Mathematics B 2008

1602

00

Supervised assessment: Rates of change, periodic functions and applications, and exponential and logarithmic functions and applications This sample demonstrates on-balance judgments within criteria. It provides information about student achievement where the indicative response matches the qualities of the A standard.

Criteria assessed • Knowledge and procedure

• Modelling and problem solving

• Communication and justification

Assessment instrument The response presented in this sample is in response to assessment items

Year 12 Mathematics B Term 1 (post-discussion) Knowledge and Procedures

Q1. (Simple/RS/RC abstract)

Differentiate these functions, with respect to 𝑥.

a. 𝑦 = (2𝑥 − 1)(𝑥2 + 3𝑥 + 5) (use the product rule or expand firstly)

b. 𝑦 = (3𝑥2 + 3)3 (use the chain rule)

c. 𝑦 =

d. 𝑦 = ln(3𝑥 − 2) + 1

e. 𝑦 = −4𝑐𝑜𝑠 (𝑥2

2)

f. 𝑦 = e 𝑠𝑖𝑛 𝑥

g. 𝑦 = (𝑥3 − 1) . 𝑠𝑖𝑛3(3𝑥)

(2+2+2+2+2+3+4 = 17 marks)

x43 −

Mathematics B 2008 Annotated indicative response and judgments within criteria

The school considers the objectives of the syllabus when designing the assessment items.

The objectives, described as syllabus standards A–E, inform the design of the items and the allocation of marks.

The matching of the standards with the items is demonstrated on: instrument-specific criteria and standards on pages 5, 6 and 7; and throughout the indicative response starting on page 8.

The marking scheme on page 18 indicates the items grouped by the type of tasks — simple, simple routine, routine complex, non-routine simple, and non-routine complex.


Queensland Curriculum & Assessment Authority March 2016

Page 2 of 18

Q2. (RS abstract)

Use algebraic techniques to solve the following equations. Give answers in exact form.

a. ln (𝑥 + 2) = 3

b. log 5 + log 4 − log 2 = m

c. 102𝑥 = 500

d. 𝑘 cos �𝜋6� + 3 = 8

(1+2+2+2=7 marks)

Q3. (Simple/RS life-related)

A population model has been designed for a small industrial town in India. The first year of the model is at the start of 2003 when 𝑡 = 0, and the population is 15000.

The population model is written below.

𝑃𝑛 = 𝑃0𝑒𝑘𝑡 (𝑡 is the time in years and 𝑘 is a constant).

a. Write down the value of 𝑃0

b. Given that the population at the start of 2008 is 18321 find the value of k (to 2 decimal places) (Recall the logarithm definition: 𝑥 = log𝑏 𝑎⟺ 𝑎 = 𝑏𝑥)

c. Find the value of the population at the start of 2011.

d. In which year will the population double?

(1+3+1+3=8 marks)

Q4. (RS/NRS abstract)

a. Find the equation of the tangent to the curve y = 2 sin �𝒙𝟑� at P(3π, 0).

b. Show that there are 2 tangents with mT = −𝟏𝟑

, 0≤ x ≤ 5π, to the curve y = 2 sin �𝑥3�

(3+4=7 marks)

Q5. (RS life-related)

The vertical displacement (height) from ground level of a point on a large rotating wheel is modelled by the equation ℎ = −5 𝑐𝑜𝑠 2𝑡, where ℎ represents the displacement in metres and 𝑡 represents the time from the start in minutes.

a. Find an expression for the vertical velocity of the point at any time.

b. Determine the vertical acceleration of the point one minute after the start. Describe the acceleration.

(2 + 3= 5 marks)

Q6. (NRS abstract)

The function 𝑓 is given by 𝑓 (𝑥) = Use algebraic techniques to find the domain of the function.

(4 marks)

.)2(n 1 −x



Page 3 of 18

Q7. (RS/RC/NRS abstract)

Solve each of the following for the domain stated, using algebraic methods where appropriate.

a. √3 sin 𝑥 − cos 𝑥 = 0 (0 ≤ 𝑥 ≤ 2𝜋)

b. 2 cos 2θ + 1 = 0 (0 ≤ θ ≤ 2𝜋)

c. �1+sin 𝑡4

= 0.35 (0 ≤ 𝑡 ≤ 2𝜋)

(3+4+3= 10 marks)

Q8. (NRS abstract)

Let log10𝑃 = 𝑥, log10𝑄 = 𝑦 and log10𝑅 = 𝑧. Express in terms of 𝑥, 𝑦 and 𝑧.

