Mathematics Guide Page 9

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1 Mathematics Guide Page 9 Part C Questions 15 to 5 4 marks each No marks are to be given if work is not shown. Eamples of correct solutions are given. However, other acceptable solutions are possible. 15 Eample of an appropriate solution /4 Interval for which the height of the helicopter is at least 5 m 4 t t t t t = 10 Answer: Note: The helicopter was at least 5 m above the ground for 10 seconds. Accept a graphical representation of the solution. Students who graphed the function and indicated the line y = 5 have shown they have a partial understanding of the problem. 16 Eample of an appropriate solution 4, 3, or 0 marks /4 (h, k) is midpoint of vertices (, 13) c = 13 b = 5 c 13 = a = a a = 1 + b y and 1 Answer: The equations of the asymptotes are = + ( ) y = ( ) Note: Accept any other equivalent response. Student who identified the values of b and c have shown a partial understanding of the problem.

2 Mathematics Guide Page Eample of an appropriate solution /4 ( cos ) 1 sin cos 3cos = 3 3cos = 3 3cos = 3 0 = cos 0 = 1 cos = or π = 3 or + 3cos + 1 ( cos + 1)( cos + 1) cos = 1 = π π Answer: The eact values are = or = π. 3 Students who were able to arrive at 0 = ( cos + 1) (cos + 1) have shown they have a partial understanding of the problem.

3 Mathematics Guide Page Eample of an appropriate solution /4 Constraints: Objective function: maimize profit y + y y y 0 y P = y 1500 Vertices P = y A (400, 800) $ B (500, 1000) A (400, 800) C (1000, 500) B (500, 1000) $ C (1000, 500) $ Answer: The maimum profit the company can make is $ Note: Deduct 1 mark if students use P = y Students who have the inequalities and the correct graph have shown they have a partial understanding of the problem. OR Student who have the inequalities and the correct vertices have shown they have a partial understanding of the problem.

4 Mathematics Guide Page 1 19 /4 Eample of an appropriate solution Solution = + = + = a a k h a y Solving for when y = = = = + = giving (4, ) Using the absolute value function = = + = = = = + = y a a a = or = 14 Answer: The plane spent 14 seconds in the air. Note: Students who found the point (4, ) have shown they have a partial understanding of the problem.

5 Mathematics Guide Page 13 0 Eample of an appropriate solution /4 Rule of distance of head to the ground Frequency = Period = 6 seconds/cycle Amplitude = 40 P = π b b y = a cos b 10 cycles 60 seconds ( h) ma + min k = = = 80 π = 6 π = 3 + k π y = 40 cos t Distance to ground at t = 5 π y = 40 cos 3 y = 60 cm ( 5) + 80 Answer: Five seconds after it is released, the head is at the height of 60 cm. Note: To account for rounding at different places, accept answers in the range of 59 to 61. Students who use an equivalent rule to arrive at the same answer should not be penalized. Students who use an appropriate method to determine any parameters have shown they have a partial understanding of the problem..

6 Mathematics Guide Page 14 1 Eample of an appropriate solution /4 Since the radius is 4 cm, diameter AB is 8 cm. Segment AC is 4 cm, then triangle ABC is a triangle. Therefore, m BAC = 60 and m ABC = 30. Since arcs are double their inscribed angles, m CDB = 10 and m AC = 60 Given m CD = 80 m DB = mcdb MCD m DB = m DB = 40 Since m DPB = m AC mbd m DPB = = 10 Accept any method to determine that measure of BAC is 60. Answer: The measure of DPB is 10. Students who use an appropriate method to correctly determine the measure of AC have shown they have a partial understanding of the problem.

7 Mathematics Guide Page 15 Eample of an appropriate solution /4 Length of CE 10 = 5 5 ( + m CE) 0 = 5 + m CE 15 = m CE Constant product theorem Length of CG m CG = 7.5 cm A chord perpendicular to a diameter is bisected. Length of DG Let m DG = m FG = 3 ( 7.5)( 7.5) = ( 3) 56.5 = = 4.3 = since the diameter is 4 Constant product theorem Answer: Note: To the nearest tenth of a centimetre, the length of DG is 4.3 cm. Do not penalize students who did not round, or rounded incorrectly. Students who have determined the length of CG have shown they have a partial understanding of the problem.

