Grade 12 Pre-Calculus Mathematics Achievement Test. Marking Guide

Transcription

1 Grade 12 Pre-Calculus Mathematics Achievement Test Marking Guide January 2015

2 Question 1 T1 Convert 13π to degrees. 5 Solution 13π π mark 8 Pre-Calculus Mathematics: Marking Guide (January 2015)

3 Exemplar 1 ½ out of 1 award full marks ½ mark for arithmetic error Exemplar 2 0 out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 9

4 Question 2 P2, P3 a) From a group of 9 people, in how many ways can you select a committee of 4 members? b) From a group of 9 people, in how many ways can you select a president, a vice president, a secretary, and a treasurer? c) Explain why the answers in a) and b) are different. Solution a) C = 126 ways mark for C mark b) P = 3024 ways mark for P mark c) Part a) is a combination because the order does not matter; part b) is a permutation because committee members have specific roles. 1 mark 10 Pre-Calculus Mathematics: Marking Guide (January 2015)

5 Exemplar 1 a) 0 out of 1 concept error (using permutations instead of combinations) b) 1 out of 1 consistent with concept error in a) c) 1 out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 11

6 Exemplar 2 a) 0 out of 1 b) 1 out of 1 c) ½ out of 1 award full marks ½ mark for lack of clarity in explanation 12 Pre-Calculus Mathematics: Marking Guide (January 2015)

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8 Question 3 R10 A population of 500 bacteria will triple in 20 hours. Using the formula given below, A = rt Pe A = population after t hours P = initial population r = rate of growth t = time in hours a) Determine the rate of growth, r. b) Determine how many hours it will take for the initial population to double with the same rate of growth. Solution a) 1500 = 500e 3 = e 20r ln 3 = ln e 20r ( 20) ln 3 = 20r ln e r ½ mark for substitution ½ mark for applying logarithms ½ mark for power rule r r = = ln ½ mark for evaluating quotient of logarithms 2 marks b) 1000 = 500e 2 = e t t t ln 2 = ln e ln 2 = t ln e t = t = ln hours ½ mark for substitution ½ mark for applying logarithms ½ mark for power rule ½ mark for evaluating quotient of logarithms 2 marks 14 Pre-Calculus Mathematics: Marking Guide (January 2015)

9 Exemplar 1 a) 1 out of 2 award full marks 1 mark for concept error in line 3 b) 2 out of 2 award full marks [work consistent with answer in a)] Pre-Calculus Mathematics: Marking Guide (January 2015) 15

10 Exemplar 2 a) 2 out of 2 award full marks b) 2 out of 2 award full marks [work consistent with answer in a)] E7 (transcription error in line 1) 16 Pre-Calculus Mathematics: Marking Guide (January 2015)

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12 Question 4 T5 Talla incorrectly solved the following trigonometric equation: Solve: 2sec x 5 = 0; 0 x 360 Talla s work: 2secx 5 = 0 sec x = 5 2 No solution, sec x cannot be greater than 1. a) Explain her error. b) Determine the correct solution. Solution a) Talla incorrectly stated that sec x cannot be greater than 1. The value of cos x cannot be greater than 1. 1 mark b) 5 sec x = 2 2 cos x = 5 x = r x = x = mark for reciprocal 1 mark for solving for x (½ mark for each value of x) 2 marks 18 Pre-Calculus Mathematics: Marking Guide (January 2015)

13 Exemplar 1 a) ½ out of 1 ½ mark for lack of clarity in explanation b) 1 out of mark for reciprocal Pre-Calculus Mathematics: Marking Guide (January 2015) 19

14 Exemplar 2 a) 0out of 1 b) ½ out of mark for reciprocal + ½ mark for value of x 1 mark for concept error in line 5 E6 (rounding error in line 5) 20 Pre-Calculus Mathematics: Marking Guide (January 2015)