(3 marks)

Q9. (RS/RC abstract)

Let 𝑓 (𝑥) = 6 sin π𝑥, and 𝑔 (𝑥) = 6𝑒−𝑥 – 3, for 0 ≤ 𝑥 ≤ 2. The graph of 𝑓 is shown on the diagram below.

There is a maximum value at B (0.5, 𝑏)

a. Write down the value of b.

b. On the same diagram, sketch the graph of 𝑔

c. Solve 𝑓 (𝑥) = 𝑔 (𝑥) ,0.5 ≤ 𝑥 ≤ 1.5.

(1+2+2= 5 marks)

Q10. (RC life-related)

The area A km2 affected by a forest fire at time t hours is given by 𝐴 = 𝐴0 e𝑘𝑡. When t = 5, the area affected is 1 km2 and the rate of change of the area is 0.2 km2 h−1.

Find the value of t when 100 km2 are affected.

(6 marks)

2

310log

QRP

0 1 2

B

x

y



Page 4 of 18

Modelling and problem solving

Q11. Routine Simple life-related

The intensity of light at a distance of m metres from the entrance of a cave can be modelled by the equation

I = Io e –km It is found that the intensity of light is halved when a person is 5 metres from the cave entrance.

If artificial light is necessary when the intensity of light is less than 10% of the intensity at the cave entrance, find out how deep a cave explorer can go before artificial light is necessary.

Justify the reasonableness of this result. Identify any assumptions that have been made.

Q12. Routine Complex life-related

A space vehicle is moving on a path that is described by the equation 43

32

7ddy += where y is the vertical distance from the launch point and d is the horizontal distance from the launch point in kilometres.

When the horizontal distance is 20 000 km, the space vehicle fires an exploratory rocket into space tangentially on a straight line course for Bega Nova (an abandoned space station). Bega Nova is located at a horizontal distance of 950 000 km and a vertical distance of 3 x 105 km from Earth.

Will the rocket collide with Bega Nova?

Q13. Routine Complex Abstract

Given that f(x) = (2𝑥 − 𝑘)3, find the value of the constant 𝑘 for which f”(1) = 24

Q14. Non-routine Complex life-related

A snowball rolling down a snow-covered slope increases in size according to the formula 𝑑 = 0.4𝑡 + 5 where 𝑑 is the diameter of the snowball in centimetres and t is the time in seconds.

Given that density = massvolume

and that packed snow has a density of 0.9 g/cm3,

find the rate of increase of the mass of the snowball after 4 seconds. Clearly state any assumptions and their associated effects.



Page 5 of 18

Instrument-specific standards matrix Standard A Standard B Standard C Standard D Standard E

Kno

wle

dge

and

proc

edur

es

The student work has the following characteristics: • recall, access, selection of

mathematical definitions, rules and procedures in routine and non-routine simple tasks through to routine complex tasks, in life-related and abstract situations Q 1c, 1d, 1e, 1f, 1g, 2, 3c, 4, 5, 6, 7, 8, 9, 10

• application of mathematical definitions, rules and procedures in routine and non-routine simple tasks through to routine complex tasks, in life-related and abstract situations Q 1c, 1d, 1e, 1f, 1g, 2, 3c, 3d, 4, 5, 6, 7, 8, 10

• numerical calculations, spatial sense and algebraic facility in routine and non-routine simple tasks through to routine complex tasks, in life-related and abstract situations appropriate selection and accurate use of technology Q 2, 3d, 4, 5, 7, 8, 10

• appropriate selection and accurate use of technology Q 3c, 3d, 5, 6, 7c, 9b, 9c, 10


mathematical definitions, rules and procedures in routine and non-routine simple tasks through to routine complex tasks in life-related and abstract situations Q 1c, 1d, 1e, 1f, 1g, 2, 3c, 4, 5, 6, 7, 8, 9, 10

