مملكة البحرين Kingdom of Bahrain

مملكة البحرين KINGDOM OF BAHRAIN

هيئة جودة التعليم والتدريب EDUCATION & TRAINING QUALITY AUTHORITY

إدارة االمتحانات الوطنية Directorate of National Examinations االمتحانات الوطنية للصف الثاني عشر Grade 12 National Examinations

امتحان مارس March 2019 2019 حل المشكالت PROBLEM SOLVING الورقة 2 تحليل وحل المشكالت Paper 2 Problem Analysis and Solution مدة االمتحان: ساعتان Duration: 2 hours اكتب اإلجابة في كراسة اإلجابة. .Write your answer on the answer booklet

األدوات اإلضافية: مسطرة، قلم رصاص، ممحاة، ,Additional materials: ruler, pencil, eraser calculator, answer booklet. آلة حاسبة، كراسة اإلجابة.

Read these instructions first:

Use a blue pen only.

Follow the instructions on the answer booklet provided.

Answer all questions.

The use of calculators is permitted.

The total number of marks for this paper is 50.

This document consists of 14 printed pages and 2 blank pages.

1 Study the information below and answer the questions that follow. You should show your working.

Mohannad owns a goldsmith’s workshop in the Kingdom of Bahrain, and sells his jewelry via his website. Once the customer selects a piece of jewelry and pays its price electronically, the piece is delivered to him free of charge within the Kingdom.

Gold comes in different karats; each referred to by a whole number smaller than or equal to 24, indicating the percentage of pure gold present. The most famous karats are: 24 k, 21 k and 18 k. 24 k is 100% pure gold, whereas the remaining karats are mixed with other metals such as copper, zinc and silver in certain percentages determined by the karat. For example: the percentage of pure gold in 18 k gold is 18 out of 24, which equals 75% (18 ÷ 24 × 100%).

Jewelry stores add an additional cost to customers per gram of gold known as “manufacturing charge”. This is an amount determined by the labour required to manufacture the piece of jewelry.

Mohannad has three types of jewelry available at his workshop; they are known as simple, elegant and heavy. The table below shows the options he offers his customers for a piece of jewelry in terms of karat, weight and manufacturing charge.

Weight of the piece of Manufacturing charge per jewelry (gram) gram (BD) Karat Simple Elegant Heavy Simple Elegant Heavy 18 8 – 9 10 – 11 13 – 14 4 6 8 21 7 – 9 10 – 12 13 – 15 6 8 11

For example: all 21 k heavy pieces of jewelry in Mohannad’s workshop weigh 13 to 15 grams and have a manufacturing charge of BD 11 per gram.

Gold prices change from day to day and there is no fixed price for a gram of gold of any karat. The price of a gram of gold for a given karat is calculated using the following formula: Price of a gram of gold for a given karat = the price of a gram of 24 k gold × the percentage of pure gold in the relevant karat

The price of a piece of jewelry, on the other hand, is calculated using the following formula: Price of a piece of jewelry = (the price of a gram of gold for the relevant karat + manufacturing charge per gram) × the weight of the piece in grams

(a) What is the percentage of other metals in a gram of 21 k gold? [1]

(b) A female customer bought a piece of 21 k gold jewelry weighing 7.2 gram from Mohannad’s website. The price per gram of 24 k gold on the day of purchase was BD 14.400. How much did this customer pay Mohannad? [2]

(c) Mohannad repurchases his gold jewelry pieces if the customers wish to sell them, and sends a representative to the relevant customers to collect the piece of jewelry. The repurchase price is determined by the weight of the piece only, without the manufacturing charge, and based on the price per gram of the relevant karat on the day of repurchase.

A customer bought a piece of jewelry from Mohannad’s website. The price per gram of 24 k gold on the day of purchase was BD 14.200, and the customer sold the piece back to Mohannad after one month when the price per gram of 24 k gold was BD 13.400. What is the highest loss the customer may have incurred in the process of purchasing and selling the piece? [2]

(d) During the last year, the price per gram of 24 k gold varied between BD 13.200 and BD 19.400.