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16 Question 5 P4 Simplify the 6th term in the expansion of: 2x 3 2 x 10 Solution 5 3 t = C ( 2x) x 5 ( x ) 243 = x = x 5 2 marks (1 mark for C, ½ mark for each consistent factor) mark for simplification (½ mark for coefficient, ½ mark for exponent) 3 marks 22 Pre-Calculus Mathematics: Marking Guide (January 2015)

17 Exemplar 1 2½ out of mark for C mark for consistent factors + ½ mark for simplification of exponent Exemplar 2 2 out of mark for consistent factors + 1 mark for simplification Pre-Calculus Mathematics: Marking Guide (January 2015) 23

18 Question 6 T1 Determine the arc length subtended by a central angle if the diameter is 19 cm and the central angle is 1.6 radians. Solution s = θr s = s = 15.2 cm ( 1.6)( 9.5) 1 mark 24 Pre-Calculus Mathematics: Marking Guide (January 2015)

19 Exemplar 1 ½ out of 1 ½ mark for procedural error Exemplar 2 1 out of 1 award full marks E5 (missing units of measure in line 3) Pre-Calculus Mathematics: Marking Guide (January 2015) 25

20 Question 7 T3, T5 Solve the following equation algebraically for x, where 0 x 2π. 2 2cos x = 3sinx Solution 2 ( x) 2 1 sin = 3sin x 2 2 2sin x = 3sinx 2 0 = 2sin x 3sin x 2 0 = ( 2sin x+ 1)( sin x 2) 1 sin x = sin x = 2 2 No Solution 7π x = 6 11π x = 6 1 mark for identity 1 mark for solving for sin x 1 mark for indicating no solution 1 mark for solving for x (½ mark for each value) 4 marks 26 Pre-Calculus Mathematics: Marking Guide (January 2015)

21 Exemplar 1 3 out of mark for identity + 1 mark for solving for sin x + 1 mark for indicating no solution Pre-Calculus Mathematics: Marking Guide (January 2015) 27

22 Exemplar 2 1out of mark for identity Exemplar 3 4out of 4 award full marks E5 (answer stated in degrees instead of radians) E7 (notation error in line 3) 28 Pre-Calculus Mathematics: Marking Guide (January 2015)

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24 Question 8 P2 In how many different ways can you arrange the letters in the word VOLLEYBALL? State your answer as a factorial. Solution 10! 4! 1 mark Note(s): award full marks for Pre-Calculus Mathematics: Marking Guide (January 2015)

25 Exemplar 1 0 out of 1 Exemplar 2 1 out of 1 Exemplar 3 0 out of 1 Exemplar 4 1 out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 31

26 Question 9 R11 Is ( x 2) a factor of the polynomial ( ) Justify your response. Solution p x = x 3x + 11x + 3x 10? Method ( ) ( ) ( ) ( ) ( ) p 2 = = = 0 The remainder is zero, so ( x ) 2 is a factor. ½ mark for p ( 2) 1 mark for the remainder theorem ½ mark for justification 2 marks Method The remainder is zero, so ( x 2) is a factor. ½ mark for x = 2 1 mark for synthetic division (or for any equivalent strategy) ½ mark for justification 2 marks Method 3 I entered y = x 3x + 11x + 3x 10 into my calculator and located the zeroes. 2 x 2 is a factor. x = was a zero, which means ( ) 1 mark for graphing calculator method 1 mark for relating the zeroes to the factors 2 marks 32 Pre-Calculus Mathematics: Marking Guide (January 2015)

27 Exemplar 1 1½ out of 2 award full marks ½ mark for procedural error in line 1 Exemplar 2 ½ out of 2 + ½ mark for p ( 2) Pre-Calculus Mathematics: Marking Guide (January 2015) 33

28 Question 10 T4 Determine the period of the sinusoidal function State your answer in radians. 1 1 y = sin x 2 3. Solution p p p 2π = b 2π = 1 3 = 6π or p = ½ mark for correct value of b ½ mark for period consistent with b 1 mark 34 Pre-Calculus Mathematics: Marking Guide (January 2015)