8 Mathematics Guide Page 16 3 Eample of an appropriate solution /4 Equation representing profit decrease for Company A Substituting (, 3.4) ( 0, 4) g ( ) = ac 4 = ac 4 = a 3.4 = 4c 0.81 = c g 0.9 = c 0 ( ) = 4( 0.9) Equation representing growth for Company B f() = ac + 15 Substituting (0, -10) 10 = ac 5 = a f ( ) 4.5 = 5c 10.5 = 5c 0.4 = c 0.9 c = 5c Point ( 10, 4.5) f() = -5(0.9) + 15 Solving for f(11) and g(11) f g ( 11) = 5( 0.9) = thousands or $5009 ( 11) = 4( 0.9) = 1.55 thousands or $155 Difference between the two companies $5009 $155 = $3754 Answer: Company B would make $3754 more than Company A. Accept answers in the interval [ 3754, 400] Accept also [ 3.7 thousands and 4.0 thousands] Note: Students who find the correct rule for either one of the two companies, have shown they have a partial understanding of the problem.

9 Mathematics Guide Page 17 4 Eample of an appropriate solution /4 To convert from general form to standard form: + y 30 + y ( 15) ( 15) 30 50y + 85 = 0 50y = ( y 5) + ( y 5) = 5 65 = 85! Formula for large circle C1: ( 15) + (y 5) = 5. Therefore the center is at (15, 5) and its radius is 5 m.! Vertical distance from center of C1 to center of C is 5 m 9 m = 16 m. Therefore the coordinates for the center of C are (15, 16) and its radius is 4 m.! Verte for the lower parabola is center of C Radius of C. Therefore coordinates for the verte (h, k) of the lower parabola are (15, 1)! Since center of C is also point on the directri for lower parabola, then the value of parameter c value is vertical distance from verte of lower parabola to center of C. Therefore c = -4 Substituting in ( h) = 4c(y k) we get: ( 15) = 16(y 1)! The y-coordinate of P is 0; the -coordinate must be found. Therefore, substitute y = 0 in the equation above: ( 15) = 16 ( 0 1) ( 15) = = ± 19 = 15 ± or 8.86 Answer: The distance from point 0 to point P is 1.14 m. Note: Students who use an appropriate method to determine the verte of the parabola, (15, 1) have shown they have a partial understanding of the problem. Do not penalize students who did not round or rounded incompletely.

10 Mathematics Guide Page 18 5 Eample of an appropriate solution /4 f f ( 3) = ( 3) = therefore a point (, y) on the ellipse is (3, 1) Verte on horizontal ais is at (4, 14) Center of the ellipse must be at (0, 14) = (h, k) a is 4 0 = 4 (-value of verte on horizontal ais h) Substituting a, h, k, and y we get: ( h) ( y k) ( 3 0) ( 1 14) 4 a + + b b b 4 b b b = 1 = 1 = 1 = = Answer: Note: The total height of the logo is = 17.0 cm, to the nearest hundredth centimetre. Do not penalize students who did not round, or rounded incorrectly. Students who have used an appropriate method to determine the equation of the ellipse have shown they have a partial understanding of the problem

11 MATHEMATICS Summative Eamination June 008 Question Booklet Secondary 5 Mathematics and Science & Technology Committee 3 hours

12 Mathematics Question Booklet Page 1 INSTRUCTIONS 1. Write the required information on the cover page of your Answer Booklet.. Answer all 5 questions in the Answer Booklet. 3. Each question is worth 4 marks. 4. You may use a calculator (with or without graphing display), and a memory aid. 5. The following materials are allowed: graph paper, ruler, compass, set square, and protractor. 6. The figures in this booklet have NOT been drawn to scale. 7. At the end of the eam period, hand in the Question Booklet and Answer Booklet. Time allotted 3 hours

13 Mathematics Question Booklet Page Part A Questions 1 to 8 In the Answer Booklet, blacken the letter that corresponds to the answer chosen. 1 Which of the following scatter plots shows the weakest correlation? A) B) C) D)

14 Mathematics Question Booklet Page 3 The manager of the Little Fry fast food restaurant noticed the following consistencies on any given day: At least 50 orders of French fries were sold. The restaurant sold no more than twice as many orders of large fries as small fries. Let l represent the number of orders of large fries s represent the number of orders of small fries Which of the following systems of inequalities represents this situation? A) s + l 50 B) s + l 50 l s l s l 0 l 0 s 0 s 0 C) s + l 50 D) l s l 0 s 0 s + l 50 l s l 0 s 0

15 Mathematics Question Booklet Page 4 3 The graph of a step function is shown below. y Which of the following rules defines this function? A) y & 1 # = 10 $! % 0 " B) y = 10[ 0] & 1 # & 1 # C) y = 10 $! D) y = 10 % 0 $! " % 0 "

16 Mathematics Question Booklet Page 5 4 Given f() = + 3 and g() = Which of the following represents (f ο g) ()? A) B) C) 1 1 D) Which epression below is equivalent to the following? 10 log a 3 log b + 1 log 9 & 15 a # A) $! % b " log B) log ( 3 a ) 10 b 3 10 log D) & 3 a! # log $ 3 % b " C) ( 135 ab) 6 An ellipse has two vertices at (4, -3) and (4, -9), and one of its foci at (0, -6). Which of the following is the equation of the ellipse? A) ( + 4) ( y 6) = 1 B) ( + 4) ( y 6) = 1 C) ( 4) ( y + 6) = 1 D) ( 4) ( y + 6) = 1

