Principles of Mathematics 12 Sample 2007/08 Provincial Examination Multiple-Choice Key

Transcription

1 Principles of Mathematics Sample 007/08 Provincial Examination Multiple-Choice Key Cognitive Processes K = Knowledge U = Understanding H = Higher Mental Processes Weightings 0% 70% 0% Question Type 44 = Multiple Choice (MC) 8 = Written Response (WR) Topics. Patterns and Relations. Shape and Space 3. Statistics and Probability Prescribed Learning Outcomes (PLOs) A B C A A3: Geometric Sequences and Series A4 A6, A3 5: Logarithms and Exponents A7 A, A6-A8: Trigonometry B B6: Shape and Space C C5: Combinatorics C5 C0: Probability Question Number Keyed Response Cognitive Process Mark Topic PLO Question Type. C U.5 A8 MC. D U.5 A7 MC 3. D K.5 A6 MC 4. C U.5 A7 MC 5. A U.5 A7 MC 6. D U.5 A MC 7. B U.5 A9 MC 8. C H.5 A9, A MC 9. D K.5 A5 MC 0. B U.5 A6 MC. C U.5 A6 MC. A U.5 A4 MC 3. C H.5 A5 MC 4. D H.5 A5 MC 5. A U.5 3 C5 MC 6. C K.5 B, B MC 7. A U.5 B3 MC 8. A U.5 B, B3 MC 9. B U.5 B, B MC 0. A H.5 B, B3 MC. C U.5 B, B MC. B U.5 A9 MC Principles of Mathematics 007/08 Sample Form A Key Page

2 Question Number Keyed Response Cognitive Process Mark Topic PLO Question Type 3. A U.5 A8 MC 4. C U.5 A7 MC 5. C H.5 A MC 6. D K.5 A MC 7. B U.5 A MC 8. C U.5 A MC 9. B U.5 A3 MC 30. C U.5 A MC 3. D H.5 A MC 3. A K.5 A3 MC 33. A U.5 A4 MC 34. C U.5 A5 MC 35. C U.5 3 C MC 36. A U.5 3 C3 MC 37. D U.5 3 C4 MC 38. D U.5 3 C5 MC 39. B U.5 3 C, C3 MC 40. A U.5 3 C6 MC 4. C U.5 3 C8 MC 4. A H.5 3 C8 MC 43. B U.5 3 C0 MC 44. C U.5 3 C5 MC Principles of Mathematics 007/08 Sample Form A Key Page

3 Principles of Mathematics Sample 007/08 Provincial Examination Written-Response Key Cognitive Processes K = Knowledge U = Understanding H = Higher Mental Processes Weightings 0% 70% 0% Question Type 44 = Multiple Choice (MC) 8 = Written Response (WR) Topics. Patterns and Relations. Shape and Space 3. Statistics and Probability Prescribed Learning Outcomes (PLOs) A B C A A3: Geometric Sequences and Series A4 A6, A3 5: Logarithms and Exponents A7 A, A6-A8: Trigonometry B B6: Shape and Space C C5: Combinatorics C5 C0: Probability Question Number Keyed Response Cognitive Process Mark Topic PLO Question Type. U 3 B5, B6 WR. U B4 WR 3. U 5 A5 WR 4. U 3 3 C0 WR 5. U 3 C0 WR 6. U 3 A8 WR 7. U A0 WR 8. H 5 A, A WR Principles of Mathematics 007/08 Sample Written-Response Key Page

4 Principles of Mathematics Sample 007/08 Provincial Examination Scoring Guide Use the following graph to answer questions and. The graph of y = f ( x) is shown below. y 5 y = f (x) 5 5 x 5 Principles of Mathematics 007/08 Sample Scoring Guide Page

5 . On the grid provided, sketch the graph of y = f ( x). (3 marks) SOLUTION y x 5 mark: vertical translation mark: absolute value mark: vertical expansion Note: Deduct mark if graph does not end at correct points Cap at marks if wrong order of transformations. Principles of Mathematics 007/08 Sample Scoring Guide Page

6 . On the grid provided, sketch the graph of y =. ( marks) f ( x) SOLUTION y x 5 mark: asymptotic behaviour mark: invariant points mark: shape from x = 0tox = Note: Deduct mark if graph does not end at correct points 0, 3 and ( 5, ) Principles of Mathematics 007/08 Sample Scoring Guide Page 3

7 3. Solve algebraically log log( x ) = log( x + ) log( x + 7). (5 marks) SOLUTION log x = log x + x + 7 mark x = x + x + 7 mark x + 34 = x mark 0 = x x 35 mark 0 = ( x 7) ( x + 5) x = 7, mark x = 5 reject mark mark x = 7 Principles of Mathematics 007/08 Sample Scoring Guide Page 4

8 Use the following information to answer questions 4 and 5. It is known that % of the population has a certain disease. A test for this disease is 90% accurate. This means that the outcome of the test is correct 90% of the time. A positive test result claims that a person has the disease. 4. Determine the probability that a randomly selected person will test positive for this disease. (Answer accurate to at least 3 decimal places.) (3 marks) SOLUTION D D mark tree diagram P N P N mark mark P( P)= ( 0.9) ( 0.0)+ ( 0.) ( 0.98) P( P)= 0.6 or 9 50 mark Principles of Mathematics 007/08 Sample Scoring Guide Page 5

9 5. Given that a randomly selected person tests positive, what is the probability that this person actually has the disease? (Answer accurate to at least decimal places.) ( marks) SOLUTION D D P N P N ( ) = P D and P P D P ( ) = 0.0 P D P ( ) P( P) ( )( 0.9) mark 0.60 mark mark P( D P) = 0.6 or 9 58 mark Principles of Mathematics 007/08 Sample Scoring Guide Page 6

10 6. A minimum value of a sinusoidal function is at of this point is at 7,7. Determine an equation of this function. 4,3. The nearest maximum value to the right (3 marks) SOLUTION mark y = sin3 x 5 ( ) + 5 mark mark mark OR y = cos3( x 4 ) + 5 Principles of Mathematics 007/08 Sample Scoring Guide Page 7

11 7. The two smallest positive solutions of cos 4x = 0.6 are x = 0.3 and x =.34. Determine the general solution for cos 4x = 0.6. ( mark) SOLUTION x = n,.34 + n, wheren is an integer mark mark Note: Deduct mark for not writing where n is an integer Principles of Mathematics 007/08 Sample Scoring Guide Page 8

12 8. Prove the identity: tan x sec x + = cosx cos x sin x (5 marks) SOLUTION LEFT SIDE RIGHT SIDE tan x sec x + mark cos x cos x mark sin x + cos x mark sin x ( cos x ) cos x + ( ) ( ) cosx cos x mark sinx cos x mark cos x sin x cos x sin x mark ( )( + cos x ) mark + cos x cos x mark sin x + cos x ( ) sin x mark sin x + cos x ( ) sin x + cos x mark LS = RS Principles of Mathematics 007/08 Sample Scoring Guide Page 9