29 Exemplar 1 ½ out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 35

30 Question 11 R1 The domain of f ( x ) is x 2. The domain of g( x ) is x 7. State the domain of f ( x) + g( x). Justify your answer. Solution Both f ( x ) and g( x ) have restricted domains, so both domains need to be considered. f ( x) g( x) 7 2 The solution is the intersection of the two domains. { xx, 7 x 2} or [ 7, 2] 1 mark for justification 1 mark for domain 2 marks 36 Pre-Calculus Mathematics: Marking Guide (January 2015)

31 Exemplar 1 1 out of mark for domain Exemplar 2 2 out of 2 award full marks E8 (domain written in incorrect order) Exemplar 3 ½ out of mark for justification ½ mark for lack of clarity Pre-Calculus Mathematics: Marking Guide (January 2015) 37

32 Question 12 T6 Prove the identity below for all permissible values of θ. 1 2 cotθ = csc θ 1 + cosθ sinθ Solution Method 1 Left-Hand Side 1 1 cosθ + Right-Hand Side 2 csc cos 1 sin 2 sin sin 1 cosθ 1 2 sin sinθ sin 1 cos 2 2 sin sin 1 cos 2 sin 1 cos 2 1 cos cotθ θ sinθ θ θ θ θ θ θ 1 cosθ ( 1 cosθ)( 1+ cosθ) 1 1+ cosθ θ θ θ θ θ θ θ 1 mark for correct substitution of identities 1 mark for algebraic strategies 1 mark for logical process to prove an identity 3 marks 38 Pre-Calculus Mathematics: Marking Guide (January 2015)

33 Question 12 T6 Solution Method Left-Hand Side 1 1+ cosθ 1 cosθ 1 cos cosθ sin ( 1 cosθ) ( 1 cosθ) 1 cosθ 2 2 sin θ sin θ csc csc θ θ cosθ 1 θ sinθ sinθ cotθ θ sinθ Right-Hand Side 2 cotθ csc θ sinθ 1 mark for algebraic strategies 1 mark for correct substitution of identities 1 mark for logical process to prove an identity 3 marks Pre-Calculus Mathematics: Marking Guide (January 2015) 39

34 Exemplar 1 1 out of mark for correct substitution of identities 40 Pre-Calculus Mathematics: Marking Guide (January 2015)

35 Exemplar 2 2½ out of 3 award full marks ½ mark for procedural error in line 4 Pre-Calculus Mathematics: Marking Guide (January 2015) 41

36 Exemplar 3 0 out of 3 42 Pre-Calculus Mathematics: Marking Guide (January 2015)

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38 Question 13 R12 Explain how the end behaviours of the graphs of polynomial functions with an even degree and with an odd degree are different. Solution When the degree is odd, the end behaviour is in opposite directions. When the degree is even, the end behaviour is in the same direction. 1 mark 44 Pre-Calculus Mathematics: Marking Guide (January 2015)

39 Exemplar 1 ½ out of 1 award full marks ½ mark for lack of clarity in explanation Exemplar 2 0 out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 45

40 Question 14 R1 Given the graphs of f ( x ) and g( x ), sketch the graph of g( x) f ( x). f ( x ) g( x) y y 1 1 x 1 1 x Solution g( x) f ( x) ( ) ( ) ( )( ) x g x f x g f x mark for subtraction of g( x) f ( x) 1 1 x 1 mark for restricting domain on graph 2marks 46 Pre-Calculus Mathematics: Marking Guide (January 2015)

41 Exemplar 1 1 out of mark for subtraction of g( x) f ( x) Exemplar 2 1 out of mark for restricting domain Pre-Calculus Mathematics: Marking Guide (January 2015) 47

42 Exemplar 3 1½ out of 2 award full marks ½ mark for arithmetic error (1 incorrect point) 48 Pre-Calculus Mathematics: Marking Guide (January 2015)