• application of mathematical definitions, rules and procedures in routine or non-routine simple tasks, through to routine complex tasks, in either life-related or abstract situations Q 1c, 1d, 1e, 1f, 1g, 2, 3c, 3d, 4, 5, 6, 7, 8, 10

• numerical calculations, spatial sense and algebraic facility in routine or non-routine simple tasks, through to routine complex tasks, in either life-related or abstract situations Q 2, 3d, 4, 5, 7, 8, 10

• appropriate selection and accurate use of technology Q 3c, 3d, 5, 6, 7c, 9b, 9c, 10


mathematical definitions, rules and procedures in routine, simple life-related or abstract situations Q 1c, 1d, 1e, 1f, 2, 3c, 4a, 5, 7a, 9a

The student work has the following characteristics: • use of stated rules and

procedures in simple situations Q 1a,1b, 3a, 3b

The student work has the following characteristics: • statements of relevant

mathematical facts

• application of mathematical definitions, rules and procedures in routine, simple life-related or abstract situations Q 1c, 1d, 1e, 1f, 2, 3c, 3d, 4a, 5, 7a

• numerical sense, spatial sense and algebraic facility in routine, simple life-related or abstract situations Q 2, 3d, 4a, 5, 7a

• selection and use of technology Q 3c, 3d, 5, 9b

• numerical sense, spatial sense and/or algebraic facility in routine or simple tasks Q 3a, 3b

• use of technology Q 3b • use of technology

Preliminary grade boundaries 72–62 61–48 47–20 19–6 5–1

Note: Preliminary grade boundaries are based on the school’s experience with similar assessment instruments and on the relative number of marks available based on the task-specific descriptors (drawn from the syllabus exit standards). All grade boundaries must be confirmed once they have been applied to student responses and matched to syllabus standards.



Page 6 of 18

Standard A Standard B Standard C Standard D Standard E

Mod

ellin

g an

d pr

oble

m s

olvi

ng

The student work has the following characteristics:





• use of problem-solving strategies to interpret, clarify and analyse problems to develop responses from routine simple tasks through to non-routine complex tasks in life-related and abstract situations Q 11, 12, 13, 14

• identification of assumptions and their associated effects, parameters and/or variables Q 11, 14

• use of problem-solving strategies to interpret, clarify and analyse problems to develop responses to routine and non-routine simple tasks through to routine complex tasks in life-related or abstract situations Q 11, 12, 13

• use of problem-solving strategies to interpret, clarify and develop responses to routine, simple problems in life-related or abstract situations Q 11

• evidence of simple problem-solving strategies in the context of problems

• evidence of simple mathematical procedures

• use of data to synthesise mathematical models and generation of data from mathematical models in simple through to complex situations Q 11, 12, 14

• use of data to synthesise mathematical models in simple situations and generation of data from mathematical models in simple through to complex situations Q 11, 12, 14

• use of mathematical models to represent routine, simple situations and generate data Q 11

• use of given simple mathematical models to generate data

• investigation and evaluation of the validity of mathematical arguments including the analysis of results in the context of problems, the strengths and limitations of models, both given and developed Q 11, 12, 13, 14

• interpretation of results in the context of simple through to complex problems and mathematical models Q 11, 12, 13, 14

• interpretation of results in the context of routine, simple problems Q 11



Page 7 of 18

Standard A Standard B Standard C Standard D Standard E

Com

mun

icat

ion

and

just

ifica

tion

The student work has the following characteristics • appropriate interpretation and

use of mathematical terminology, symbols and conventions from simple through to complex and from routine through to non-routine, in life-related and abstract situations Q 1c, 1d, 1e, 1f, 2, 3c, 3d, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14

• organisation and presentation of information in a variety of representations Q 1c, 1g, 5, 6, 7c, 8, 9b, 10, 11, 12, 14

• analysis and translation of information from one representation to another in life-related and abstract situations from simple through to complex and from routine through to non-routine Q 1f, 2, 3c, 3d, 4, 5, 6, 7, 8, 9b, 9c, 10, 11, 12, 13, 14

• use of mathematical reasoning to develop coherent, concise and logical sequences within a response from simple through to complex and in life-related and abstract situations using everyday and mathematical language Q 1g, 2, 3c, 3d, 4, 5, 6, 7, 8, 9c, 10, 11, 12, 13, 14

