Solution a b notes in b give 1 mark if only a

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Solution a) b) Note(s): in b) give 1 mark if only a correctly labelled tree diagram is provided 2 marks 3 marks ( )( ) ( )( ) 0.6 0.85 0.4 0.9 0.87 + = ½ mark for complements: 0.85 and 0.9 1 mark for multiplying probabilities (½ mark for each product) ½ mark for consistent sum ( ) ( )( ) ( )( ) ( )( ) 0.6 0.15 9 P bus/late 0.692 0.6 0.15 0.4 0.1 13 = = + or ½ mark for conditional probability fraction ½ mark for consistent answer 1 mark for ( ) P late 1 mark for ( ) P bus and late car bus on time late 0.4 on time late 0.6 0.90 0.10 0.85 0.15
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Marking Guide—Pre-Calculus Mathematics (40S)—January 2007 Page 19 EXEMPLARS a) b) Mark: 1 out of 3 Note(s): gave 1 mark for P (bus and late) Mark: ½ out of 2 Note(s): gave ½ mark for complements no mark for sum, notation error
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Page 20 Marking Guide—Pre-Calculus Mathematics (40S)—January 2007 Question No. 6 Outcome: A5 Solve for x , where x \ . Give the general solution in radians correct to 3 decimal places: ( ) 3 cos 2 5 x = − Solution Method 1 Method 2 Note(s): in methods 2 and 3, if general solution is given with “ k + π ” award full marks 4 marks 2 2.214 2 , 4.069 2 I 1.107 , 2.034 x k k k x k k = + π + π = + π + π 2 marks (1 mark for each solution) 1 mark for dividing by 2 1 mark for general solution 2 marks (½ mark for each solution) 1 mark for dividing by 2 1 mark for general solution 4 marks 1 3 cos 2.2143 5 2 2.2143, 4.0689, 8.4945, 10.3521 1.107 2 2.034 2 I 4.249 2 5.176 2 x x k k k k k = = = + π + π + π + π
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Marking Guide—Pre-Calculus Mathematics (40S)—January 2007 Page 21 Question No. 6 Outcome: A5 Method 3 Method 4—Graphing Calculator [ ] 1 2 cos2 3 5 Find intersections on 0, 1.107 2.034 1.107 y x y x x x k k k Ι = = π = = = + π = 2.034 + π 1 mark for showing calculator entries 1 mark for explanation 1 mark for x -coordinates of intersections 1 mark for general solution 4 marks 2 2 2 2 2cos 1 cos2 3 2cos 1 5 2 2cos 5 1 cos 5 1 cos 5 1.107 2 2.034 2 I 4.249 2 5.176 2 x x x x x x x k k k k k = = − = = = ± = + π + π + π + π ½ mark for identity ½ mark for solving for cos x 2 marks (½ mark for each solution) 1 mark for general solution 4 marks
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Page 22 Marking Guide—Pre-Calculus Mathematics (40S)—January 2007 EXEMPLAR Mark: 3 out of 4 Note(s): deducted 1 mark for no general solution 2.035 is a correct quadrant III angle 4.249 is a correct quadrant II angle
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Page 24 Marking Guide—Pre-Calculus Mathematics (40S)—January 2007 Question No. 7 Outcome: E1 Using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, how many 3-digit numbers can be made if there are no repetitions of digits and the second digit is even?
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