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44 Question 15 R6 Given f ( x) = 3x + 7, evaluate f 1 ( ) Solution 2. f f f 1 1 Let 1 ( ) ( 2) ( ) ( ) y = f x f x = 3x+ 7 y = 3x+ 7 x = 3y + 7 x 7 = 3y x 7 y = 3 x 7 ( x) = = 3 2 = 3 1 mark for switching x and y 1 ½ mark for f ( x) ½ mark for f 1 ( 2) 2 marks 50 Pre-Calculus Mathematics: Marking Guide (January 2015)

45 Exemplar 1 2 out of 2 E7 (transcription error in line 6) Exemplar 2 0 out of 2 Pre-Calculus Mathematics: Marking Guide (January 2015) 51

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47 Booklet 2 Questions Pre-Calculus Mathematics: Marking Guide (January 2015) 53

48 Answer Key for Multiple-Choice Questions Question Answer Learning Outcome 16 C P4 17 B T1 18 B R9 19 C P3 20 C R13 21 B T6 22 A R8 23 D T2 24 C R12 25 A R5 54 Pre-Calculus Mathematics: Marking Guide (January 2015)

49 Question 16 P4 How many terms are there in the expansion of ( ) x + 3? a) 9 b) 10 c) 11 d) 12 Question 17 T1 11π A co-terminal angle for θ = in the domain 2π θ 0 would be: 3 a) b) c) d) 5π 3 π 3 π 3 5π 3 Question 18 R9 x The x-intercept of the graph of y = 3 1 is: a) 1 b) 0 c) 1 d) 2 Pre-Calculus Mathematics: Marking Guide (January 2015) 55

50 Question 19 P3 If C = C, the value of n must be: n 5 n 3 a) 3 b) 5 c) 8 d) 15 Question 20 R13 What is the domain of the function f ( x) = ( x+ ) a) { xx, x 1} b) { xx, x 1} c) { xx, x 1} d) { xx } 1? Question 21 T6 Identify a non-permissible value of x for the expression a) 0 1 cos 2x. b) c) π 4 π 2 d) π 56 Pre-Calculus Mathematics: Marking Guide (January 2015)

51 Question 22 R8 The expression 1 2log x log y as a single logarithm is: 3 a) b) 2 log x 3 y 2 log 3 x y c) log x 2 3 y 2 d) log 3 ( x y) Question 23 T2 The point ( ) P θ lies on the unit circle. What are the coordinates of the point if θ = 300? a) b) 1 3, , 2 2 c) 3 1, 2 2 d) 1 3, 2 2 Pre-Calculus Mathematics: Marking Guide (January 2015) 57

52 Question 24 R12 What is the degree of the polynomial function represented by the graph below? a) 2 y b) 3 c) 4 d) 5 x Question 25 R5 When the point ( 4, 3) is reflected over the line y = x, the coordinates of the new point are: a) ( 3, 4) b) ( 3, 4 ) c) ( 4, 3) d) ( 4, 3) 58 Pre-Calculus Mathematics: Marking Guide (January 2015)

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54 Question 26 R4, R9 Sketch the graphs of: a) x 1 y = 4 b) x 1 y = 2 4 Solution a) ( 1, 4) 1 y ½ mark for decreasing exponential function ½ mark for y-intercept ( 0, 1 ) ½ mark for consistent point of an exponential function ½ mark for horizontal asymptote at y = x 2 marks b) ( 1, 8) y 1 mark for a vertical stretch by a factor of 2 of the graph consistent with a) 1 mark 2 x 60 Pre-Calculus Mathematics: Marking Guide (January 2015)

55 Exemplar 1 a) 1½ out of 2 + ½ mark for decreasing exponential function 0, 1 + ½ mark for y-intercept ( ) + ½ mark for consistent point of an exponential function E9 (missing arrowhead) b) 1out of 1 consistent with a) Pre-Calculus Mathematics: Marking Guide (January 2015) 61

56 Exemplar 2 a) 1½ out of 2 + ½ mark for decreasing exponential function 0, 1 + ½ mark for y-intercept ( ) + ½ mark for consistent point of an exponential function b) 1 out of 1 62 Pre-Calculus Mathematics: Marking Guide (January 2015)