(d1) A customer sent Mohannad an e-mail complaining about non- delivery of his piece of jewelry despite receiving a confirmation of the payment via the website during last year. The customer told Mohannad that his piece of jewelry weighs 10.4 gram, and that he paid an exact amount of BD 262. Mohannad discovered that the customer was planning a fraud. Explain how Mohannad was able to discover that. [2]

(d2) During the last year, the friends Hamad and Hassan bought a number of pieces of gold jewelry from Mohannad’s website. Hamad bought a number of identical pieces of 21 k gold, and Hassan bought a number of identical pieces of 18 k gold on the same day, which differ than Hamas’s pieces of jewelry in terms of type and manufacturing charge. The total weights of their pieces of jewelry were equal, and they paid the same amount of money.

What was the price per gram of 24 k gold on the day on which the two friends bought their jewelries? Explain your answer. [2]

(e) The chairwoman of a women’s association wanted to buy pieces of gold jewelry to honour the members of the association on the Bahraini Women’s Day. She decided that each head of committee receives an identical piece of jewelry at 11.5 gram of 21 K gold and that each member of a committee receives an identical piece of jewelry at 8.5 gram of 18 K gold. The association has more than one committee and all committees have the same number of members.

The chairwoman bought the pieces of gold jewelry from Mohannad’s website and paid BD 1530.500. The price per gram of 24 k gold on the day of purchase was BD 12. How many pieces of jewelry did the chairwoman buy? [3]

(f) Al-Saeed Jewelry Store requests some pieces of gold jewelry from Mohannad’s website every year. The total weight of the pieces of jewelry requested is 450 gram divided equally between 18 k and 21 k, then equally divided between the three types of simple, elegant and heavy jewelry pieces of each karat.

The store owner sent Mohannad an e-mail expressing his desire to obtain the same total weight in the upcoming year under the same terms; with the addition of another karat value for the same three types of simple, elegant and heavy jewelry pieces.

The store owner realizes that Mohannad would only accept to manufacture gold jewelries of a new karat value if: • The percentage of pure gold is higher than 62.5%. • The manufacturing charge per gram for the new karat value of all three types is higher than that of 18 k gold and lower than that of 21 k gold, and is in whole dinars.

Mohannad approved the store owner’s request and stated that the initial total cost will be BD 8500 based on the current price per gram of 24 k gold which is BD 15.

Determine the new karat value requested by the owner of Al-Saeed Jewelry Store, specifying the manufacturing charge per gram of all three types of jewelry pieces: simple, elegant and heavy, for this new karat value. [3]

BLANK PAGE

2 Study the information below and answer the questions that follow. You should show your working.

Bahrain Bus Company manages the transportation through all main locations in Bahrain. The schedule below gives the daily destination routes of the buses, starting from the first station to the last station on each route. In addition, it gives the first three and the last three departure times from the first station on the route according to the day of the week. The remaining departure times from the first station are uniformly scheduled at a fixed interval of 15, 20, 30, or 60 minutes depending on the route. As an example, the buses of the A1 Route are scheduled to depart regularly from Isa Town every 15 minutes from 05:00 onwards, so on a daily basis, the 4th bus leaves Isa Town at 05:45, the 5th bus at 06:00 and so on.

“A” Route A1 Route: Isa Town  Salmaniya  Manama  Muharraq  Airport 1st 2nd 3rd Last Bus Bus Bus Bus every Daily 05:00 05:15 05:30 22:00 22:15 22:30 15 minutes

A2 Route: Airport  Muharraq  Manama  Salmaniya  Isa Town 1st 2nd 3rd Last Bus Bus Bus Bus every Daily 05:00 05:15 05:30 22:00 22:15 22:30 15 minutes

“U1” Route (Loop) Isa Town  Sanad  Alba  Riffa  University  Isa Town 1st 2nd 3rd Last Bus Bus Bus Bus Sun. every to 05:00 05:30 06:00 21:30 22:00 22:30 30 minutes Thu. Fri. every and 05:00 06:00 07:00 20:00 21:00 22:00 60 minutes Sat.

