and ( f'(4) > 0 ), which one of these statements must be true?

A. f is strictly decreasing on [0, 2]
B. f does not have an inverse function
C. f is positive on [4, 5]
D. f has a local minimum at x = 3

Question 11 [2024 Exam 2 Section A Q11]

Twelve students sit in a classroom, with seven students in the first row and five in the second row. Three students are chosen randomly from the class.
The probability that exactly two of the three students chosen are in the first row is

A. (frac{7}{44})
B. (frac{21}{44})
C. (frac{5}{22})
D. (frac{245}{576})

Question 12 [2024 Exam 2 Section A Q12]

The graph of ( y = f(x) ) is shown below

Which of the following options best represents the graph of ( y = f(2x + 1) )?

Question 13 [2024 Exam 2 Section A Q13]

The function ( f: (0, infty) to mathbb{R}, f(x) = frac{x}{2}+frac{2}{x} ) is mapped to a function ( g ) by:
1. dilation by a factor of 3 from the y-axis
2. translation by 1 unit in the negative y-direction
The function ( g ) has a local minimum at

A. (6, 1)
B. ((frac{2}{3}, 1))
C. (2, 5)
D. ((2, -frac{1}{3}))

Question 14 [2024 Exam 2 Section A Q14]

Let ( h ) be the probability density function for a continuous random variable ( X ), where

[ h(x) = begin{cases} dfrac{x}{6} + k, & text{for } -3 leq x < 0 -dfrac{x}{2} + k, & text{for } 0 leq x leq 1 0, & text{elsewhere} end{cases} ]

and ( k ) is a positive real number.
The value of ( Pr(X < 0.5) ) is

A. (frac{1}{2})
B. (frac{15}{16})
C. (frac{3}{16})
D. (frac{49}{48})

Question 15 [2024 Exam 2 Section A Q15]

The points of inflection of the graph of ( y = 2 - tan(pi(x - tfrac{1}{4})) ) are

A. ((k + tfrac{1}{4}, 2), k in mathbb{Z})
B. ((k - tfrac{1}{4}, 2), k in mathbb{Z})
C. ((k + tfrac{1}{4}, -2), k in mathbb{Z})
D. ((k - tfrac{3}{4}, -2), k in mathbb{Z})

Question 16 [2024 Exam 2 Section A Q16]

Suppose that a differentiable function ( f: mathbb{R} to mathbb{R} ) and its derivative ( f' : mathbb{R} to mathbb{R} ) satisfy ( f(4) = 25 ) and ( f'(4) = 15 ).
Determine the gradient of the tangent line to the graph of ( y = sqrt{f(x)} ) at ( x = 4 ).

A. (sqrt{15})
B. (frac{1}{10})
C. (frac{15}{2})
D. (frac{3}{2})

Question 17 [2024 Exam 2 Section A Q17]

Consider the algorithm below which prints the roots of the cubic polynomial ( f(x) = x^3 - 2x^2 - 9x + 18 ).

begin{aligned} &texttt{define } f(x) &quad texttt{return } (x^3 - 2x^2 - 9x + 18) &texttt{c } gets f(0) &texttt{if } c < 0 texttt{ then} &quad texttt{c } gets -c &texttt{end if} &texttt{while } c > 0 texttt{ do} &quad texttt{if } f(c) = 0 texttt{ then} &quadquad texttt{print c} &quad texttt{end if} &quad texttt{if } f(-c) = 0 texttt{ then} &quadquad texttt{print -c} &quad texttt{end if} &quad texttt{c } gets c - 1 &texttt{end while} end{aligned}

The algorithm prints in order:

A. -3, 3, 2
B. -3, 2, 3
C. 3, 2, -3
D. 3, -3, 2

Question 18 [2024 Exam 2 Section A Q18]

Find the value of ( x ) which maximises the area of the trapezium shown below.

A. 10
B. 5(sqrt{2})
C. 7
D. (sqrt{10})

Question 19 [2024 Exam 2 Section A Q19]

Consider the normal random variable ( X ) such that ( Pr(X < 10) = 0.2 ) and ( Pr(X > 18) = 0.2 ).
The value of ( Pr(X < 12) ) is closest to

A. 0.134
B. 0.297
C. 0.337
D. 0.365

Question 20 [2024 Exam 2 Section A Q20]

The function ( f: mathbb{R} to mathbb{R} ) has average value ( k ) on the interval [0, 2] and satisfies ( f(x) = f(x + 2) ) for all ( x in mathbb{R} ).
The value of the definite integral ( int_2^6 f(x),dx ) is

A. 2k
B. 3k
C. 4k
D. 6k

End of Section A

Section B
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.

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 2 [2024 Exam 2 Section B Q2]

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

[ f(t) = begin{cases} 12 + 30t & 0 leq t leq frac{1}{3} 22 & t > frac{1}{3} end{cases} ]

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 3 [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 + 700cosleft(frac{pi t}{6}right) + 400cosleft(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 4 [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

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

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

Question 5 [2024 Exam 2 Section B Q5]

The graph below shows the compositions ( g circ f ) and ( f circ g ), where ( f(x) = sin(x) ) and ( g(x) = sin(2x) ).

i. The graph of ( y = (g circ f)(x) ) has a local maximum whose ( x )-value lies in the interval ( [0, frac{pi}{2}] ).
Find the coordinates of this local maximum, correct to one decimal place. 1 mark

ii. State the range of ( g circ f ) where ( x in [0, 2pi] ). 1 mark

i. Find the derivative of ( f circ g ). 1 mark

ii. Show that the equation ( cos(sin(2x)) = 0 ) has no real solutions. 2 marks

iii. Find the ( x )-values of the stationary points of ( f circ g ) where ( x in [0, 2pi] ). 1 mark

iv. Find the range of ( f circ g ) where ( x in [0, 2pi] ). 1 mark

i. Write a single definite integral that gives the area bounded by the graphs of ( y = (f circ g)(x) ) and ( y = (g circ f)(x) ) in the interval ( [0, 2pi] ). 1 mark

ii. Hence, state the area bounded by the graphs of ( y = (f circ g)(x) ) and ( y = (g circ f)(x) ) in the interval ( [0, 2pi] ), correct to two decimal places. 1 mark

d. Let ( f_1 : (0, 2pi) rightarrow mathbb{R}, , f_1(x) = sin(x) ).
Find all values of ( x ) in the interval ( (0, 2pi) ) for which the composition ( f_1 circ g ) is defined. 2 marks

End of examination questions

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