17 Mathematics Question Booklet Page 6 7 Consider vectors v and s below. ur v ur s What is the resultant vector of v s? A) B) C) D) 8 In the diagram on the right, m ACD = 48. m AED = 60 A 60 E B? D 48 C What is the measure of BC? A) 4 B) 60 C) 36 D) 96

18 Mathematics Question Booklet Page 7 Part B Questions 9 to 14 Write your answer in the space provided in the answer booklet. Show your work, where required. 9 Given the function ( ) = f. What is the rule for its inverse? 10 Solve the following logarithmic equation: ( + 3) = 3 log ( 4) log 11 Prove the following trigonometric identity. sin 1 cos + ( sin ) = 1 cos 1 Consider v = 8 and AB, whose vertices are A(-, 4) and B(8, 8). The scalar product of the two vectors is 104. What is the measure of the angle between the two vectors?

19 Mathematics Question Booklet Page 8 13 In the adjacent circle with centre O, CD is perpendicular to AB at point D m BD = 4 cm m OC = 13.5 cm A O 13.5 cm What is the area of triangle ABC? Round your answer to the nearest tenth. 4 cm D B C 14 Sally s Z-score on a math test was The marks of all the students in Sally s math class are listed below. The standard deviation of the class is Billy s mark in a different math class was 3 higher than Sally s. His class average was 78 and the standard deviation for his class was 5. What was Billy s Z-score?

20 Mathematics Question Booklet Page 9 Part C Questions 15 to 5! Show all your work as well as your answer. The work shown is taken into consideration when marks are awarded.! Your written information must be legible, complete, and clearly stated in correct language so the marker understands eactly what you have done. Even if your answer is correct, no marks will be given unless acceptable work is shown. 15 In designing software for a helicopter simulation game, a computer programmer uses the following rule as a model to determine the height of the helicopter at any given 4 time:" h ( t) = t where h is the height in metres and t is the elapsed time in seconds. For how many seconds was the helicopter at least 5 m above the ground? 16 On a Cartesian plane, the vertices of a hyperbola are located at (, 8) and (, 18). The foci of the hyperbola are located at (, 0) and (, 6). What equations can be used to represent the asymptotes of the hyperbola? 17 Solve the following trigonometric equation. Give the eact value(s). sin 3 cos = 3 [ 0, π] 18 A company produces different games. The two most popular are the Memory game and the Construction game. The number of games that the company stocks is based on past sales, which indicate that it sells at most twice as many Construction games as Memory games. The company cannot have more than 1500 of these games in stock. It costs $5 to produce the Memory game and $10 for the Construction game. The company epects to spend a minimum of $ to produce these games. The Memory game sells for $35 while the Construction game sells for $50. Let represent the number of Memory games in stock y represent the number of Construction games in stock What is the maimum profit the company can make selling these games?

21 Mathematics Question Booklet Page Anthony received a remote-controlled airplane for his birthday. The plane s altitude, as a function of time, is represented by a square root function followed by an absolute value function. The plane s altitude follows a square root function until it first reaches metres, at which point the altitude can be described by an absolute value function. Anthony begins by putting his plane into take-off position from an altitude of 6 metres. One second after take-off, the plane is 4 metres above the ground. The plane reaches its maimum altitude of 6 metres 8 seconds after take-off. y Altitude (m) 6 (1, 4) 8 Time (seconds) How much time did the plane spend in the air? 0 The diagram below depicts the head of a Jack-in-a-bo used in the display window of a department store. The head is connected to a motor, and its up-and-down movement follows a sinusoidal curve. The head is compressed to 40 cm at t = 0 and it reaches a maimum height of 10 cm. It bounces with a frequency of 10 cycles per minute. y 10 cm 40 cm At what height is the head, 5 seconds after it is released?

22 Mathematics Question Booklet Page 11 1 The circle below with center O has a radius of 4 cm. The measure of segment AC is 4 cm, and the measure of arc CD is 80. Segment CE is an altitude of triangle ABC whose side AB passes through the center of circle O. A E O C B 80 D P What is the measure of DPB? Show and justify all of your work.

