(1) 2017 SAC Matrices - Further Maths Practice SAC PDF

Preview

Summary

Further Maths Practice SAC...

Description

2017%VCE%FURTHER%MATHEMATICS% % MATRICES% % SCHOOL%ASSESSED%COURSEWORK%(SAC)% % % NAME:%____________________________% %

%

FORM:%__________%

%

TEACHER%(CIRCLE):%

%%%%%%%BOD%

%

TRO% %

HOE!

! !

% INSTRUCTIONS:% ! • The!answers!to!the!questions!must!be!completed!in!the!spaces!provided.! •

The!time!allowed!to!complete!the!SAC!is!a!MAXIMUM%of!120%MINUTES.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! All!work!must!be!completed!during!this!time!allocation.!

•

ALL!answers!must!be!correct!to%TWO%DECIMAL%PLACES%unless!otherwise!stated.!

•

You!must!NOT!take!this!paper!with!you.!!

•

You!may!use!ONE!BOUND!reference!during!the!SAC.!!

•

You!may!use!a!CAS%CALCULATOR%and!a!scientific!calculator.!

!!!! GOOD%LUCK%☺ ☺!

!

!

Page 1 of 13

QUESTION 1 – LANGUAGE CLASSES SUBJECT SELECTION Students in Year 7 or 8 can either pick from either three electives languages: Spanish (S), French (F) and German (G). Each student must select one elective. The matrix (N) below outlines the student numbers in each language class.

S F G ⎡25 29 34⎤ Year!7 ⎥ N ! = !⎢ ⎣32 37 18⎦ Year!8

a) How many students have chosen Spanish?

(1 mark) b) How many students in total are in Year 7?

(1 mark)

c) What does !!N 2,1 represent?

(1 mark)

d) What was the most selected language elective? (1 mark)

!

Page 2 of 13

TEXTBOOKS Each subject requires a textbook. The cost (C) of each textbook is listed in the matrix below.!

⎡38.20⎤ S ⎢ ⎥ C = ⎢41.00⎥ ! F ⎢29.50⎥ G ⎣ ⎦ e) What is the order of Matrix C? (1 mark) f) Evaluate 𝑁!×!𝐶

! (1 mark) g) What information does 𝑁𝐶 show?

(1 mark) h) Explain why 𝐶!×!𝑁!is not defined.

(2 marks) The bookshop is having an end of year sale and reduces all the textbooks by 10%. i)

Show a scalar multiplication to matrix C that gives the new sale prices and calculate the new sale prices of each textbook. Round each price to the nearest cent.

(2 marks) !

Page 3 of 13

QUESTION 2 – INTERNATIONAL FOOD STALL As part of the language program students are required to host an international food stall at the school fete. The Spanish class sells churros, the French class sells Nutella crepes and the German class sells pretzels. In the table below are the number of each food item sold and the takings from the school fete over Friday, Saturday and Sunday. Day

Churros

Nutella Crepes

Pretzels

Total Takings

Friday

49

61

37

$277.35

Saturday

44

31

48

$225.75

Sunday

61

84

63

$383.90

a) Write a system of three simultaneous linear equations to represent the price of each food item sold. Let c be the cost of each churro n be the cost of each Nutella crepe p be the cost of each pretzel

(3 marks) b) Write this system of simultaneous equations in matrix form.

(3 marks) c) For the square matrix in (b), find its determinant.

(1 mark) !

Page 4 of 13

d) Solve the matrix equation from (b) to find the price of each food item to the nearest cent.

(3 marks) !

Another stall at the fete sells sandwiches and drinks. One family bought 3 sandwiches and 3 drinks at a cost of $12. Another family bought 4 sandwiches and 4 drinks at a cost of $16. e) Explain in matrix terms and demonstrate why it is not possible to actually determine the cost of one sandwich and one drink.

(2 marks)

!

Page 5 of 13

QUESTION 3 – ANAGRAMS The language teachers love playing around with anagrams – letter combinations that can generate a number of different words. For example, the letters A, C, D, E and R can form the words CADRE, CARED, CEDAR or RACED. They know that permutation matrices can be used to rearrange the letters in a word. ⎡ ⎢ ⎢ a) If matrix W = ⎢ ⎢ ⎢ ⎣

L E A S T

⎡ 0 0 ⎤ ⎢ ⎥ ⎢ 0 0 ⎥ and matrix P = ⎢ 1 0 ⎥ ⎢ 0 1 ⎥ ⎢ 0 0 ⎥ ⎣ ⎦

0 1 0 0 0

0 0 0 0 1

1 0 0 0 0

⎤ ⎥ ⎥ ⎥ , then what word is formed ⎥ ⎥ ⎦

by the matrix product P × W ?

(1 mark) b) In the matrix provided below, fill in the element values for matrix Q so that the matrix product Q × W gives the word SLATE. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

..... ..... ..... ..... .....

..... ..... ..... ..... .....

..... ..... ..... ..... .....

..... ..... ..... ..... .....

..... ..... ..... ..... .....

⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ×⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

L E A S T

⎤ ⎥ ⎥ ⎥ = ⎥ ⎥ ⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

S L A T E

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2 marks)

⎡ ⎢ ⎢ c) Explain why the matrix product Q4 × W gives the matrix ⎢ ⎢ ⎢ ⎣

L E A S T

⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(1 mark) !

Page 6 of 13

QUESTION 4 – SUBJECT SELECTION At the beginning of each term the students can choose to remain in their current language class or select a new language class for that term. The following information was obtained regarding the movement between language classes: • • • • • •

34 % 14% 25% 50% 8% 41%

of current Spanish students take (remain with) Spanish next term of current Spanish students change to French next term of current French students change to German next term of current French students take (remain with) French next term of German students change to French next term of German students change to Spanish next term

a) Find the percentage of students who are taking German this term and will continue to take German next term.

