Principles of Mathematics 12 June 2004 Provincial Examination

Transcription

1 Principles of Mathematics June 004 Provincial Examination ANSWER KEY / SCORING GUIDE CURRICULUM: Organizers Sub-Organizers. Problem Solving A Problem Solving and Cross Topic Problems. Patterns and Relations B C/D C/D 3. Shape and Space E F 4. Statistics and Probability G G G Geometric Sequences and Series Logarithms and Exponents Trigonometry Conics Transformations Combinatorics Probability Statistics Part A: Multiple Choice Q K C S CO PLO Q K C S CO PLO. A U.5 C4. D H.5 D. C U.5 C3. A K.5 3 E 3. B U.5 C5 3. A U.5 3 E3 4. C K.5 D6, F4 4. C H.5 3 E 5. D U.5 C7 5. A K.5 3 F 6. D U.5 C6 6. A U.5 3 F3 7. B U.5 D5 7. D U.5 3 F6 8. A U.5 C8 8. C H.5 3 F6 9. B H.5 D6 9. D K.5 4 G6 0. A U.5 B 30. A U.5 4 G8. D U.5 B 3. C H.5 4 G4. B K.5 B3 3. C U.5 4 G7 3. D U.5 B 33. D H.5 4 G7 4. D H.5 B3 34. B U.5 4 G 5. B K.5 D3 35. B U.5 4 G3 6. C K.5 C 36. B U.5 4 G8 7. D U.5 C 37. B H.5 4 G 8. C U.5 C 38. A U.5 4 G 9. C U.5 C 39. C U.5 4 G 0. A H.5 D 40. B H.5 4 G Multiple Choice = 60 s 046pmak - - July, 004

2 Part B: Written Response Q B C S CO PLO a. U 3 F6 b. U 3 3 F4. 3 U 4 3 E 3. 4 U 5 D, D U 4 4 G 5a. 6 U 4 G8, G 5b. 7 U 4 G8, G U 4 D H 4 C8 Written Response = 30 s Multiple Choice = 60 (40 questions) Written Response = 30 (7 questions) EXAMINATION TOTAL = 90 s LEGEND: Q = Question Number K = Keyed Response C = Cognitive Level B = Score Box Number S = Score CO = Curriculum Organizer PLO = Prescribed Learning Outcome 046pmak - - July, 004

3 . The graph of y = f ( x) is shown below. y x 5 046pmak July, 004

4 a) On the grid provided, sketch the graph of y = f (-x) - 3. ( s) y x reflection over y-axis translation down 3 5 Note: Deduct if graph does not stop on the right hand side. 046pmak July, 004

5 b) On the grid provided, sketch the graph of y =. (3 s) f ( x ) y x for vertical asymptote or asymptotic behaviour x = 4 ( ) for horizontal asymptote on right or asymptotic behaviour y = 0 ( ) for invariant point ( 5, - ) for invariant point ( 3, ) for horizontal portion - 3 x - for shape from - x 3 Note: Deduct ( ) if graph does not stop at - 3,. Note: Arrowheads not necessary for full s. 046pmak July, 004

6 . Determine the equation in standard form of the parabola with vertex ( 5, -), passing through the point (, 0), and having a horizontal axis of symmetry. (4 s) Grid is provided for rough work only. y x 5 046pmak July, 004

7 ( ) + x = a y -k h ( ) + x = a y + 5 s ( ) + = a a = - s 4 x = - 3 ( y + ) or x - 5 = - 3 y + 4 or - 4 ( x -5) = y + 3 ( ) ( ) 046pmak July, 004

8 3. The population of a nest of ants can multiply threefold (triple) in 8 weeks. If the population is now 000, how many weeks will it take for the population to reach ants? (Solve algebraically using logarithms. Answer accurate to at least decimal places.) (5 s) = ( ) t 8 t 8 5 = 3 t mk mk mk 8 log5 = log3 t log5 = log3 8 8log5 t = log3 t = weeks s were given for showing how to solve an exponential equation using logs correctly. t 8 = ( ) log 3 t 8 5 = 3 t 5 = s 8 8log 5 = t 3 log5 8 = t log 3 mk mk mk = t 046pmak July, 004