“X2” Route X2A Route: Manama  City Centre  Budaiya 1st 2nd 3rd Last Bus Bus Bus Bus every Daily 05:10 05:30 05:50 21:50 22:10 22:30 20 minutes

X2B Route: Budaiya  City Centre  Manama 1st 2nd 3rd Last Bus Bus Bus Bus every Daily 04:55 05:15 05:35 21:35 21:55 22:15 20 minutes

“X3” Route X3A Route: Budaiya  Jidhafs  Central Market  Manama 1st 2nd 3rd Last Bus Bus Bus Bus every Daily 05:05 05:25 05:45 21:45 22:05 22:25 20 minutes

X3B Route: Manama  Central Market  Jidhafs  Budaiya 1st 2nd 3rd Last Bus Bus Bus Bus every Daily 05:00 05:20 05:40 21:40 22:00 22:20 20 minutes

“M2” Route (Loop) Muharraq  Samaheej  Hidd  Arad  Airport  Muharraq 1st 2nd 3rd Last Bus Bus Bus Bus Sun. every to 04:50 05:20 05:50 21:20 21:50 22:20 30 minutes Thu. Fri. every and 04:50 05:10 05:30 21:50 22:10 22:30 20 minutes Sat.

In addition, the duration it takes the bus to travel between the stations

on these routes are shown below.

From To Duration (minutes) Airport Muharraq 10 Alba Riffa 18 Arad Airport 10 Budaiya City Centre 15 Budaiya Jidhafs 22 Central Market Jidhafs 19 Central Market Manama 15 City Centre Budaiya 20 City Centre Manama 17 Hidd Arad 10 Isa Town Salmaniya 22 Isa Town Sanad 12 Jidhafs Central Market 20 Manama Central Market 14 Manama City Centre 10 Manama Muharraq 15 Muharraq Airport 14 Muharraq Manama 17 Muharraq Samaheej 24 Riffa University 13 Salmaniya Manama 16 Samaheej Hidd 15 Sanad Alba 10 University Isa Town 20

The Bus Company issues three types of tickets according to the number of stations the passenger would like to travel. The passengers buy the tickets when they get into the bus from the bus driver assistant. Each type of ticket has a validity duration. Once the ticket is issued, the passenger can leave the bus and then within the validity period of time he can get back on the bus and continue travelling on the same route without any further payment.

The following table shows for each type of ticket, the number of stations

allowed to travel, their validity duration and their price:

Ticket Number of Validity duration Price type stations (minutes) (fils) Yellow 1 --- 300 Green 2 45 400 Red 3 or more 60 500

For example, a passenger can buy a green ticket for 400 fils to travel from Salmaniya to Muharraq on the A1 Route. His ticket remains valid for 45 minutes. He can leave the bus in Manama after 16 minutes since his ticket was issued if he wishes and then he can get back on the same route within the remaining 29 minutes to get to Muharraq, even if the time to the next destination takes longer than the remaining time of the 29 minutes.

(a) How many journeys are taken by the U1 Route buses from Isa Town to Sanad every Friday? [1]

(b) At what time does the 7th bus of the X3A Route leave Budaiya? [1]

(c) A passenger takes the 3rd bus of the M2 Route on Friday to travel from Samaheej to Hidd. At what time does he arrive at Hidd? [2]

(d) A student studying at University of Bahrain lives in Riffa. He had an exam at 14:30 on Saturday. What was the last bus he could have taken to reach the university in time for the start of his exam? [2]

(e) Nabeel lives in Isa Town and has decided to go to Jidhafs next Saturday. He has planned to be at the Isa Town bus station at 08:55 and to stop in Salmaniya for 25 minutes.

What is the least cost Nabeel can pay for his trip from Isa Town to Jidhafs? Explain your answer. [3]

(f) Rashid lives in Budaiya and he works in pharmaceutical marketing. He distributes medicines to pharmacies in particular areas. Rashid always takes the first bus after his arrival at any station. The table below shows the pharmacies that he will go to -not necessary in order-, the locations of them and the time it will take him to do his work and return to the bus station at that location.

Pharmacy Location Duration (minutes) Health Care Manama 40 Treatments City Centre 35 Fast Recovery Jidhafs 25 Ibn Seena Central Market 50

Last Tuesday, Rashid arrived at Budaiya bus station at 10:00 and started his journey by using X2B Route first and then X3B Route to get him back to Budaiya. However, he has discovered that if he used, X3A Route first and then X2A Route, he will finish his work and return to Budaiya at 14:40 which is 20 minutes earlier.

How long does it take the bus to travel from Jidhafs to Budaiya? [3]

(g) Ramadan is a pipefitter who lives in Isa Town and uses the bus for his transportation. He always plans his work ahead to ensure the lowest possible cost of his bus travel.