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58 Question 27 R Determine all of the zeroes of the function p( x) = x 5x 2x+ 24, given one of the factors of p( x ) is ( x 3. ) Solution = x 5x 2x ½ mark for x = 3 1 mark for synthetic division (or for any equivalent strategy) 2 x 2x 8 = 0 ( x )( x ) = 0 zeroes: 3, 4, 2 ½ mark for consistent zeroes 2 marks 64 Pre-Calculus Mathematics: Marking Guide (January 2015)

59 Exemplar 1 1½ out of 2 + ½ mark for x = mark for synthetic division E2 (changing an equation to an expression) E7 (transcription error in line 2) Pre-Calculus Mathematics: Marking Guide (January 2015) 65

60 Question 28 R13 Given the graph of f ( x ), y 1 1 x sketch the graph of y = f ( x). Solution y 1 mark for restricting domain ½ mark for shape between invariant points ½ mark for shape to the right of the invariant points 1 1 x 2 marks 66 Pre-Calculus Mathematics: Marking Guide (January 2015)

61 Exemplar 1 0 out of 2 Exemplar 2 1 out of mark for restricting domain Pre-Calculus Mathematics: Marking Guide (January 2015) 67

62 Question 29 T4 Sketch the graph of at least one period of the function y 2sin( 4x) =. Solution y 2 1 π 2 π 4 1 π 4 π 2 x 2 1 mark for amplitude 1 mark for period 1 mark for reflection in the x-axis 3 marks 68 Pre-Calculus Mathematics: Marking Guide (January 2015)

63 Exemplar 1 3 out of mark for amplitude + 1 mark for period + 1 mark for reflection in the x-axis E9 (scale values on y-axis not indicated) Exemplar 2 2 out of mark for amplitude + 1 mark for reflection in the x-axis Pre-Calculus Mathematics: Marking Guide (January 2015) 69

64 Exemplar 3 2 out of 3 award full marks 1 mark for concept error, sketched y = 2 cos( 4x) 70 Pre-Calculus Mathematics: Marking Guide (January 2015)

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66 Question 30 R8 Evaluate: Solution 1 log 144 log 4 + 2log ( ) 2 + ( ) log 144 log 4 log log 12 log 4 + log log 3 4 log mark for power rule ½ mark for product rule ½ mark for quotient rule 1 mark for evaluating a logarithm 3 marks 72 Pre-Calculus Mathematics: Marking Guide (January 2015)

67 Exemplar 1 2½ out of mark for power rule + ½ mark for quotient rule + 1 mark for evaluating a logarithm Pre-Calculus Mathematics: Marking Guide (January 2015) 73

68 Exemplar 2 0 out of mark for power rule 1 mark for concept error in line 1 Exemplar 3 3 out of mark for power rule + ½ mark for product rule + ½ mark for quotient rule + 1 mark for evaluating a logarithm E7 (transcription error in line 4) 74 Pre-Calculus Mathematics: Marking Guide (January 2015)

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70 Question 31 R14 Match each function with its correct description. a) The graph of this function has a vertical asymptote at x = 1. b) The graph of this function has a point of discontinuity (hole) at x = 3. c) The graph of this function has a horizontal asymptote at y = 4. d) The domain of this function is x. Solution Place the appropriate letter in this column. f ( x) = 2 ( ) g x ( ) h x x x = x + 3 = ( x )( x+ ) ( x 3) d) c) b) ( ) k x = 4( x 3) ( x+ 3)( x+ 1) a) ½ mark for each correct answer 2 marks 76 Pre-Calculus Mathematics: Marking Guide (January 2015)

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72 Question 32 R3 = 1. 2 The point ( 3, 4) is on the graph of y f ( 3x) State the coordinates of the corresponding point on the graph of y = f ( x). Solution ( 9, 8) ½ mark for each coordinate 1 mark 78 Pre-Calculus Mathematics: Marking Guide (January 2015)