• coherent, concise and logical justification of procedures, decisions and results Q 6, 9c, 10, 11, 12, 14

• justification of the reasonableness of results Q 11

The student work has the following characteristics • appropriate interpretation and

use of mathematical terminology, symbols and conventions in simple or complex and from routine through to non-routine, in life-related or abstract situations Q 1c, 1d, 1e, 1f, 2, 3c, 3d, 4, 5, 6, 7, 8, 9a, 10, 11, 12, 13, 14

• organisation and presentation of information in a variety of representations Q 1c, 1g, 5, 6, 7c, 8, 9a, 9b, 10, 11, 12, 14

• analysis and translation of information from one representation to another in life-related or abstract situations, simple or complex, and from routine through to non-routine Q 1f, 2, 3c, 3d, 4, 5, 6, 7, 8, 9b, 9c, 10, 11, 12, 13, 14

• use of mathematical reasoning to develop coherent and logical sequences within a response in simple or complex and in life-related or abstract situations using everyday and/or mathematical language Q 1g, 2, 3c, 3d, 4, 5, 6, 7, 8, 9c, 10, 11, 12, 13, 14

• coherent and logical justification of procedures, decisions and results Q 6, 9c, 10, 11, 12, 14

The student work has the following characteristics • appropriate interpretation

and use of mathematical terminology, symbols and conventions in simple routine situations Q 1c, 1d, 1e, 1f, 2, 3c, 3d, 4a, 5, 7a, 9a, 11

• organisation and presentation of information Q 1c, 5, 9a, 9b, 11

• translation of information from one representation to another in simple routine situations Q 1f, 2, 3c, 3d, 4a, 5, 7a, 9a, 9b, 11

• use of mathematical reasoning to develop sequences within a response in simple routine situations using everyday or mathematical language Q 2, 3c, 3d, 4a, 5, 7a, 11

• justification of procedures, decisions or results Q 11

The student work has the following characteristics • use of mathematical

terminology, symbols or conventions in simple or routine situations Q 1a, 1b, 3a, 3b

• presentation of information

The student work has the following characteristics • use of mathematical

terminology, symbols or conventions

• presentation of information



Page 8 of 18

Indicative response — Using marks Instrument-specific standard descriptors

Knowledge and procedures Question 1 (a–g) Q1a.

𝑦 = (2𝑥 − 1)(𝑥2 + 3𝑥 + 5) = 2𝑥3 + 5𝑥2 + 7𝑥 − 5 𝑦’ = 6𝑥2 + 10𝑥 + 7

or

Let 𝑢(𝑥) = (2𝑥 − 1) then 𝑢’(𝑥) = 2 Let 𝑣(𝑥) = (𝑥2 + 3𝑥 + 5) then 𝑣’(x) = 2𝑥 + 3

By the product rule 𝑦’ = (2𝑥 − 1) (2𝑥 + 3) + (𝑥2 + 3𝑥 + 5)x2

Marking scheme Use of stated rule to expand expression

(1 mark) Use power and constant multiple rule for differentiation

(1 mark) or Define 𝑢 and 𝑣 and find 𝑢’ and 𝑣’

(1 mark) Use product rule

(1 mark)

Define 𝑢 and find 𝑢’ (1 mark)

Use of chain rule and substitution

(1 mark)

Change from surd to index form, define 𝑢 and find 𝑢’

(1 mark) Use of chain rule and substitution

(1 mark)

Define u and find u’ (1 mark)

Use chain rule and recall and apply derivative of natural logarithm and constant

(1 mark)

Use of stated rules and procedures in simple situations

Q1b. 𝑦 = (3𝑥2 + 3)3 Let u = (3𝑥2 + 3) then 𝑢’ = 6𝑥 Hence 𝑦 = 𝑢3 and 𝑦’ = 3𝑢2 Using chain rule and substitute for 𝑢 Derivative = 3(3𝑥2 + 3)2 × (6𝑥)

Recall, access, selection of mathematical definitions, rules and procedures

Q1c.

