For all Queensland schools

2017 Senior External Examination

Mathematics B Monday 30 October 2017Paper One — Question book 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 — Resource book• Paper One — 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

DirectionsYou may write in this book during perusal time.Paper One has six questions. Attempt all questions.

AssessmentPaper 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 sessionTake this book when you leave.

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. The ages of nine adults attending an art class are:

57 22 19 25 24 27 36 25 72

Calculate the mean and standard deviation of the data.

b. Two chemists performed multiple measurements to determine the volume of a chemical solution left after a reaction. Their results, in millilitres, were recorded and then presented as five-number summaries as shown in the table below.

i. Draw parallel boxplots of the data, on the graph paper at the back of your response book.

ii. Use the boxplots to compare and contrast the data.(KP)

c. a, b, c and d are four positive integers where .

The mean of the four numbers is 26.The median is 27.The mode is 27.The range is 30.

Determine the values of the four positive integers. Show full working.

(MP)

Five-number summary Chemist A Chemist B

Minimum 24.6 22.5

Q1 24.9 23.6

Median 26.5 26.6

Q3 27.1 27.7

Maximum 28.0 28.4

a b c d

1

Question 2 a. , where

i. Draw the graph on the graph paper at the back of your response book. Determine, giving reasons, if is both continuous and a function.

ii. State the domain and range of .

iii. Calculate .(KP)

b. By completing the square, express in the form where a, b and c are constants. Hence determine the transformations required to convert to

.(KP)

c. Jake travelled in his car for 450 km at a constant speed. If Jake had travelled 15 km/hr faster,the trip would have taken 1 hour less.i. If x km/h is Jake’s constant speed, show that the time saved is given by

.

ii. Find Jake’s constant speed by solving the equation. Show full working.(MP)

f x

=x2 2x– 2+

x 1+ 3 x 1–

1 x 6

f 2– f+ 3

y a= x b 2 c+ +

450x

--------- 450x 15+--------------- 1=–

2

Question 3 a. Using an algebraic method, solve for x in terms of π if 2 cos x + 1 = 0 and −2π ≤ x ≤ 2π.

(KP)

b. Show that .(KP)

c. The mathematical model relates the water depth, y metres, to the time t hours,

after high tide at the local wharf.

i. Determine the amplitude, tidal period and the greatest and least water depth of the model. (KP)

ii. A boat is ready to sail from this wharf. On the morning of departure, high tide occurs at 2 am. To avoid an overhead power cable, the boat can only leave if the water depth at the wharf is 8.5 metres or less. Also, in order for the ship to be able to sail through the shallow harbour entrance, the water level must be at least 2 metres above the low tide level.

Use a graphics calculator to determine what the earliest and latest times are for the ship to leave the wharf in the morning and safely negotiate the harbour entrance.

(KP)

d. Two identical buildings, each 50 m tall, are erected on a straight road running from east to west. From a point due south of the most easterly building, the angle of elevation to the top of that building is 23°. From the same point, the angle of elevation to the top of the other building is 17°.

Find the distance between the buildings. Identify any assumptions made and the effect these would have in the calculation of this distance.

(MP)

3

Question 4 a. Given that and

i. solve

ii. find the inverse function,

iii. determine the composite function, .(KP)

b. Solve algebraically the simultaneous equations and .(KP)

c. Determine the value of the constant, a, given that the tangent to at passes

through .(MP)

4

Question 5 a. Show from first principles that the derivative of is .

(KP)

b. Determine in each of the following.

i.

ii.

iii.(KP)

c. The displacement, x metres, of a particle from its starting point, after t seconds, is given by theequation .

i. Find the initial position of the particle.ii. Determine the velocity of the particle when t = 2 seconds.

(KP)

d. A ladder, AB, of length metres is placed so that it rests on a one metre high fence, PQ, that is located two metres from the upright wall, AC. One end of the ladder is placed on the ground at B and the other end against the wall at A.

i. Given QB = x metres, show that .

ii. Using calculus methods, find the minimum length of the ladder, AB.(MP)

x 3t 4 t 2+=

Anot to scale

C Q2 m

1 m

Bx m

P

5

Question 6 a. A box contains six blue marbles and nine red marbles. A marble is randomly drawn from the box,

its colour noted and then the marble is put back in the box before another marble is drawn.i. If two marbles are drawn from the box as described, what is the probability of obtaining two

blue marbles?ii. If X is the random variable that represents the number of blue marbles drawn from the box in

10 trials, calculate . (KP)

b. A plant nursery has two sites where seedlings are grown. The costs at each site are the same, but the seedlings are of different varieties.

At the first nursery site, 848 seedlings are produced in a month. The mean height of the seedlings is 51.0 mm and the standard deviation is 5.36 mm. At the second site, 936 seedlings are grown in the same time period with a mean height of 52.0 mm and a standard deviation of 6.5 mm. Seedlings with a height between 44.0 mm and 66.0 mm are saleable. All seedlings are sold and all sell for the same price.

If the heights of the seedlings are normally distributed, determine if the claim that the first nursery site is more profitable is reasonable. Show full working.

(MP)

End of Paper One

P 5 X 8

6

Ass

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8

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9

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Copyr

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www.

e State of Queensland (Queensland Curriculum and Assessment Authority) 2017

ight enquiries should be made to:

ger hing Unit

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ensland Curriculumssessment Authorityx 307, Spring Hill QLD 4004 Australia

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