73 Exemplar 1 0 out of 1 Exemplar 2 ½ out of 1 + ½ mark for the y-coordinate Pre-Calculus Mathematics: Marking Guide (January 2015) 79

74 Question 33 R12 Sketch the graph of y 2( x 1)( x 3)( x 1) = +. Solution y 1 mark for x-intercepts 1 mark for y-intercept 1 mark for end behaviour 3 marks x 6 80 Pre-Calculus Mathematics: Marking Guide (January 2015)

75 Exemplar 1 1 out of mark for x-intercepts Pre-Calculus Mathematics: Marking Guide (January 2015) 81

76 Exemplar 2 2 out of mark for x-intercepts + 1 mark for y-intercept Exemplar 3 2 out of mark for x-intercepts + 1 mark for end behaviour 82 Pre-Calculus Mathematics: Marking Guide (January 2015)

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78 Question 34 T6 2 1 sin x sin2x π a) Verify that the equation = is true for x =. cos x 2sin x 3 π b) Explain why verifying the equation for x = is insufficient to conclude that the equation is an 3 identity. Solution a) 2 π π 1 sin sin2 3 3 π π cos 2sin LHS= RHS 1 mark for exact values (½ mark for sin 3 π, ½ mark for cos 3 π ) 1 mark for simplification (½ mark for LHS, ½ mark for RHS) 2 marks b) Proving that it is true for one value does not mean that it is true for all values. 1 mark 84 Pre-Calculus Mathematics: Marking Guide (January 2015)

79 Exemplar 1 a) 1½ out of 2 award full marks ½ mark for arithmetic error in line 4 b) 0 out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 85

80 Exemplar 2 a) 2 out of 2 b) ½ out of 1 award full marks ½ mark for lack of clarity 86 Pre-Calculus Mathematics: Marking Guide (January 2015)

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82 Question 35 P3 Evaluate: P 7 2 P 7 5 Solution 7! 7 2! 7! 7 5! ( ) ( ) 7! 5! 7! 2! 2! 5! ½ mark for substitution ½ mark for simplification 1 mark for expanding factorials 2 marks 88 Pre-Calculus Mathematics: Marking Guide (January 2015)

83 Exemplar 1 1 out of 2 award full marks 1 mark for concept error (used combinations) Exemplar 2 2 out of 2 award full marks E1 (final answer not stated) Pre-Calculus Mathematics: Marking Guide (January 2015) 89

84 Question 36 R4 Use the graph of y = f ( x) to sketch the graph of y f ( x) = y y = f ( x) 1 1 x Solution y 1 mark for horizontal compression 1 mark for vertical translation 2 marks 1 1 x 90 Pre-Calculus Mathematics: Marking Guide (January 2015)

85 Exemplar 1 1 out of mark for vertical translation E9 (coordinate point labelled incorrectly) Pre-Calculus Mathematics: Marking Guide (January 2015) 91

86 Exemplar 2 1 out of mark for horizontal compression E9 (incorrect endpoint) 92 Pre-Calculus Mathematics: Marking Guide (January 2015)

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88 Question 37 R10 Solve the following equation: Solution Method 1 ( x ) log log 3 = log x ( x ) log log 3 = log ( x ) 3( x 2) log = log = x x x 3x+ 6 = x x = 3 No solution 1 mark for product rule 1 mark for equating arguments ½ mark for solving for x ½ mark for rejecting extraneous root 3 marks Method 2 ( x ) log log 3 = log x log ( x+ 2) + log 3 log x = ( x ) log 0 4 x = 0 3x = x x = 3 x = 3 1 mark for logarithmic rules (½ mark for product rule; ½ mark for quotient rule) 1 mark for exponential form ½ mark for solving for x ½ mark for rejecting extraneous root 3 marks 94 Pre-Calculus Mathematics: Marking Guide (January 2015)