𝑦 =

Let 𝑢 = (3 − 4x) then 𝑢’ = −4

Hence 𝑦 = 𝑢12 and 𝑦’ = 1

2 𝑢−12

Use chain rule and substitute for 𝑢

(– 4)

Application of mathematical definitions, rules and procedures

Q1d.

y = ln(3x − 2) + 1

Let u = (3x – 2) then u’ = 3

Hence y = ln (u) + 1 and y’ = 1u

Use chain rule and substitute for u

y’ = 3(3x−2)

21

)43(43 xx −=−

21

)43(21

dd −

−= xxy

• The matching of the standards with the items is demonstrated in the left-hand column, as also indicated on the Instrument-specific criteria and standards on pages 5, 6 and 7.

• The marking scheme is in the right-hand column, with both columns indicating the relationship between the standards and the marks.



Page 9 of 18

Instrument-specific standard descriptors Recall, access and selection of mathematical definitions, rules and procedures Application of mathematical definitions, rules and procedures

Q1e.

y = −4 cos(.5x2)

Let u = (.5x2) then u’ = x

Hence y = −4 cos (u) and y’ = 4 sin (u)

Use chain rule and substitute for u

y’ = 4 sin(.5x2) × x

Define 𝑢 and find 𝑢’

(1 mark) Use chain rule and recall and apply derivative of the power and constant multiple rule and trigonometric function

(1 mark)

Define u and find 𝑢’ (1 mark)

Use chain rule and recall and apply derivative of exponential functions

(1 mark) and trigonometric functions

(1 mark)

Q1f.

y = esin x

Let u = sin x then u’ = cos x

Hence y = eu and y’ = eu

Use chain rule and substitute for u dydx

= (cos x)(esin x) Recall, access and selection of mathematical definitions, rules and procedures Application of mathematical definitions, rules and procedures

Q1g.

y = (x3 − 1)(sin3 3x)

Let u(x) = (x3 − 1) then u’(x) = 3x2

Let v(x) = (sin3 3x) and using chain rule v’(x) = 3(sin23x) 3 cos 3x

Using product rule dydx

= (x3 − 1)3 sin2(3x). 3 cos 3x + sin3(3x).3x2

Define 𝑢 and find 𝑢’

(1 mark) Define 𝑣 and find 𝑣’

(1 mark) Use chain rule and recall and apply derivative of power and constant multiple rule and trigonometric functions

(1 mark) Use product rule

(1 mark)

(17 marks) Recall, access, selection of mathematical definitions, rules and procedures

Question 2. Q2a.

𝑙𝑛 (𝑥 + 2) = 3

Using log laws and definitions

x + 2 = e3

x = e3 − 2

Use the definition (change from log form to index form) and rearrange

(1 mark)

Application of mathematical definitions, rules and procedures Numerical calculations, spatial sense and algebraic facility

Q2b.

Using log laws

m = log 5 + log 4 − log 2 = log 20 − log 2

m = log 10 = 1

Use two log properties — addition and subtraction

(1 mark) Use definition (change from log form to index form) and solve for m

(1 mark)



Page 10 of 18

Instrument-specific standard descriptors

Q2c.

102x = 500

Using relationship between log and exponentials

2x = log10500

x = log5002

Use definition (change from index form to log form)

(1 mark) Rearrange and solve for x

(1 mark)

Q2d.

𝑘 × √32

+3 = 8

√3 2

k = 5

k = 10√3

Marking scheme Recall value of common acute angle

(1 mark) Rearrange and solve for k

(1 mark)

(7 marks)

Use of stated rules and procedures in simple situations

Question 3. Q3a.

When t = 0

P0 = 15000

Substitution procedure

(1 mark)

Numerical sense, spatial sense and algebraic facility Selection and use of technology

Q3b.

18321 = 15000𝑒5𝑘

𝑒5𝑘 = 1832115000

Using relationship between log and exponentials

5𝑘 = 𝑙𝑜𝑔𝑒 �1832115000

�

𝑘 = .039999 = .04 (2 𝑑𝑝)

Substitution procedure

(1 mark) Use of stated law (index to log form)

(1 mark) Algebraic facility Use of technology

(1 mark)

Recall, access, selection of mathematical definitions, rules and procedures Application of mathematical definitions, rules and procedures Numerical calculations, spatial sense and algebraic facility Selection and use of technology

Q3c.

Using function find P(8)

P(8) = 15000e(8×.04)

P(8) = 20657 (to nearest whole number)

Q3d.