(1 mark) b) Fill in the rest of the transition matrix (T) for this situation. State values as proportions. !

S!!!!!! F!!!!!! G ⎡ ___ ___ ___ ⎤ S ⎥ ⎢ 0.14 0.50 0 . 08 T ! = !⎢ ⎥ F! ⎢ ___ 0.25 ___ ⎥ G ⎦ ⎣ !! (2 marks) At the beginning of the 2017 (Term 1) in Year 7 there are 25 students taking Spanish, 29 students taking French and 34 students taking German. c) Write this information as an initial state matrix S1 (Term 1, Year 7).

(1 mark) !

Page 7 of 13

d) Calculate S 2 .

(1 mark)

e) Calculate the number of students expected to be taking French in Term 1 Year 8 (there are four terms in a school year).

(1 mark)

f) How many students are expected to be taking French in the long term? Justify your answer with appropriate working.

(2 marks)

!

Page 8 of 13

After Term 4 the transition of students between classes changes as the German teacher takes long service leave and therefore German will no longer be offered. The transition matrix from Term 4 to Term 5 is shown below and represented by T*.!

S!!!!!! F!!!!!! G ⎡0.46 0.30 0.55⎤ S ⎢ ⎥ 0.54 0 . 70 0 . 45 T * ! = !⎢ ⎥ F ⎢ 0 0 0 ⎥⎦ G ⎣ !! !!! g) Explain why the third row is all zeros.!

(1 mark) h) Consider the students who choose French in Term 4 but were planning to choose German in Term 1 2018 ( S5 ). What percentage of these students are now going to remain studying French in Term 1 2018?

(1 mark) The numbers from the beginning of the 2017 (Term 1) in Year 7 remain the same. There are 25 students taking Spanish, 29 students taking French and 34 students taking German. i) If the German teacher is unavailable at the end of Term 4, show that there is expected to be 46 students taking French in Term 1 2018 ( S5 ). ! ! ! ! !

(2 marks)

!

Page 9 of 13

QUESTION 5 – DICTIONARIES The local book supplier has developed a matrix formula for determining the number of Spanish and French dictionaries that should be ordered each year.

S 0 is a column matrix that lists the number of dictionaries sold in 2015.

⎡505⎤ Spanish S0 = ⎢ ⎥ ! ⎣316⎦ French !! O1 is a column matrix listing the number of dictionaries to be ordered for 2016. O1 is given by the matrix formula

⎡ 80 ⎤ ⎡ 0.85 0 ⎤ and O1 = A S0 + B where A = ⎢ B=⎢ ⎥ ⎥ 0.78 ⎦ ⎣ 75 ⎦ ⎣ 0

a) Determine O1 , correct to the nearest integer (whole number).

(1 mark)

⎡499 ⎤ Spanish represents the number of dictionaries sold in 2016, ⎥! ⎣303 ⎦ French

b) Given that S1 = ⎢

determine O2 using the same matrix equation.

(1 mark)

!

Page 10 of 13

The matrix formula above only allows the supplier to predict the number of dictionaries that should be ordered one year ahead. A new matrix formula enables the supplier to determine the number of dictionaries to be ordered two or more years ahead. The new matrix formula is On + 1 = C On + D where On is a column matrix listing the number of Spanish and French dictionaries to be ordered for year n.

⎡ 72 ⎤ ⎡ 0.85 0 ⎤ and For this matrix equation, C = ⎢ D=⎢ ⎥ ⎥ 0.85 ⎦ ⎣ 50 ⎦ ⎣ 0 The number of dictionaries ordered in 2015 was given by

⎡500⎤ Spanish O1 = ⎢ ⎥ ! ⎣320⎦ French !!

c)

Use the new matrix formula to predict, correct to the nearest integer, the number of each language dictionary that the book supplier should order in the years 2016 – 2019 (inclusive).

(2 marks)

!

Page 11 of 13

QUESTION 6 – TABLE TENNIS COMPETITION STUDENT TABLE TENNIS COMPETITION Four language students Amy (A), Ben (B), Cameron (C) and Danielle (D) participate in a table tennis competition. Each student plays each other and there is a winner and loser from each match. The results from the competition are recorded in the dominance matrix below

LOSER

WINNER !!

A B C D

A ⎡0 ⎢ ⎢0 ⎢1 ⎢ ⎣0

B C D 1 0 1⎤ ⎥ 0 0 1⎥ 1 0 1⎥ ⎥ 0 0 0⎦

a) Draw a directed graph that displays the results of the competition.

(2 marks) b) Using the sum of one-step and two-step dominances, rank the players from first to last. Show your calculations (working out) below.

1…………………………………………………………. 2. ………………………………………………………… 3. ………………………………………………………… 4. ………………………………………………………… (3 marks) !

Page 12 of 13

TEACHER TABLE TENNIS COMPETITION Four teachers from the language department also participate in a table tennis competition. Each teacher plays each other once and there is a winner and loser from each match. The one step and two step dominances are listed in the table below. Teacher Aaron (A) Brad (B) Chris (C) David (D)

One-Step Dominances 3 2 1 0

Two-Step Dominances 2 1 0 0

c) Use the table to construct a one-step dominance matrix.

(2 marks) d) Using the sum of one-step and two-step dominances, rank the players from first to last. 1…………………………………………………………. 2. ………………………………………………………… 3. ………………………………………………………… 4. ………………………………………………………… (2 marks) e) Write down the results for the following matches below. Matches

Result Won…………………………..

Aaron versus Brad Lost……………………………

Won………………………….. Brad versus Chris Lost…………………………… (2 marks) !

Page 13 of 13...