Ser6 Fmathe 1B - further maths prac exam PDF

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Practice Written Examination 1

QATs VCE®Further Mathematics

NAME:

VCE®FURTHER MATHEMATICS Units 3 & 4 Practice written examination 1 Reading time: 15 minutes Writing time: 1 hour 30 minutes MULTIPLE-CHOICE QUESTION BOOK Section A - Core B - Modules

Number of questions 24 32

Number of questions to be answered 24 16

Number of modules

Number of modules to be answered

4

2

Marks 24 16 Total 40

- Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, one bound reference, one approved graphics calculator or approved CAS calculator or CAS software and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. - Students are NOT permitted to bring into the examination room: blank sheets of paper and/or white out liquid/tape. Materials supplied - Question book of 41 pages. - Answer sheet for multiple-choice questions. - Formula sheet. - Working space is provided throughout the book. Instructions - Ensure that your name is clearly written on the answer sheet, and that the modules answered are correctly identified. - Unless otherwise indicated, the diagrams in this book are not drawn to scale. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic

©2017 Published by QATs. Permission for copying in purchasing school only.

Ser6FMATHE1B 1

Practice Written Examination 1

QATs VCE®Further Mathematics

SECTION A Instructions for Section A Answer all questions in pencil on the answer sheet provided for multiple-choice questions. Choose the response that is correct for the question. A correct answer scores 1, an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question.

Data analysis Use the following dot plot to answer Questions 1 and 2.

n = 22

Question 1 The mode of the dot plot is A. 9 B. 3 C. 1 D. 4 E. 8

©2017 Published by QATs. Permission for copying in purchasing school only.

Ser6FMATHE1B 2

Practice Written Examination 1

QATs VCE®Further Mathematics Question 2 The interquartile range of the dot plot is A.

9

B.

5.5

C. 3 D. 5 E. 4

Use the following information to answer Questions 3 and 4. The voting preference (Labor, Liberal, Greens) and the gender (Male, Female) of 130 adults was recorded. The results are displayed in the two-way frequency table below. Gender

Voting Preference Labor Liberal Greens Total

Male 31 25 10 66

Female 29 18 17 64

Question 3 The percentage of females whose voting preference is Labor is closest to A.

22%

B.

48%

C.

45%

D.

47%

E.

28%

©2017 Published by QATs. Permission for copying in purchasing school only.

Ser6FMATHE1B 3

Question 4 The variables voting preference (Labor, Liberal, Greens) and gender (Male, Female) are A.

both ordinal variables.

B.

both nominal variables.

C.

a nominal variable and an ordinal variable respectively.

D.

an ordinal variable and ordinal variable respectively.

E.

a discrete variable and a nominal variable respectively.

Use the following information to answer Questions 5 and 6. The scores for a physics and a maths exam at university are normally distributed. The following table shows the mean score and standard deviation for the physics and maths exam. Physics Maths

Mean 67 72

Standard Deviation 8 6

Question 5 If 2400 students took the physics exam, the number of students that were expected to get a score of at least 59 is closest to A.

1200

B.

792

C.

2016

D.

2280

E.

1608

Question 6 If a particular student achieved scores of 71 on the physics exam and 75 on the maths exam, then which one of the following statements is true? A.

Both exams have the same standardised score.

B.

Physics has a higher standardised score.

C.

Maths has the higher standardised score.

D.

The student scored in the top 13.5% of students on both exams.

E.

The student scored in the bottom 50% on the Physics exam and the top 50% on the Maths exam.

Question 7 The dot plot below shows the heights, in cm, of 11 year 8 students.

Which of the following box plots best represents this data? A.

B.

C.

D.

E.

Question 8 The parallel boxplot below shows the distribution in height, in cm, of students in a metropolitan school and students in a country school.

Country students

Metropolitan students

Which of the following statements is not true? A.

There are no outliers for the data set.

B.

The shortest Metropolitan student is shorter than the shortest Country student.

C.

The median height of Country students is great than the median height of Metropolitan students.

D.

50% of Country students are taller than 75% of Metropolitan students.

E.

75% of Country students are taller than 50% of Metropolitan students.

Question 9 The number of hours studied, x, by students and the scores achieved, y, on an English paper were recorded. The mean and standard deviation of each variable was calculated and is summarised in the table below. Hours studied, x Score achieved, y

Mean 6.8 68

Standard Deviation 2.2 8.8

If the correlation coefficient for the data is r = 0.85, then a least squares regression line that describes the association between the variables hours studied and score achieved is closest to

A.

score achieved  3.4  45 hours studied

B.

hours studied  44.9  3.4 score achieved

C.

score achieved  44.9  3.4 hours studied

D.

score achieved  44.9  34 hours studied

E.

hours studied  44.9  34 score achieved

Question 10 A large study of adult males shows that there is a negative association between the amount of time spent exercising each week and the weight of an adult male. From this information is can be concluded that A.

male adults who tend to spend more time exercising each week tend to have a lower weight.

B.

male adults who have a lower weight will spend more time exercising.