Find t when P(t) = 30000

30000 = 15000e.04t

2 = e.04t

. 04t = loge2

t = 17.32

In the 18th year or 2020

Select mathematical procedure (find 𝑃𝑛 when t=8)

(1 mark) Select and apply mathematical procedure (find t when 𝑃𝑛 = 2𝑥15000

(1 mark) Rearrange and select mathematical rule (change from index to log form)

(1 mark) Use of calculator to solve for t

(1 mark)

(8 marks)



Page 11 of 18

Instrument-specific standard descriptors Recall, access, selection of mathematical definitions, rules and procedures Application of mathematical definitions, rules and procedures Numerical calculations, spatial sense and algebraic facility

Question 4.

Q4a.

y’ = 23

cos �x3�

gradient of tangent = y’(3π) = 23

cosπ = −23

Equation of tangent at P (3π, 0)

(y – 0) = −23

(x − 3π)

3y = −2x + 6π

Q4b.

equate derivative to −13

−1 3

= 23

cos �x3�

−1 2

= cos �x3�

so x3

= 2π3

. 4π3

, 8π3

, 10π3

x = 2π, 4π, 8π, 10π

For given domain x = 2π, 4π

There will be two tangents at the points where 𝑥 = 2𝜋, 4𝜋 where the gradient is −1

3

Marking scheme Recall definition for gradient of a curve

(1 mark) Evaluate the gradient

(1 mark) Apply mathematical definition for a straight line

(1 mark) Recall definition for gradient at a point

(1 mark) Substitute correctly

(1 mark) Apply rules for solving trigonometric equations

(1 mark) Recall and apply mathematical definition of a domain

(1 mark) (7 marks)

Select and apply mathematical definition for velocity

(2 marks) Apply mathematical definition for acceleration

(1 mark) Substitution to solve for acceleration at the given time

(1 mark) Use of technology

(1 mark) (5 marks)

Recall, access, selection of mathematical definitions, rules and procedures Application of mathematical definitions Numerical calculations Selection and use of technology

Question 5.

Velocity = h’ = 10 sin (2t)

Acceleration = h’’ = 20 cos (2t)

h”(1) = 20 cos (2)

= −8.32 m/𝑚𝑖𝑛2

The point on the wheel is decelerating at the time 1 minute



Page 12 of 18

Instrument-specific standard descriptor Recall, access, selection of mathematical definitions, rules and procedures

Question 6.

ln (x – 2) ≥ 0 since we need to find its square root

ln(1) = 0

→ x – 2 ≥ 1

→x ≥ 3

Marking scheme Recall every non-negative real number has a unique non-negative square root

(1 mark) Recall rule log𝑎 1 = 0

(1 mark) Apply to inequation

(1 mark) Rearrange and solve inequation

(1 mark)

(4 marks) Application of mathematical definitions, rules and procedures

Question 7.

Q7a.

√3 sin x = cos x

tan x = 1√3

x = π6

, �π + π6� = 7π

6

Recall form of trigonometric equation

(1 mark) Recall unit circle and trigonometric ratios

(1 mark) Solve for x

(1 mark)

Numerical calculations, spatial sense and algebraic facility

Q7b.

cos(2θ) = −12

Using quadrant rules

2θ = �π − π3� = 2π

3, �π + π

3� = 4π

3

2π + 2π3

= 8π3

, 2π + 4π3

=10π3

θ = π3

, 2π3

, 4π3

, 5π3

Apply procedure for solving trigonometric equations with multiple angles

(1 mark) Recall unit circle and trigonometric ratios

(1 mark) Generate co-terminal angles

(1 mark) Rearrange to solve for unknown

(1 mark)

Application of mathematical definitions, rules and procedures Numerical calculations, spatial sense and algebraic facility Selection and use of technology

Q7c.

Squaring LHS and RHS 1+sin (t)

4= .1225

1 + sin(t) = .49

sin(t) = −.51

t = sin−1(−.51)

Using GDC

t = π + .535 = 3.67,2π − .535 = 5.75

Alternatively students may use technology and graphically find the solution (point of intersection between two functions)

Rewrite equation in a different form

(1 mark) Recall form of trigonometric equation and unit circle rules and use calculator to solve

(2 marks) (10 marks)



Page 13 of 18

Instrument-specific standard descriptor Recall, access, selection of mathematical definitions, rules and procedures Application of mathematical definitions, rules and procedures

Question 8.