9 4. In the Canadian Junior Hockey League, 60% of the players are from Eastern Canada and 40% are from Western Canada. From this league, 8% of the Eastern players and % of the Western players go on to play in the NHL. If a randomly chosen NHL player who came from the Canadian Junior Hockey League is selected, what is the probability that he is from Western Canada? (4 s) P( W NHL) = ( ) P( NHL) P W and NHL E NHL W NHL = ( 0.4) ( 0.) ( 0.4) ( 0.) + ( 0.6) ( 0.8) = = 0.3 Note: Deduct for rounding error. 046pmak July, 004

10 Ø Ø Ø 5. A multiple-choice test has 48 questions. Each question has four choices, only one of which is correct. If a student answers all the questions by randomly guessing, determine the probability that the student will correctly answer between 0 and 3 questions inclusive by using the following methods. a) Use the binomial distribution to obtain this probability. (Answer accurate to at least 4 decimal places.) ( s) binomcdf ( 48, 0.5, 3) - binomcdf ( 48, 0.5, 9) = binompdf ( 48, 0.5, 0) + binompdf ( 48, 0.5, ) + binompdf ( 48, 0.5, ) terms ( ) + binompdf ( 48, 0.5, 3) = ( ( { })) ª sum binompdf 48, 0.5, 0,,, 3 mk C Ê ˆ Ê 3ˆ Ê ˆ Ê 3 ˆ Ê ˆ Ê 3 ˆ Ê ˆ Ê 3ˆ 0 Ë 4 Ë C Ë 4 Ë C Ë 4 Ë C 3 Ë 4 Ë ( 4 terms) = pmak July, 004

11 b) Use the normal approximation to the binomial distribution to obtain an estimate of this probability. (Answer accurate to at least 4 decimal places.) ( s) m s = = ( ) = np = npq = (. )(. ) = normalcdf 9.5, 3.5,, ( ) = continuity correction 046pmak - - July, 004

12 6. A mass is supported by a spring so that it rests 50 cm above a table top, as shown in the diagram below. The mass is pulled down to a height of 0 cm above the table top and released at time t = 0. It takes 0.8 seconds for the mass to reach a maximum height of 80 cm above the table top. As the mass moves up and down, its height h, in cm, above the table top, is approximated by a sinusoidal function of the elapsed time t, in seconds, for a short period of time. table top Determine an equation for a sinusoidal function that gives h as a function of t. (4 s) mk mk Ø Ø h = -30 cos p.6 t + 50 OR p h = 30 sin ( t ) for proper phase shift expression ( t = t - 0) OR p h = 30 cos ( t ) OR p h = -30sin ( t -. ) OR p h = -30sin ( t ) pmak - - July, 004

13 7. Prove the identity: csc qsinq - sec qcos q = sec q (4 s) both for LEFT SIDE RIGHT SIDE sec q Ø sin q Ø Ø Ø sin q- ( -) s Æ -+ Æ mk sec q LS = RS Note: Deduct if LS and RS are not identical. 046pmak July, 004

14 7. Prove the identity: csc qsinq - sec qcos q = sec q (4 s) both for LEFT SIDE RIGHT SIDE sec q s Æ Ø Ø Ø Ø sin q- ( sin q cos q-sin q) -+ sin q + sin q Æ cos q+sin q Æ sec q LS = RS Note: Deduct if LS and RS are not identical. 046pmak July, 004

15 7. Prove the identity: csc qsinq - sec qcos q = sec q (4 s) both for LEFT SIDE RIGHT SIDE sec q Ø Ø Ø Ø sin q- ( sin q - sin q) s Æ - + sin q Æ cos q- + sin q ( ) - cos q+sin q - Æ sec q LS = RS Note: Deduct if LS and RS are not identical. 046pmak July, 004

16 7. Prove the identity: alternate solution 3 csc qsinq - sec qcos q = sec q (4 s) LEFT SIDE sin0 cos 0 = - sin q RIGHT SIDE sec q = = sin0 - sin qcos 0 sin qcos q sin( 0 - q) sin qcos q s sin q = sin qcos q = = sec q LS = RS Note: Deduct if LS and RS are not identical. END OF KEY 046pmak July, 004