For all Queensland schools
2017 Senior External Examination
Mathematics B Monday 30 October 2017
Paper Two — Question book 1:15 pm to 4:25 pm
Time allowed
• Perusal time: 10 minutes
• Working time: 3 hours
Examination materials provided
• Paper Two — Question book
• Paper Two — Response book
Equipment allowed
• QCAA-approved equipment
• ruler graduated in millimetres
• protractor
• graphics calculator
• additional non-CAS calculator
Equipment not allowed
• Calculators with computer algebra system (CAS) functionality
Directions
You may write in this book during perusal time.
Paper Two has five questions. Attempt all questions.
Assessment
Paper Two 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
Take this book when you leave.
Planning space
1 Paper Two has five 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. The graph of the function is drawn below.
The graph has equation
i. Given that x = 3 is an x-intercept, determine the value of a.
ii. Determine the other x-intercept.
iii. For what values of x is increasing?
(KP) b. Use the trapezoidal rule with four partitions of equal width to evaluate , correct to
three decimal places.
(KP) c. The function has stationary points at x = –2 and x = 1. If ,
determine the nature of the stationary points. Show full working.
(MP)
not to scale
f x
g x = ax
2– 4x + b
2
Question 2 a. Determine:
i.
ii.
(KP) b. The region formed by the intersection of the curves , and x = 3
is shown below.
i. Use a graphics calculator to determine the coordinates of the point of intersection of the curves at A.
ii. Calculate the shaded area. Show full algebraic working.
(KP) c. The acceleration, in m/s
2, of a projectile at time t seconds is given by .
If the projectile has an initial velocity of 200 m/s, develop a mathematical model for the velocity of this projectile.
Show full working.
(MP)
not to scale
dv
--- dt = 400e
–0.5t3
Question 3
a. Express 1.254 radians in degrees.
(KP) b. The velocity, in m/s, of a particle at time t seconds is given by .
i. Calculate the initial velocity of the particle.
ii. Use a graphics calculator to determine the first time the particle has a velocity of 12 m/s.
iii. Determine the particle’s displacement at t = 2 seconds if its initial displacement is 0 m.
(KP)
c. The graph of is shown below.
If the point marked A on the graph is an x-intercept, examine the validity of the claim that the shaded area is given by
Show full working.
(MP) v = 15 6 – cos t
not to scale
2 sin 2
0
Ax + 2 sin x dx
4
Question 4
a. Without using your calculator, i. simplify
ii. evaluate
(KP) b. Using algebraic methods, solve:
i.
ii.
(KP) c. The rate at which the current I, reduces in a circuit at time t seconds is given by
When t = 2 seconds, the current is 150 amps.
i. Solve the differential equation for I as a function of t.
ii. Determine how long it takes for the current in this circuit to reduce to zero. Show full working.
(MP)
Question 5
a. Over what period of time would $12 000 have to be invested to amount to $15 000 at 4.8% p.a.
compounding monthly?
(KP) b. Susan has taken out a loan of $400 000 at 5.6% p.a. compounding monthly over 30 years with
monthly repayments. Ten years later Susan receives an inheritance of $200 000.
Explain whether this would be enough for Susan to pay off her loan.
(MP)
End of Paper Two
dI
--- dt – 200t 1 + t
2---
=
5
Assessment standards from the Mathematics B Senior External Syl labus 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 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.
6
(continued) CriterionABCDE Modelling and problem solving (MP)
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 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 sometimesdemonstrates mathematical thinking which includes: •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.
7
(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 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 sometimesdemonstrates 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 rarelydemonstrates use of the basic conventions of language and mathematics.
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Manager Publishing Unit
Email: [email protected]
Queensland Curriculum
& Assessment Authority
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