2019 Senior External Examination
Mathematics B
Paper One — Question book Thursday 24 October 2019 9 am to 12:10 pm Time allowed
• Perusal time: 10 minutes
• Working time: 3 hours
Examination materials provided
• Paper One — Question book
• Paper One — Response book
• Paper One — Resource book Equipment allowed
• QCAA-approved equipment
• ruler graduated in millimetres
• protractor
• graphics calculator
• additional calculator Equipment not allowed
• calculators with computer alegbra system (CAS) functionality Directions
Do not write during perusal time.
Paper One has six questions. Attempt all questions.
Assessment
Paper One assesses the following assessment criteria:
• Knowledge and procedures (KP)
• Modelling and problem solving (MP)
• Communication and justification (CJ) Assessment standards are at the end of this book.
After the examination session
The supervisor will collect this book when you leave.
Paper One has six questions. Attempt all questions.
Each question assesses Knowledge and procedures (KP), Modelling and problem solving (MP),
or a combination of both. Communication and justification (CJ) will be assessed by an overalljudgment of your responses to all questions.
Write your responses in the response book. Show full working to meet the standards for each
criterion.Question 1
a. The ages of nine adults in a group are:
57, 26, 19, 25, 24, 28, 36, 25, 72 i. State the mode.
ii. Calculate the mean and standard deviation of the data.
b. Two sprinters have the following training times in seconds for a 100 metre sprint.
Sprinter A 12.3 12.4 12.5 12.3 12.4 12.8 13.1 12.1 12.7 12.5
Sprinter B 12.1 12.0 12.0 14.2 12.1 12.2 12.4 12.1 12.3 12.4
i. Respond to this question on page 22 of your response book. Represent the data as parallel boxplots.
ii. The coach of the two sprinters believes that Sprinter A is a more consistent sprinter.
Evaluate the validity of the coach’s opinion by referring to the data sets.
(KP)
(KP)
(MP)
a. The following equation describes the relationship f (x)
2
2 2, 3 1
( ) 1, 1 6
− + − ≤ ≤
= + < <
x x x
f x x x
i.
Determine the domain and range of f (x) .
ii.
Determine if f (x) is both continuous and a function. Give appropriate reasons.
b. The graph of y a x b c = − + where a, b and c are constants is drawn below.
i. Determine the constants a, b and c.
ii. Describe the set of transformations that converts y x y a x b c = y x y a x b c into = = = − + − + .
c. A cyclist travels at a constant speed to a town 40 km away, then on the return trip reduces speed by 4 km/h. If the cyclist had travelled at 16 km/h for the whole trip, the total time taken would have been one hour less.
i. If
x km/h is the cyclist’s constant speed, show that40 40 6 + 4 = x x − . ii. Solve this equation to find x.
(KP)
(MP)
(KP)
Question 3
a. Solve cos x = − 0.5 for x in terms of π where − π ≤ ≤ π 2 x cos 2 x = − 0.5 − π ≤ ≤ π 2 x 2 .
b. The angle of elevation from a point A on the ground to the top of a flagpole mounted on a building is 72° and to the top of the building is 70°. The flagpole is 4 metres tall.
Calculate the height of the building, BC, correct to the nearest metre.
c. A show ferris wheel is 100 metres in diameter and at the lowest point a ferris wheel carriage is 1 metre above ground level. The wheel takes 25 minutes to make one complete rotation without stopping.
Passengers enter the carriages at the bottom of the wheel.
Respond to the following on page 24 of your response book.
i. Develop a mathematical model that determines the height of the bottom of a carriage above the ground at time t minutes. Graph the developed model.
Respond to the following by using a graphics calculator.
ii. The show fireworks display will be visible to a passenger when the bottom of their cabin is at a height of 65 metres or more from ground level. Determine how long passengers will be able to view the fireworks in one full rotation of the wheel.
Comment on any assumptions made and their effect.
(KP)
(KP)
(KP)
(MP)
a. Given that f (x ) = 4
x –2 and g (x) = 2
x2+ 7x : i. solve g (x ) = –6
ii. find the inverse function f
–1(x)
iii. determine and simplify the composite function f °
g (x).b. Solve algebraically the simultaneous equations
3 2 15 and 2 13 2
− = x+ + =y
x y
.
c. A basketballer throws a ball from a height of 2 metres at the ring 28 metres away, 3 metres above the ground. The ball goes in the ring after reaching a maximum height of 4 metres.
i. Develop a mathematical model of the form y = ax bx c
2+ + given that a, b and c are constants and y is the height of the ball above the ground when its horizontal displacement from the basketballer is x metres.
ii. Use the developed model to determine how far from the basketballer the ball reaches its greatest height.
