2018 Senior External Examination
Mathematics B
Paper One — Question book
Monday 29 October 2018 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
You may write in this book 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
Planning space
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 overall judgment 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 set of four coins is thrown 16 times and the number of heads appearing face-up per throw is presented below.
a. For the given data:
i. state the mode
ii. use your calculator to find the mean and standard deviation.
(KP) b. Determine the five-number summary to construct a boxplot using the graph paper at the back of your
response book (from page 21).
(KP) c. The four coins are tossed two more times. The mean of the 18 scores is now exactly 2. Determine
all possible ways this could occur. Examine the validity of the claim that the interquartile range does not change.
(MP)
2
Question 2
a. The relation y = f (x) is drawn below.
i. Determine the domain and range of f (x).
ii. State if this relation is continuous or not. Give a reason for your choice.
iii. Using the graph, determine f (–1)+f (2).
(KP) b. By completing the square, express y=2x2−4 5x+ in the form y a x b= ( + )2+c where a, b and c
are constants. Hence, determine the transformations required to convert y x= 2 to y=2x2−4x+5. (KP) c. A triangle has a base of length (2x + 3) cm and an altitude of (x – 1.5) cm. If the area of the triangle
is 18 cm2, determine x.
(MP)
Question 3
a. Simplify sin sin 1 cos 1 cos
x x
x− x
− + by using the Pythagorean identity.
(KP) b. Solve cos x= −0.86 for x across the domain − π ≤ ≤ π2 x 2 .
(KP) c. A yacht is located at point Y and is sailing on a bearing 032° T towards a lighthouse at point L
1.5 km from point Y. From Y, the yacht’s navigator spots a boat at point B bearing 120° T. The bearing of the lighthouse from the boat at B is 332° T.
i. Calculate the distance between the yacht and the boat.
(KP) ii. From the boat, the top of the lighthouse has an elevation of 2°. Determine the height of the
lighthouse above sea level.
(MP)
4
Question 4
a. Given that f x( ) 2= x−3 and ( ) 4 2 5 g x = x+ − i. solve g x( ) 0=
ii. find g x−1( ), the inverse function
iii. determine the composition function, f ◦ g(x).
(KP) b. Using an algebraic method, solve simultaneously y=3 10x+ and 4x y+ =24.
(KP) c. The graph of y=f(x) has a stationary point at x= –1 and the point A (1, 2) lies on the curve.
If f (x) = axebx, find the values of the constants a and b and hence determine f (x).
(MP)
Question 5
a. Show from first principles the derivative of f x( ) 5 3= + x2.
(KP) b. Determine dy
dx in each of the following:
i. y=y=y= 444−−−xxx222
ii. n
6 y x
= x iii. y e= 2xsinx
(KP) c. A cylindrical juice container (closed at both ends) is made from sheet metal and holds 850 mL of
juice. Assume that the cost per centimetre of making the seams around the top and bottom and up one side of the container is k times the cost per square centimetre of the sheet metal where 0 <k< 1.
Note: 1 mL of fluid occupies 1 cm3 of space.
i. Show that the cost, C, of the sheet metal used, including the seams, is given by:
2
2
850 425
0.02 0.01 2
C a r k r
r r
= π + + π + π
2
2
850 425
0.02 0.01 2
C a r k r
r r
= π + + π + π
2
2
850 425
0.02 0.01 2
C a r k r
r r
= π + + π + π
2
2
850 425
0.02 0.01 2
C a r k r
r r
= π + + π + π
2
2
850 425
0.02 0.01 2
C a r k r
r r
= π + + π + π
where r is the radius of the base in centimetres and the cost of the sheet metal is $a per
square metre. (KP)
ii. Determine the value of k if the cost, C, of the sheet metal used (including the seams),
is optimised when r = 4.2 cm. (MP)
Quesiton 6
a. A box contains 4 green marbles and 6 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 green marbles selected from the box in 12 trials, calculate P(X≤ 5).
(KP) b. A six-sided die is loaded so that the outcome of rolling an even number compared to the rolling of an
odd number is in the ratio 3:2. The die is rolled 100 times.
The binomial variable X represents the number of times an even number appears on the uppermost face of the die when it is rolled. Using the normal approximation for a binomial event, determine P(55 <X≤ 70).
(MP)
End of Paper One
6 1 2008 Subject — Paper One — Question/Response book
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Assessment standards from the Mathematics B Senior External Syllabus 2006 CriterionABCDE Knowledge and procedures (KP)The overall quality of a candidate’s achievement across the full range within the
contexts of application, technology and complexity, and across topics, consistently demonstrates: •accurate recall, selection and use
of definitions and rules •accurate use of technology •recall and selection of procedures and the
ir 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 thinkingwhich includes: •interpreting, clarifying and analysing a range of situations 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 situat
ions 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 sometimesdemonstrates
mathematical thinking which includ
es following basic
procedures and/or using strategies.
The overall quality of a candidate’s achievement rarely demonstrates
mathematical thinking which includes following basic procedures
and/or using strategies.
8 3 2008 Subject — Paper One — Question/Response book
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(continued) CriterionABCDE
Communication and justification (CJ)
The overall quality of a candidate’s achievement
across the full range within each context
consistently 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 logical arguments in support of
conclusions, results and/or propositions •justification of procedures •recognition of the effects of assumptions •evaluation of the validity of argum
ents.
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 reasonin
g to develop simple logical arguments.
The overall quality of a candidate’s achievement sometimesdemonstrates
evidence of the use of the basic conventions of language and mathematics and o
ccasional use of mathematical reasoning.
The overall quality of a candidate’s achievement rarelydemonstrates use of
the basic conventions of language and mathematics.
© The State of Queensland (Queensland Curriculum and Assessment Authority) 2018 Copyright enquiries should be made to:
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