Paper Two — Question book

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2018 Senior External Examination

Mathematics B

Paper Two — Question book

Monday 29 October 2018 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 calculator

Equipment not allowed

• calculators with computer alegbra 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

The supervisor will collect this book when you leave.

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Planning space

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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. Given that y=(x²−1)(x²+2), use algebraic methods to calculate:

i. the x-intercepts

ii. any stationary points and determine their nature

iii. the greatest and least values of this function in the interval –2 ≤ x ≤ 3. (KP)

b. Use the trapezoidal rule with three partitions of equal width to evaluate ⁴

1

x−1dx

∫

^,

correct to three decimal places. (KP)

c. The height, H metres, of crab apple trees, t years after planting, is modelled by H = - 9 6 45+

2 5

. t

i. Calculate the height of a crab apple tree when t = 2 years. (KP)

ii. Calculate the average rate of growth of a typical crab apple tree across 2 ≤ t ≤ 5 years. (KP) iii. Show that dH

dt > 0 and d H²₂

dt < 0 for t> 0 and comment on the significance of these results. (KP) iv. A scientist claims that crab apple trees do not grow to more than 9.6 metres in height.

Examine the validity of this claim according to the given model. (MP)

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2

Question 2

a. Determine:

i.

∫

(4x+3)⁶dx ii.

0

sin5x cosxdx

π

∫

+

0

sin5x cosxdx

π

∫

+ ^(KP)

b. The region formed by the intersection of the line x y+ =2 and the curve y= − +x² 3x+2 is illustrated.

i. Using algebraic methods, determine the coordinates of the points of intersection, A and B.

ii. Write the integral that represents the shaded area. (KP)

c. The point Q (4, –20) lies on the curve y=2a x b+ where a and b are constants. The tangent at Q has gradient –3. Determine the y-intercept of the curve. (MP)

Question 3

a. Express 150.6° in radians. (KP)

b. The acceleration in m/s² of a particle at time t seconds is given by dv

dt =e³^t+ +2t 1. i. Determine the initial acceleration of the particle.

ii. If the particle has a velocity of 4 m/s when t = 2 seconds, calculate its velocity when

t = 3 seconds. (KP)

c. A curve has equation y x= log_ex. The point A on this curve is a stationary point.

i. Determine the coordinates of A. Show full working. (KP)

ii. Given that log ²log ²

2 ^e 4

e x x x

x xdx= − +C

∫ ∫

^x^log^e^xdx ⁼ ^x²^log₂ ^e^{x x}⁻ ₄² ⁺^C where C is a constant of integration, calculate the area enclosed by the curve, the x-axis and the ordinates at point A and x= 2. Show full

working. (MP)

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Question 4

a. Simplify using appropriate index and logarithmic laws:

i. 125 2

5

6 1

3 x x

x

×( )⁻

ii. log₂16 4− log₂8 (KP)

b. Solve for x:

i. 5²^x− ×2 5 3 0^x− =

ii. log (₄ 3x+10)=log₄x+1 (KP)

c. Scientists measuring the level of carbon dioxide in the atmosphere recorded the following information.

Year C

Carbon dioxide levels in parts per million (ppm)

2014 396

2015 400

2016 402

2017 405

2018 409

Scientists create two equations to model the data. The first is the linear model C = at + b, and the second is an exponential model C = ge^ht,where t is the time in years.

i. Use a graphics calculator to determine the values of the constants a, b, g and h. (KP) ii. Compare and contrast the effectiveness of the linear and exponential equations in

modelling the relationship between C and t. Examine the validity of both models

in predicting the amount of carbon dioxide in the atmosphere in 2025. (MP)

Question 5

a. How long will it take for $5 400 invested at 4.8% per annum compounding half-yearly

to grow to $7 200? (KP)

b. Five years ago, David took out a 25-year loan for $300 000 with monthly repayments.

The interest rate was 7.2% per annum compounding monthly. After five years, the interest rate changes to 6.8% per annum compounding monthly.

David believes that if he continues to make the same repayment (as for the first five years) each month, he will reduce the time taken to repay the loan by at least two years.

Evaluate the validity of David’s claim. (MP)

End of Paper Two

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4 1 2008 Subject — Paper One — Question/Response book

T:\exexams\2017\Papers\Subjects\Maths B\Standards\Standards - Maths B.fm August 14, 2018 2:18 pm

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.

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2 2008 Subject — Paper One — Question/Response book 5

(continued) CriterionABCDE

Modelling and problem solving (MP)

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.

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.

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6 3 2008 Subject — Paper One — Question/Response book

(continued) CriterionABCDE

Communication and justification (CJ)

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.

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.

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