(KP) (KP)
(MP)
Question 5
a. Determine 5
2log 3
= + = 2 1
+
x e
dy y e x y x
dx in each of the following. x
i. 5
2log 3
= + = 2 1
+
x e
dy y e x y x
dx x
2
ii. 3
5 log
= + = 2 1
+
x e
dy y e x y x
dx x
b. The displacement of a particle from its starting point is modelled by the equation x =
t2– 4t where x is the displacement in metres after t seconds.
i. Determine the average speed of the particle across the interval 3 ≤ t ≤ 6 seconds.
ii. Calculate the velocity of the particle when its displacement is 5 metres from its starting position.
c. A cone has a slant height of 15 cm and a base radius of x cm.
i. Show that the volume of the cone is given by
2225
23
π −
= x x
V .
ii. Using calculus methods, determine the value of x that maximises the volume of the cone.
(KP)
(KP)
(KP)
(MP)
a. A box has 7 green marbles and 3 black marbles. A marble is randomly selected from the box, its colour noted, and then put back in the box. If X is the random variable that represents the number of black marbles selected in 12 trials, calculate P (X ≤ 4).
b. The lengths of a particular species of fish are normally distributed. When fishing, it is only legal to keep fish between 16 cm and 40 cm. Sampling indicates that 18% of fish are undersized and 12% are oversized. In a sample of 1000 fish, what number are expected to fall between lengths of 25 cm and 35 cm?
End of Paper One
(KP)
(MP)
Assessment standards from the Mathematics B Senior External Syllabus 2006
ABCDE overall quality of a didate’s achievement across ll range within the contexts ication, technology and exity, and across topics, :accurate recall, selection and use of definitions and rules accurate use of technology recall and selection of procedures and their accurate and proficient use effective transfer and application of mathematical procedures.
The overall quality of a
candidate’s achievement across a range within the contexts of application, technology and complexity, and across topics, generally demonstrates: • accurate recall, selection and use of definitions and rules
• accurate use of technology • recall and selection of procedures and their accurate use.
The overall quality of a
candidate’s achievement in the contexts of application, technology and complexity generally demonstrates: • accurate recall and use of basic definitions and rules
• use of technology • accurate recall, selection and use of basic procedures.
The overall quality of a
candidate’s achievement in the contexts of application, technology and complexity sometimes demonstrates: • accurate recall and use of some definitions and rules • use of technology • use of basic procedures.
The overall quality of a candidate’s achievement rarely demonstrates knowledge and use of procedures.
(continued)
ABCDE The overall quality of a candidate’s achievement across the full range within each context, and across topics, generally demonstrates mathematical thinking which includes: • interpreting, clarifying and analysing a range of situations and identifying assumptions and variables • selecting and using effective strategies • selecting suitable procedures required to solve a range of problems … andsometimes demonstrates mathematical thinking which includes: • suitable synthesis of procedures and strategies to solve problems • initiative and insight in exploring the problem • identifying strengths and limitations of models.The overall quality of a
candidate’s achievement across a range within each context, and across topics,
generally demonstrates mathematical thinking which includes: • interpreting, clarifying and analysing a range of situations and identifying assumptions and variables
• selecting and using effective strategies
• selecting suitable procedures required to solve a range of problems
… andsometimes demonstrates mathematical thinking which includes: • suitable synthesis of procedures and strategies.
The overall quality of a candidate’s achievement demonstrates mathematical thinking which includes: • interpreting and clarifying a range of situations
• selecting strategies and/or procedures required to solve problems.
The overall quality of a candidate’s achievement sometimes demonstrates mathematical thinking which includes: • following basic procedures and/or using strategies.
The overall quality of a candidate’s achievement demonstrates mathematical
thinking which includes following basic procedures and/or using strategies.
(continued)
ABCDE overall quality of a consistently :accurate use of mathematical terms and symbols accurate use of language organisation of information into various forms suitable for a given use use of mathematical reasoning to develop logical arguments in support of conclusions, results and/or propositions justification of procedures recognition of the effects of assumptions evaluation of the validity of arguments.
The overall quality of a
candidate’s achievement across a range within each context generally demonstrates: • accurate use of mathematical terms and symbols
• accurate use of language • organisation of information into various forms suitable for a given use
• use of mathematical reasoning to develop simple logical arguments in support of conclusions, results and/or propositions
• justification of procedures.
The overall quality of a
candidate’s achievement in all contexts
generally demonstrates: • accurate use of basic mathematical terms and symbols
• accurate use of language • organisation of information into various forms
• use of some mathematical reasoning to develop simple logical arguments.
The overall quality of a candidate’s achievement sometimes demonstrates evidence of the use of the basic conventions of language and mathematics and occasional use of mathematical reasoning.
The overall quality of a candidate’s achievement rarely demonstrates use of the basic
conventions of language and mathematics.
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