89 Exemplar 1 2½ out of mark for product rule + 1 mark for equating arguments + ½ mark for solving for x Exemplar 2 2½ out of 3 award full marks ½ mark for procedural error in line 2 Pre-Calculus Mathematics: Marking Guide (January 2015) 95

90 Exemplar 3 1out of mark for product rule E2 (changing an equation to an expression) Exemplar 4 2out of mark for logarithmic rules + 1 mark for exponential form 96 Pre-Calculus Mathematics: Marking Guide (January 2015)

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92 Question 38 R14 Determine the coordinates of the point of discontinuity (hole) for the graph of the function ( 2 x)( x 3) y =. x 2 Solution ( ) x 2 y = 1 ( 2, 1) ( x 3) ( 2 3) y = y = 1 mark for point of discontinuity (hole) at ( 2, 1 ) (½ mark for x = 2, ½ mark for consistent y-coordinate) 1 mark 98 Pre-Calculus Mathematics: Marking Guide (January 2015)

93 Exemplar 1 ½ out of 1 + ½ mark for x = 2 Exemplar 2 ½ out of 1 + ½ mark for x = 2 Pre-Calculus Mathematics: Marking Guide (January 2015) 99

94 Question 39 T3 Evaluate and simplify 5π π sec tan. 6 6 Solution π 1 mark for sec (½ mark for value, ½ mark for quadrant) 6 π 1 mark for tan (½ mark for value, ½ mark for quadrant) 6 2 marks 100 Pre-Calculus Mathematics: Marking Guide (January 2015)

95 Exemplar 1 1½ out of 2 5π + 1 mark for sec 6 π + ½ mark for quadrant of tan 6 Pre-Calculus Mathematics: Marking Guide (January 2015) 101

96 Exemplar 2 ½ out of 2 π + ½ mark for value of tan Pre-Calculus Mathematics: Marking Guide (January 2015)

97 Exemplar 3 1 out of 2 5π + ½ mark for quadrant of sec 6 π + ½ mark for quadrant of tan 6 Pre-Calculus Mathematics: Marking Guide (January 2015) 103

98 Question 40 R13 Sketch the graph of the following function: y = 2 x 3 Solution Method 1 y 1 1 ( 3, 0) ( 4, 2) x 1 mark for shape (graph of a radical function) 1 mark for vertical reflection 1 mark for horizontal shift 1 mark for vertical stretch 4 marks Method 2 ( 4, 1) ( 7, 2) ( 3, 0) ( ) 4, 2 ( 7, 4) y y = x 3 = x 3 y = 2 x 3 1 mark for invariant points where y = 0 and y = 1 (½ mark for each point) 1 mark for domain of [ 3, ) ½ mark for shape between invariant points ½ mark for shape to the right of the invariant points 1 mark for applying transformations (vertical stretch, vertical reflection) 4 marks 104 Pre-Calculus Mathematics: Marking Guide (January 2015)

99 Exemplar 1 3 out of mark for shape + 1 mark for vertical reflection + 1 mark for vertical stretch Exemplar 2 3 out of mark for shape + 1 mark for horizontal shift + 1 mark for vertical stretch Pre-Calculus Mathematics: Marking Guide (January 2015) 105

100 Exemplar 3 2 out of mark for shape + 1 mark for horizontal shift 106 Pre-Calculus Mathematics: Marking Guide (January 2015)

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102 Question 41 R14 Sketch the graph of f ( x) = 2x + 3. x + 2 Solution ( 3, 3) ( 1, 1) y x 1 mark for vertical asymptote 1 mark for horizontal asymptote ½ mark for graph left of the vertical asymptote ½ mark for graph right of the vertical asymptote 3 marks 108 Pre-Calculus Mathematics: Marking Guide (January 2015)

103 Exemplar 1 1 out of mark for vertical asymptote Pre-Calculus Mathematics: Marking Guide (January 2015) 109

104 Exemplar 2 1½ out of mark for vertical asymptote + ½ mark for graph right of vertical asymptote 110 Pre-Calculus Mathematics: Marking Guide (January 2015)