C.

male adults who spend more time exercising will have a lower weight.

D.

male adults who spend more time exercising will have a higher weight.

E.

male adults who tend to have a higher weight tend to spend less time exercising.

Question 11 The following least squares regression line

4 shoe size

height 135  is used to describe the association between a person’s shoe size and height (cm). A person had a shoe size of 9 and a height of 176cm. When the least squares regression line is used to predict this person’s height, the residual is closest to A.

-5

B.4 C. D.5 E.6

-4

Question 12 The table below gives the maximum heart rate , MHR, in beats per minute and age for 16 people completing a fitness test. A scatterplot of the data is also shown. Age 15 16 17 18 19 32 24 25 28 23 21 34 29 25 27 17

MHR 205 197 195 190 172 162 177 168 162 168 175 160 162 163 161 190

The scatterplot is non-linear. A reciprocal transformation is applied to the variable age to linearise the data. The equation of the least squares regression line that linearises the data is closest to A.

MHR  119.4 

1218.4 age

1

B.

MHR  0.004  0.00007 age

C.

MHR  228.4 

2.3 age

D.

1 MHR  119.4 1218.4 age

E.

MHR 119.4 1218.4 age

Question 13 The time series below shows the number of hours a learner driver practices for each month (Jan=1, Feb=2, etc.).

Using three-mean smoothing, the smoothed data value for the month of October is A.

3

B.

3.7

C.

6

D.

5.3

E.

4.7

Question 14 The seasonal index for sales of air conditioners in winter is 0.75. To correct for seasonality, the actual air conditioner sales in winter should be A.

increased by 25%

B.

reduced by 25%

C.

reduced by 33%

D.

increased by 33%

E.

increased by 133%

Use the following information to answer Questions 15 and 16. Question 15 The table below shows the seasonal indices for the number of visitors each month to a swimming pool for a 1 year period. Month Number of visitors

Jan 11567

Feb 9584

Mar 9021

April 8566

May 8576

June 6895

July 5877

Aug 7123

Sep 8347

Oct Nov 8839 8999

Seasonal index

1.4

1.26

1.15

0.90

0.90

0.75

0.65

0.79

0.85

1.05

The seasonal index for October is A.

0.85

B.

0.95

C.

1.05

D.

0.58

E.

1.0

Question 16 In July, there were 5877 visitors to the swimming pool. The deseasonalised value of the number of visitors to the pool for the month of July is closest to A.

9041

B.

3820

C.

3821

D.

9042

E.

9024

Dec 10640

1.35

Recursion and financial modelling Question 17 Given the recursion relation below

To  2500,

Tn1  0.8Tn  200

The first four terms of this recurrence relation are A.

2500, 2000, 1600, 1280, …….

B.

2500, 2200, 1960, 1768, ……

C.

2500, 2700, 2900, 3100, ……

D.

2500, 2160, 1888, 1670, ……

E.

2500, 2300, 2100, 1900, ……

Question 18 Which of the following recurrence relations will generate a sequence whose values will grow geometrically by 12%? A.

V0 

B.

V0 

800,

800, C.

V0  V0 

800, E.

800,

Vn1  0.12Vn Vn1  Vn 1.12

800, D.

Vn1  Vn 12

V0 

Vn1  12Vn Vn1  1.12Vn

Question 19 Nick invests $25 000 in a savings account for three years. His savings account has an interest of 4.25% per annum compounded quarterly. The effective annual interest rate for this investment would be closest to A.

4.32%.

B.

4.31%

C.

4.06%

D.

3.03%

E.

4.33%

Question 20 Consider the recurrence relation below.

V0  $6000,

Vn1  Vn  200

The recurrence relation is best described as A.

an interest only loan of $6000.

B.

a reducing balance loan of $6000 with periodic repayments of $200.

C.

a flat rate depreciation of an asset initially valued at $6000.

D.

a unit cost depreciation of an asset originally valued at $6000.

E.

an annuity investment with periodic additions of $200 made to the investment.

Question 21 The initial value of a car is $26 000. The car depreciates each year using a reducing balance method. If the car is valued at $18 000 after 6 years, then the annual rate of depreciation is closest to A.

$1500

B.

$1300

C.

$1400

D.

$1000

E.

$8000

Question 22 Hugo invests $120 000 in a perpetuity that will provide $5000 per year to provide funding for a research grant. The rate per annum and value of the perpetuity after 6 years respectively is A.

4% and $120 000

B.

4.17% and $5000

C.

4.17% and $150 000

D.

4% and $5000

E.

4.17% and $120 000

Question 23 John takes out a $30 000 reducing balance loan. The interest rate on the loan is 4.5% per annum, compounded quarterly. John makes repayments of $1000 per compounding period. The balance of the loan after 3 years is closest to A.

$8461

B.

$21 539

C.

$27 990

D.

$4139

E.

$47 081

Question 24 Wendy invests in an annuity that earns interest at a rate of 6.1% per annum, compounded quarterly. Each quarter Wendy receives payments from the annuity. The balance of the annuity will be $141 236.21 after 3 years and $110 145.11 after 8 years. The quarterly payment that Wendy receives from the annuity is closest to A.