23 Mathematics Question Booklet Page 1 In the adjacent circle with centre O, AB is a tangent to the circle at point A m AB = 10 cm m BC = 5 cm FD is a diameter CE FD m OG = m DG F A O G D C B E What is the length of DG? Round your answer to the nearest tenth of a centimetre. 3 Company A has seen a decrease in profit since its competitor, Company B, opened its doors. The decrease can be estimated using an eponential function in the form of g() = ac. The profit of Company B can be estimated according to an eponential function in the form of f() = ac y Profit (thousands of $) Comparison of Profit Company B 4-10 (, 3.4) (10, 4.5) Company A (Years since Company B opened its doors) Based on these estimates, how much more profit would Company B make than Company A, 11 years after it opened its doors? Round your answer to the nearest dollar.

24 Mathematics Question Booklet Page 13 4 Courtney has been hired to paint lines on a field. The lines consist of a large circle C1, whose centre coincides with the centre of the rectangular field, two small (congruent) circles, and two (congruent) parabolas. The rectangular field, along with the lines Courtney must paint, are shown on the Cartesian plane below, which is scaled in metres. The equation of the large circle (C1) drawn on the field is: In addition, + y 30 50y + 85 = 0 " Circles C1 and C are tangent to one another. Their centres are vertically aligned 9 metres apart. " The lower parabola is tangent to C at its verte, which is directly below the center of C. " The center of C is a point on the directri of the lower parabola. Courtney must begin the paint job at point P and she needs to know how far away from point 0 she should start. y C1 C 0? P What is the distance from point 0 to point P? (Round your answer to the nearest hundredth metre.)

25 Mathematics Question Booklet Page 14 5 Lickety Splitz Cotton Candy Parlour is designing a logo for its letterhead. (3,y) The sides of the paper cone are defined by the function: ( ) = f 4 The cotton candy on top of the cone is in the shape of an ellipse whose centre is directly above the verte of the cone. One verte on the horizontal ais of the ellipse is at (4, 14). The ellipse intercepts the cone at = 3. (Note that all units are in centimetres). What is the total height of the logo? Round your answer to the nearest hundredth of a centimetre.

26 Name: Class: Teacher s Name: MATHEMATICS FOR TEACHER USE ONLY Part A /3 Summative Eamination June 008 Part B /4 Part C /44 Total /100 Answer Booklet Secondary 5 Mathematics and Science & Technology Committee

27 Mathematics Answer Booklet Page 1 Part A Questions 1 to 8 Blacken the letter that corresponds to the answer chosen. Each question is worth 4 marks [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] [A] [B] [C] [D] Part B Questions 9 to 15 Write your answer in the space provided. 9 The rule for the inverse of the function is Answer : 4 0

28 Mathematics Answer Booklet Page Prove the following trigonometric identity. Show all your work. sin 1 cos + ( sin ) = 1 cos

29 Mathematics Answer Booklet Page Show all your work. Answer: The measure of the angle between the two vectors is.

30 Mathematics Answer Booklet Page Show all your work. A O 13.5 cm 4 cm D B C Answer: The area of triangle ABC is cm.

31 Mathematics Answer Booklet Page Show all your work. Answer: Billy s Z-score was.

32 Mathematics Answer Booklet Page 6 Part C Questions 15 to 5! Show all your work as well as your answer. The work shown is taken into consideration when marks are awarded.! Your written information must be legible, complete, and clearly stated in correct language so the marker understands eactly what you have done. Even if your answer is correct, no marks will be given unless acceptable work is shown. Note: Diagrams have not necessarily been drawn to scale.

33 Mathematics Answer Booklet Page Show all your work. Answer: The helicopter was at least 5 m above the ground for seconds.

34 Mathematics Answer Booklet Page Show all your work. Answer: The equations of the asymptotes are: and.

35 Mathematics Answer Booklet Page Show all your work. sin 3 cos = 3 [ 0, π] Answer: The eact value(s) is (are).

36 Mathematics Answer Booklet Page Show all your work Answer: The maimum profit the company can make is $.

37 Mathematics Answer Booklet Page Show all your work. y Altitude (m) 6 (1, 4) 8 Time (seconds) Answer: The plane spent seconds in the air.

38 Mathematics Answer Booklet Page Show all your work. y 10 cm 40 cm Answer: Five seconds after it is released, the head is at the height of cm.

39 Mathematics Answer Booklet Page Show all your work. A E O C B 80 D P Answer: The measure of DPB is.

40 Mathematics Answer Booklet Page Show all your work. A B F O C G D E Answer: To the nearest tenth of a centimetre, the length of DG is cm.

41 Mathematics Answer Booklet Page Show all your work. y Profit (thousands of $) Comparison of Profit Company B 4-10 (, 3.4) (10, 4.5) Company A (Years since Company B opened its doors) Answer: Company B would make $ more than Company A.

42 Mathematics Answer Booklet Page Show all your work. y C1 C 0? P Answer: The distance from point 0 to point P is m.

43 Mathematics Answer Booklet Page Show all your work. (3,y) Answer: The total height of the logo is cm, to the nearest hundredth centimetre.

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