105 Exemplar 3 2½ out of mark for vertical asymptote + 1 mark for horizontal asymptote + ½ mark for graph right of vertical asymptote E10 (graph crosses asymptote) Pre-Calculus Mathematics: Marking Guide (January 2015) 111

106 Question 42 R1 a) Given the functions f ( x) = 4 + x and g( x) = 3x 6, evaluate ( ( 5) ) b) Is it possible to evaluate g( f ( 5) )? Justify your answer. Solution f g. a) g g f ( ) ( ) ( 5) = 21 ( ) = + 5 = = 25 = 5 1 mark for g ( 5) ( ) 1 mark for consistent value of f g( 5) 2 marks b) No, because ( ) f f f x is undefined when x = 5 or ( 5) = 4+ ( 5) ( 5) = 1 1 mark for justification 1 mark f ( 5) is undefined because you cannot evaluate the square root of a negative number. 112 Pre-Calculus Mathematics: Marking Guide (January 2015)

107 Exemplar 1 a) 1out of mark for g ( 5) E7 (notation error in line 1) b) ½ out of 1 award full marks ½ mark for arithmetic error Pre-Calculus Mathematics: Marking Guide (January 2015) 113

108 Exemplar 2 a) 1½ out of 2 award full marks ½ mark for arithmetic error in line 2 E7 (notation error in line 1) b) Yes it is possible : 1out of 1 award full marks E7 (transcription error in line 1) 114 Pre-Calculus Mathematics: Marking Guide (January 2015)

109 Exemplar 3 a) 2 out of 2 award full marks b) No because and you can t take the square of ½ out of 1 award full marks ½ mark for terminology error in line 2 Pre-Calculus Mathematics: Marking Guide (January 2015) 115

110 Question 43 R7 Identify which of these values is greater. Justify your answer. Solution log 80 or log = = 125 log 80 is less than = = 81 log 30 is more than 3 3 log 30 is greater 1 mark for justification 3 1 mark 116 Pre-Calculus Mathematics: Marking Guide (January 2015)

111 Exemplar 1 1 out of 1 Exemplar 2 0 out of 1 Pre-Calculus Mathematics: Marking Guide (January 2015) 117

112 Exemplar 3 1out of Pre-Calculus Mathematics: Marking Guide (January 2015)

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114 Question 44 Given 3 cos α =, where α is in quadrant IV, and 5 sin α β. the exact value of ( ) T3, T6 2 cos β =, where β is in quadrant II, determine 3 Solution x + y = r 3 α 5 y 2 9+ y = 25 y 2 = 16 y =± 4 y = 4 ½ mark for value of y 4 sin α = 5 y 3 β x + y = r 2 4+ y = 9 y 2 = 5 y = ± 5 y = 5 ½ mark for value of y sin β = 5 3 ( ) sin α β = sin αcos β cosαsin β = = 15 ½ mark for sin α ½ mark for sin β 1 mark for substitution into correct identity 3 marks 120 Pre-Calculus Mathematics: Marking Guide (January 2015)

115 Exemplar 1 3 out of 3 award full marks E7 (notation error in line 2) Pre-Calculus Mathematics: Marking Guide (January 2015) 121

116 Exemplar 2 2½ out of 3 + ½ mark for y = 4 + ½ mark for y = 5 + ½ mark for sin β + 1 mark for substitution into correct identity E1 (final answer not stated in line 7) 122 Pre-Calculus Mathematics: Marking Guide (January 2015)

117 Exemplar 3 2½ out of 3 award full marks ½ mark for arithmetic error in line 6 E7 (notation error in line 4) Pre-Calculus Mathematics: Marking Guide (January 2015) 123

118 Question 45 P1 Determine the number of possible sandwiches from the following menu. MENU Select one item from each column: Bread White Rye Brown Sauce Mayo Mustard Meat Turkey Ham Roast Beef Chicken Vegetable Tomato Onion Lettuce Solution sandwiches 1 mark 124 Pre-Calculus Mathematics: Marking Guide (January 2015)