GEOMETRY AND TRIGONOMETRY

Transcription

1 Name: Teacher: GEOMETRY AND TRIGONOMETRY Duration: 150 minutes Timeline for classes: Part A: 50 minutes for 20 multiple choice questions (answers to be submitted in the answer sheet). Each question is worth 1 mark. Part B: 100 minutes to answer three response analysis questions (total of 37 marks) Question 1: 11 marks Question 2: 9 marks Question 3: 17 marks Materials allowed: 1 x Bound reference; CAS calculator; ruler; pens/pencils. Note that the bound reference will be collected after the completion of Part A and will be returned to students upon commencing Part B 1

2 Outcomes: Mark allocation noted in brackets (total marks: 20) FURTHER MATHEMATICS UNIT 4 SAC 4 Outcome 1 (15 marks) Define and explain key terms and concepts as specified in the content from the areas of study, and use this knowledge to apply related mathematical procedures to solve routine application problems. Outcome 3 (5 marks) Select and appropriately use technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches in the area of study Data analysis and the selected module from the Applications area of study. Students need to demonstrate: Comprehensive and correct use of mathematical conventions, symbols and terminology in all formulations, presentations, manipulations, computations and descriptions; Thorough and relevant definition and explanation of key concepts with comprehensive identification of conditions or restrictions that apply in different contexts; Consistent application of accurate mathematical skills and techniques to obtain correct results; Critical and appropriate selection of technology for efficient and systematic production of solutions and presentations for given contexts; Skilled use of technology to enable thorough analysis and interpretation of results in tabular, graphical and numerical forms. Completion of SAC The total marks allocated is 57. Each question relates to a specific outcome and therefore your final marks relate to the two outlined above. Marks will be adjusted so that they reflect the marks allocated for each outcome and therefore students final mark will be out of 20. The final mark will be given to students once calculated based on the outcomes and after comparing the results of all three classes. After SACs have been marked, teachers will decide if a student has demonstrated enough knowledge throughout their SAC to be given a satisfactory. If a student can t demonstrate through her SAC that she understands the content covered then the student must demonstrate her knowledge by either: 1. Showing the teacher their up-to-date workbook; or, 2. Completing a short test. At the end of part students must sign an agreement form stating that they have submitted their SAC and that all work is their own. 2

3 PART A: MULTIPLE CHOICE RESPONSE SHEET Name: Date: Teacher: Please circle your response. Please note: questions that have more than one response given or it is unclear as to which response has been chosen will not receive any marks. Question 1 A B C D E 2 A B C D E 3 A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E 10 A B C D E 11 A B C D E 12 A B C D E 13 A B C D E 14 A B C D E 15 A B C D E 16 A B C D E 17 A B C D E 18 A B C D E 19 A B C D E 20 A B C D E 3

4 Name: Teacher: Part A: Multiple Choice Questions Select one correct answer for each of the following questions in the answer sheet provided 1) In Δ PQR, the size of PRQ is closest to: A. 28 B. 34 C. 37 D. 42 E. 48 2) A child s slide is 3.2 m high and 5.6 m long. The angle that it makes with the ground to the nearest degree is: A. 35º B. 32º C. 30º D. 31º E. 17º 3) The length MN is closest to: A. 136 m B. 13 m C. 124 m D. 10 m E. 90 m 4

5 4) The figure shown has a semicircular end attached to a rectangle, one side of which has length 6 cm, from which a smaller rectangle 2 cm by 3 cm has been removed. The perimeter of the figure, correct to the nearest cm, is: A. 34 B. 35 C. 36 D. 41 E. 45 5) A square of length 8 cm is placed inside a square of length x cm as shown in the diagram below. The value of x is closest to A. 8 cm B cm C. 16 cm D cm E cm 6) In the diagram to the right, AB is parallel to CD. In A. 40 B. 50 C. 60 D. 80 E. 100 Δ ABC, BAC is equal to: 5

6 7) In the figure below, the value of x can be found by using the formula: A.! =!"!"#!"!"#!!" B. C. D. E.! =!"!"#!"!"#!"! =!"!"#!"!"#!"! =!"!"#!"!"#!"! =!"!"#!"!"#!!" 8) In triangle MNP the value of x is: A. 20 B. 30 C. 35 D. 45 E. 60 6

7 9) The ratio of the side lengths of a small cube to a large cube is 1:50. The ratio of volume of the large cube to the small cube is: A. 1: 2500 B. 2500:1 C. 250:1 D : 1 E. 1: The following information relates to questions10-12 Part of a contour map is shown. The scale on the map is 1: ) The vertical distance between points A and C is: A. 200 m B. 150 m C. 50 m D. 100 m E. 20 m 11) The actual distance between D and F is closest to: A m B. 100 m C. 412 m D. 500 m E. 400 m 7