$1100

B.

$3400

C.

$2900

D.

$3500

E.

$338 END OF SECTION A TURN OVER

SECTION B - Modules Instructions for Section B Select two modules and answer all questions within the modules selected in pencil on the answer sheet provided for multiple-choice questions. Show the modules you are answering by shading the matching boxes on your multiple-choice answer sheet and writing the name of the module in the box provided. Choose the response that is correct for the question. A correct answer scores 1, an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question. Module

Page

Module 1: Matrices.............................................................................................................................17 Module 2: Networks and decision mathematics..................................................................................23 Module 3: Geometry and measurement...............................................................................................29 Module 4: Graphs and relations...........................................................................................................33

Module 1: Matrices Before answering these questions you must shade the ‘Matrices’ box on the answer sheet for multiple-choice questions and write the name of the module in the box provided. Question 1 Matrix A, shown below, represents the number of hours spent studying Maths (M), History (H) and Italian (I) in a given week by Bruce (B), Clark (C), Damian (D), Edward (E) and Francis (F).

M 3 5  5  4 1

H I 4 4 B 4 3 C  2 4 D  2 2 E 3 4  F 

The element in row i and column j of matrix A is denoted as aij. The element a32 is the number of hours of A. Italian studied by Francis. B. Italian studied by Clark. C. History studied by Clark. D. Italian studied by Damian. E. History studied by Damian. Question 2 If

Then the value of c is equal to A. 4 B. -4 C. 0 D. 2 E. 1

b

1

 3

4

2

2

0

 c

1



 0 1  7

6

Question 3 The order of Matrix X is 2 3. The element in row i and column j of matrix is xij and is determined by the rule xij  2i  j Matrix X is represented by option

A.

1   1

2 3 0 1

3 1   2 B. 0   1 1  1 0   C.  3  5 4

D.

E.

1

1

0

3 

2 1  

1 0 

3 5 2 4

Question 4 The cost of 5 sausage rolls and 2 pies is $19.50. The cost of 3 sausage rolls and 1 pie is $11. Let x be the cost of a sausage roll and y be the cost of a pie.

x Then the matrix   is equal to  y

219.50 5 11  

A.  1

3

 1

2  19.50 5  11   

B. 

3 5

2  x    3 1   

C. 

5 3 

D. 

2 19.50   1 11  

1 2 19.50   3 5 11   

E. 

Question 5 If matrix M, has an order of 3 4 . Matrix P has 6 rows and matrix N is of order m n . If the matrix product

 1 4 6 2 MNP 7 0 5 2  2 3 8 4 Then the order of matrix N is A. 4 3 B. 4 6 C. 3 4 D. 6 4 E. 3 6

Question 6 6 players, Alan (A), Belinda, (B), Chris (C), Dane (D), Erin ( E) and Fraser (F) play in a scissors, paper, rock tournament. Each player plays each other once. The results of the tournament are shown in the matrix below where a 1 means that the player in that row defeated the player in that column.

Losers A

Winners

B C D E

A B 0 0 1 0  0 0  1 1 0 0  1 1

C D E F 1 0 1 0 1 0 1 0  0 0 1 0  1 0 1 1 0 0 0 0  1 0 1 0

F The tournament ranks the players on their two-step dominance score with the player with the highest score declared the winner. Using the two-step dominance ranking system, the final rankings of the tournament are A. Dane, Fraser, Belinda, Alan, Chris, Erin. B. Erin, Chris, Alan, Belinda, Fraser, Dane. C. Dane, Fraser, Belinda, Alan, Erin, Chris. D. Fraser, Dane, Alan, Belinda, Chris, Erin. E. Dane, Belinda, Fraser, Alan, Chris, Erin.

Question 7 A local council surveys its citizens about how they travel to work. The citizens either cycle (C), use public transport (P) or use their own motor vehicle (M) to travel to work. The transition diagram below shows the way the citizens travel to work from one week to the next.

A transition matrix that provides the same information as the transition diagram is A.

B.

this week C P M 0.75 0.20 0.05 C 0.05 0.80 0.15 P next week   0.10 0.30 0.60 M this week C P M 0.75 0.05 0.10 C 0.20 0.80 0.30 P next week   0.05 0.15 0.60 M

C.

D.

this week C P M 0.75 0.10 0.05 C 0.20 0.30 0.80 P next week   0.05 0.60 0.15 M this week C P M 0.75 0.05 0.10 C 0.20 0.60 0.30 P next week   0.05 0.15 0.80 M

E.

this week C P M 75 5 10 C   20 80 30 P next week  5 15 60 M

Question 8 230 students at a secondary school are surveyed about their breakfast preferences: 80 students prefer fruit (F) 90 students prefer toast (T) 60 students prefer cereal (C) The transition matrix, T, shown below displays how the students’ preferences change from one month to the next.

this month F T C 0.60 0.05 0.15 F  0.10 0.75 0.05 T next month    0.30 0.20 0.80...