Using log laws

log10 �P

QR3�2

= 2(log10P − log10QR3)

= 2(log10P − (log10Q + 3log10R))

Substitute information

= 2(x – y – 3z)

Recall properties of logarithms

(1 mark) Apply properties of logarithms to generate expression containing given values

(1 mark) Apply procedure to find expression

(1 mark)

(3 marks)

Recall, access, selection of mathematical definitions, rules and procedures Selection and use of technology

Question 9.

Q9a.

𝑏 = 6

Q9b.

𝑥 = 1.05

Recall definition for the amplitude of a trigonometric function

(1 mark) Select functions on calculator to sketch g(x)

(2 marks) Recall procedure for graphically finding solution to an equation

(1 mark) Select function on calculator to solve for point of intersection

(1 mark) (5 marks)

1 2

By

x



Page 14 of 18

Instrument-specific standard descriptor Recall, access, selection of mathematical definitions, rules and procedures Application of mathematical definitions, rules and procedures Numerical calculations, spatial sense and algebraic facility Selection and use of technology

Question 10.

𝐴 = 𝐴0𝑒𝑘𝑡

When 𝑡 = 5,𝐴 = 1

1 = 𝐴0𝑒5𝑘 (1)

𝐴′ = 𝑘𝐴0𝑒𝑘𝑡

When 𝑡 = 5,𝐴’ = .2

. 2 = 𝑘𝐴0𝑒5𝑘 (2)

From (1) 𝐴0 = 1𝑒5𝑘

Substitute into (2) . 2 = 𝑘. 1 . 2 = 𝑘

OR

(2)(1)

= 𝑘 = .2

Sub into (1)

1 = 𝐴0𝑒1

Therefore 𝐴0 = 1𝑒

So 𝐴 = 𝑒.2𝑡

𝑒

Find t when 𝐴 = 100

100𝑒 = 𝑒 .2𝑡

. 2𝑡 = 𝑙𝑜𝑔𝑒(100𝑒)

. 2𝑡 = 𝑙𝑜𝑔𝑒100 + 1

𝑡 = 28.03 hours

Marking scheme Substitute given value into equation

(1 mark) Apply definition for rate of change

(1 mark) Substitute given value into equation

(1 mark) Apply the method of simultaneous solutions to solve unknown

(1 mark) Use given information to substitute into equation and select mathematical rule (change from index to log form)

(1 mark) Use of calculator to solve for t

(1 mark)

(6 marks)



Page 15 of 18

Instrument-specific standard descriptor


Use of problem solving strategies

Use of mathematical models

Interpretation of results

Identification of assumptions

Question 11 (routine simple life-related)

The intensity of light at a distance of m metres from the entrance of a cave can be modelled by the equation:

I = Io e –km It is found that the intensity of light is halved when a person is 5 metres from the cave entrance.

If artificial light is necessary when the intensity of light is less than 10% of the intensity at the cave entrance, find out how deep a cave explorer can go before artificial light is necessary.

Justify the reasonableness of this result.

Identify any assumptions that have been made.

Expected response

Intensity of light at entrance (𝑚 = 0) 𝐼 = 𝐼0

Intensity of light 5 metres from cave entrance (𝑚 = 5) I = 𝐼02

𝐼02

= 𝐼0𝑒−5𝑚 12

= 𝑒−5𝑚

𝑚 =ln (.5)−5

𝑚 = .1386 Find m when 𝐼 = 𝐼0

10

𝐼010

=𝐼0𝑒−.1386𝑚

𝑚 = 16.613

Therefore a cave explorer would need to use artificial light when they are just over 16.6 metres from the entrance.

The result seems reasonable — it is a greater distance than the result for 50% light intensity (given), and reflects the exponentially decreasing trend of the function.

It is assumed that there are no other light sources that are contributing to the intensity of the light (the light source is at the entrance of the cave)

The matching of the standards with the items is demonstrated in the left-hand column, as also indicated on the Instrument-specific criteria and standards on pages 6, 7 and 8.