8 12) The average slope between points D and F is closest to: A B. 4 C D. 1 E ) Daniel is about to kick the winning goal for his soccer team. He is 8 m from one goal post and 6 m from the other. The angle through which he can shoot is 18º. The distance between the goal posts is closest to: A. 14 m B. 4 m C. 10 m D. 3 m E. 2 m 2 14) Δ ABD is similar to Δ CDE. The ratio of CE to AB is 4:9. The area of Δ CDE is 28 cm. The area 2 of Δ ABD, in cm, is A. 36 B. 63 C D E

9 The following information relates to questions A bushwalker walks on a bearing of 030º from point A to point B, which is 3 km away. She then sets off on a bearing of 120º towards point C, which is 5 km away. 15) From the information given, the angle of B in ABC is: A. 30º B. 60º C. 90º D. 120º E. 240º 16) The distance between points A and C is: A. 5.8 km B. 4.3 km C. 8.0 km D. 2.5 km E. 4.0 km 17) The bearing of B from C is: A. 060º T B. 330º T C. 300º T D. 240º T E. 030º T 9

10 The following information relates to questions The square- based pyramid has a base length of 10 cm and the length of the sloping edge is 8 cm (ie. PB = 8 cm) 18) The height of the pyramid, length PO, is closest to: A. 14 cm B. 3.7 cm C. 10 cm D. 13 cm E. 141 cm 19) The angle that a sloping edge makes with the horizontal (ie. OBP) is closest to: A. 62º B. 28º C. 32º D. 58º E. 45º 20) If the volume of the pyramid is 125 cm 3 and a larger scale model of the pyramid is made, what would be the volume of the larger pyramid if its base length was to be 80 cm? A cm 3 B cm 3 C cm 3 D cm 3 E. 6.4 m 3 10

11 Name: Teacher: All answers correct to two decimal places Part B: Response Analysis Questions Question 1: (11 marks) In the diagram above, ON is due north and X and Y are two points on the verandah of a large resort hotel. The distance from X to Y is 120 m. When I look at a tree, T, from position X, the angle of elevation is 15º When I look at the same tree from position Y, the angle of elevation is 36º. a. Find XTY (1 mark) b. What is the true bearing of the tree from X? (1 mark) c. What is the true bearing of the tree from Y? (1 mark) 11

12 d. Find the distance from X to the tree, T. (2 marks) FURTHER MATHEMATICS UNIT 4 SAC 4 e. I wish to make a path from the verandah to the tree. What is the length of the shortest path, TW, that can be made, where W is a point on the verandah? (2 marks) f. How far is W from Y? (2 marks) g. The resort owner wishes to have the area around the path leading from the verandah to the tree and inside the triangle XTY, sown with grass seed as shown below. The area of the path around the verandah is m 2. Calculate the area to be sown with grass seed. (2 marks) 12

13 Question 2: (9 marks) FURTHER MATHEMATICS UNIT 4 SAC 4 The owners of a park want to build a pavilion. The roof of the pavilion is in the shape of a rectangular pyramid with a slope length of 6.3 m. The design of the pavilion is shown below. Note the diagram is not to scale. a. Determine the length EB. (2 marks) b. Given that the length of the line EF is 2.39 m, how tall is the pavilion from its base to the top of roof (F)? (1 mark) 13

14 c. Given that the total surface area of the entire pavilion above is m 2, what is the total surface area of the roof alone? (2 marks) d. Find the volume of the whole pavilion? (2 marks) e. If the owners of the park decided the pavilion didn t need to be so large and reduced the dimensions by half (i.e. the base of the pavilion is now 5 m by 3 m and the slope of the roof is 3.15 m), what percentage reduction of volume has been achieved? (2 marks) 14

15 Question 3: (17 marks) FURTHER MATHEMATICS UNIT 4 SAC 4 The laws of geometry are evident in the life of bees. The following diagram show a cross-sectional view of a beehive with its regular hexagonal compartments a. Determine the value of an interior angle of each hexagonal compartment, as shown in the diagram as AOB. (1 mark) b. Show that the triangle AOB is an equilateral triangle. (1 mark) c. Determine the area of each hexagonal compartment. (2 mark) 15

16 d. Determine the volume of honey that would be stored in each compartment, given that each compartment is 12 mm deep. (2 mark) The beekeeper decides to trial a synthetic artificial beehive where length of each side of the hexagonal compartment is 30 mm. e. What is the scale factor for the new hive compared to the original? (1 mark) f. What is the area of each hexagonal compartment in the new hive? (2 marks) g. What is the volume of each hexagonal compartment in the new hive? (2 marks) 16

17 The beekeeper wishes the bees to collect the nectar from two prime locations a field of lavender and group of wattle trees. A map of their respective locations is shown. h. Find the distance the bees have to travel to the lavender field from the hive box. (2 marks) i. What is the bearing of the lavender field from the hive box? (2 marks) j. How far north (to the nearest metre) must the beekeeper move the hive box so that it is directly west from the wattle trees? (2 marks) 17