Next Monday, Ramadan has some work to do in two different locations on the U1 Route, away from Isa Town, each of which will take at most 20 minutes. He plans to take the 7th bus that leaves Isa Town at 08:00 and he has found that the least total cost for the bus travels to finish his work and return to Isa Town is 900 fils.

List all the possibilities of the pairs of locations in which Ramadan has work to do next Monday. [3]

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3 Study the information below and answer the questions that follow. You should show your working.

Naval Wars is a game played by one player against the computer. It consists of 10 rounds. In each round of the game, there are two roles, defender and attacker. The player and the computer have to exchange their roles during each round.

The defender hides his submarines S and G, by placing them either horizontally or vertically on a 6 × 6 level grid. A location on the grid is identified by its column letter and its row number. The defender must use 7 different locations on the grid for the two submarines, which can be adjacent.

For example, the symbol of the location marked with “X” on the grid in the figure below is B4. The position of a submarine is a sequence of locations it is occupying on the grid. For example, the position of the Submarine S on the grid below could be described as either D1-E1-F1 or F1-E1-D1.

Submarine S 6 5 Submarine G 4 X 3 2 1

A B C D E F

The attacker must locate, hit and sink the hidden submarines. Upon hitting a location that is occupied by a submarine, it is shaded grey. The submarine sinks when all the locations that it occupies have been hit.

In each round, there are 36 missiles available for each player. Each missile must target a different location in the opponent’s grid. The attacker continues firing missiles until both submarines have been sunk and the round is over.

At the end of the round, points are awarded according to the following

rules: • 3 points are given to the round winner if he fires 3 or more missiles less than his enemy. • 2 points are given to the round winner if he fires 2 missiles less than his enemy. • 1 point is given to the round winner if he fires 1 missile less than his enemy. • No points are given when the number of missiles fired in the round by the player and the computer are equal. • No points are given if both the player and the computer each used more than 30 missiles.

The player that gains the most points at the end of 10 rounds is the winner. In the case of a tie, the computer is the winner.

(a) The scores of the first 5 rounds between Husam and the computer are listed in the following table:

Number of missiles fired Round number Husam Computer 1 24 26 2 28 31 3 19 30 4 35 31 5 22 17

(a1) How many points had Husam scored in total and how many had the computer scored in total from the 1st round till the end of the 5th round? [2]

(a2) At the end of the 6th round, Husam had scored 11 points and the computer had scored 3 points. What was the minimum number of points that Husam needed to score in the 7th round to guarantee his win at the end of the game whatever the outcome of the last three rounds? [2]

(b) In one round, Submarine S is shown as in the figure below because Husam has already sunk it. The locations containing an X are those that Husam has fired a missile at, but did not hit anything.

6 5 X 4 X X 3 X X 2 X 1 X X A B C D E F

(b1) Husam fired his next missile knowing it would definitely hit Submarine G. At which location did he fire the missile? [2]

(b2) The computer fired 17 missiles, but Husam won the round. What was the highest score he could have got in this round? Justify your answer. [2]

(c) In the next round, Husam fired missiles at 19 locations marked with ‘X’ on the grid shown below. The first 18 missiles did not hit any submarine; however, the 19th missile hit Submarine G.

6 X X X X 5 X X 4 X X X X 3 X X X 2 X X 1 X X X X A B C D E F

What is the only location that the 19th missile hit the Submarine G? [2]

(d) The computer is programmed so that in the current round it will fire a missile at row 1, then a missile at row 2 and so on up to row 6, then will start again at row 1.

Husam has placed his submarines as shown in the figure below.

6 5 4 3 2 1 A B C D E F

What is the least number of missiles the computer has to fire in order to ensure hitting one of Husam's submarines? Justify your answer. [2]

(e) Husam began a new round. He wanted to hide the Submarine S first. In how many different ways could he place it on the grid? [2]

(f) Husam decided to start the next round by placing Submarine G on the grid before Submarine S.

(f1) In which columns and rows could he place Submarine G to have the maximum possible number of ways to place Submarine S? Give all the possibilities. [2]

(f2) What is the difference between the maximum number and the minimum number of choices there could be for placing Submarine S depending upon where Submarine G has been placed? [4]

PS12/02E

امتحان مارس March 2019 2019 حل المشكالت PROBLEM SOLVING الورقة 2 تحليل وحل المشكالت Paper 2 Problem Analysis and Solution

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