Paper Two — Question book Thursday 24 October 2019 1:15 pm to 4:25 pm Time allowed

(1)

2019 Senior External Examination

Mathematics B

Paper Two — Question book Thursday 24 October 2019 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

Do not write 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.

(2)

Planning space

(3)

1 2019 Mathematics B — Question book — Paper Two

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 = (x + 2) (1 + x

²

), use algebraic methods to calculate:

i.

any x-intercepts

ii.

any stationary points and determine their nature

iii.

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

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

²

0

sin d

∫

^{x x}

^.

c. The function y = g (x) has stationary points at x = –2 and x = 1.

If

dg ax x b= ²−4 +

dx

, determine the nature of the stationary points. Show full working.

Question 2 a. Determine:

i. ∫

(^x²+3^e^x−2)^dx

∫

cos5^x+ ^{x dx}

ii.

( 2+3 −2) cos5 +

∫

^x ^e^x ^dx

∫

^x ^{x dx}

b. In the following diagram, the shaded region is the area between the curve y = 2 – (x – 1)

³

, the x-axis and the ordinates at x = –1 and x = 3.

i. Using algebraic methods, determine the x-intercept at A.

ii. Calculate the shaded area. Show full working.

c. Solve

–2

3 6

4 =

∫

+

b dx

x

for the constant b. Show full algebraic working.

What limitations exist on x and b?

(KP) (KP)

(MP)

(KP)

(MP)

(4)

Question 3

a. Express 2.5 radians in degrees correct to one decimal place.

b. The velocity in m/s of a particle at time, t seconds, is given by

dx =12 5sin− t

dt

.

i. Determine the initial velocity of the particle.

ii. If the particle has a displacement of 4 m when t = 2 seconds, calculate its displacement when t = 3 seconds.

c. The point P is the y-intercept of the curve y = 1 –

xe ^–x

. The tangent and normal to the curve at P meet the x-axis at B and C respectively. Use algebraic methods to determine the area of ΔPCB. Show full working.

Question 4

a. Simplify the following equations using appropriate index or logarithmic laws:

i.

2 ¹ 2 2 2

2 4 log 24 log 3 log 9 8

+

−

× + −

n n

n

ii.

1

2 2 2

2

2 4 log 24 log 3 log 9 8

+

−

× + −

n n

n

.

b. Solve the following equations for

x:

i.

5^{3 2}^x⁻ =125 log₄x+log (₄ x− =3) 1

ii.

3 2 4 4

5^x⁻ =125 log x+log (x− =3) 1

.

c. The following table provides data about how the volume, V, of bottled water in billions of litres sold in Germany has been increasing since 2004 (T years since 2004).

Years since

2004 (T) Volume in

litres (V) log

_eV correct to

four decimal places

10 1.78 × 10

¹⁰

23.6025

14 2.58 × 10

¹⁰

23.9736

i. Find the constants a and b such that log

_eV = aT + b.

ii. Express

V as an exponential function and use this function to estimate the volume of

bottled water sold in 2019 (T = 15). Comment on the assumptions made in developing this model and identify any strengths or limitations in extending this model to predict the volume of bottled water sold in 2025.

(KP)

(MP)

(KP)

(MP)

(5)

Question 5

a. How long will it take for $6200 invested at 3.6% per annum compounding half-yearly to grow to $8900?

b. Susan is buying a car on terms using dealer finance.

i. Susan will pay $650 monthly over 5 years at 7.2% p.a. compounding monthly.

Calculate the amount that Susan borrowed.

ii. After 6 months, Susan pays $2000 off the balance. Susan believes that she will reduce the time taken to pay off the loan by at least four months. Is Susan’s claim reasonable?

End of Paper Two

(KP)

(MP)

(6)

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 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.

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.

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.

(7)

(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 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.

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 rarely demonstrates mathematical

thinking which includes following basic procedures and/or using strategies.

(8)

(continued)

CriterionABCDE

Communication and justification (CJ)

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.

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.

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.

(9)

(10)

Manager Publishing Unit

Email: publishing@qcaa@qld.edu.au

Queensland Curriculum

& Assessment Authority

PO Box 307, Spring Hill QLD 4004 Australia Level 7, 154 Melbourne Street, South Brisbane T +61 7 3864 0299