Page 16 of 18


Question 12 (routine complex life-related)

A space vehicle is moving on a path which is described by the equation 𝑦 = 𝑑

23 + 7𝑑

34 where 𝑦 is the vertical distance from the launch point and 𝑑 is

the horizontal distance from the launch point in kilometres.

When the horizontal distance is 20 000 km, the space vehicle fires an exploratory rocket into space tangentially on a straight line course for Bega Nova (an abandoned space station). Bega Nova is located at a horizontal distance of 950 000 km and a vertical distance of 3 x 105 km from Earth.

Will the rocket collide with Bega Nova?

Expected Response

𝑦 = 𝑑2 3� + 7𝑑3 4�

𝐹𝑖𝑛𝑑 𝑇𝑎𝑛𝑔𝑒𝑛𝑡: 𝑃𝑜𝑖𝑛𝑡:𝑑 = 20000,𝑦 = 12509.36

𝑦′ =23𝑑−1 3� +

214𝑑−1 4�

𝑚𝑇 =23

20000−1 3� +214

20000−1 4� = 0.466

Equation:

𝑦 − 12509.36 = 0.466(𝑥 − 20000)

𝑦 = 0.466𝑥 + 3189.95

To check if the rocket will hit the BN station, sub x-coordinate (950000 ) into the equation:

𝑦(950 000) = 0.466 × 950 000 + 3189.95 𝑦(950 000) = 445890 ≠ 300 000

The rocket will NOT hit the station BN.






Page 17 of 18


Marking scheme — Indicative response

Use of problem solving strategies Interpretation of results

Question 13 (routine complex abstract)

Given that 𝑓(𝑥) = (2𝑥 − 𝑘)3, find the value of the constant 𝑘 for which 𝑓”(1) = 24

Expected response




Question 14 (non-routine complex life-related)

A snowball rolling down a snow covered slope increases in size according to the formula 0.4 5d t= + where d is the diameter of the snowball in centimetres and t is the time in seconds.

Given that density = massvolume

and that packed snow has a density

of 0.9 𝑔/𝑐𝑚3, find the rate of increase of the mass of the snowball after 4 seconds. Clearly state any assumptions and their associated effects.

Expected response

Let r = radius of the snowball.

Then r = 0.2t + 2.5

Assuming the snowball is spherical in shape, its volume can be found using the formula

V = 34

3rπ

= 34 (0.2 2.5)

3tπ +

density = massvolume

∴ mass (𝑀)= density × volume

𝑀 = 34 (0.2 2.5)0.9

3tπ +

×

𝑀 = 31.2 (0.2 2.5)tπ + g dMdt

= 23.6 (0.2 2.5) 0.2tπ + ×

= 20.72 (0.2 2.5)tπ +

When t = 4, dMdt

= 20.72 (0.2(4) 2.5)π +

= 24.63 ≈ 24.6 g/s



Page 18 of 18

After 4 seconds, the mass of the snowball is increasing at a rate of approximately 24.6 g/s.

Assumptions:

• The snowball is spherical in shape and the effect is that the surface is in contact with snow at a constant rate.

• The snowball rolls downhill unhindered during the first 4 seconds. If hindered it will not pick up as much snow and thus the rate of mass increase will be less.

• The downhill slope, and therefore the speed of the snowball, is consistent during the first 4 seconds. If not consistent, the mass will either increase or decrease depending on the rate of speed.

Instrument-specific standard descriptor Identification of assumptions and their associated effects

Marking scheme Knowledge and procedures

Simple RS (Routine Simple) RC (Routine Complex) NRS (Non-routine simple)

1 4 9 4

2 7

3 4 4

4 3 4

5 5

6 4

7 3 4 3

8 3

9 3 2

10 6

Totals 8 34 16 14

Total / 72


Routine Simple Routine Complex Non-routine complex

Question Eleven X

Question Twelve X

Question Thirteen X

Question Fourteen X

The four modelling and problem solving questions, Qs 11–14, are designed to assess the principles of complexity and initiative along the continuum. This is indicated by an ‘X’ to show whether the task is routine simple, routine complex, or non-routine complex. The Xs are coloured to indicate that the tasks are: Life-related or abstract.

Documents

Mathematics B 2008