= (A1) 12 M15/5/MATSD/SP2/ENG/TZ1/XX/M. 1. (a) (i) H 0 : age and opinion (about the reduction) are independent.

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1 2 M5/5/MATSD/SP2/ENG/TZ/XX/M. (a) (i) H 0 : age and opinion (about the reduction) are independent. (A) Notes: Accept not associated instead of independent. (ii) H : age and opinion are not independent. (A)(ft) Notes: Follow through from part (a)(i). Accept associated or dependent. Award (A)(ft) for their correct H worded consistently with their part (a)(i). (b) 2 (A) [ mark] (c) (M) Note: Award (M) for 30 seen. The following (A) cannot be awarded without this statement. = (A) = 2.5 (AG) Note: Both an unrounded answer that rounds to the given answer and rounded must be seen for the (A) to be awarded. Accept 2.54 or 2.53 as an unrounded answer. (d) (i) χ 2 statistic = 0.3 ( ) (G2) Note: Accept 0 as a correct 2 significant figure answer. (ii) p-value = ( ) (G) [3 marks] (e) since p-value < 0.0, H 0 should not be accepted (R)(A)(ft) since χ 2 2 statistic > χ critical value, H 0 should not be accepted (R)(A)(ft) Note: Do not award (R0)(A). Follow through from their answer to part (d). Award (R0)(A0) if part (d) is unanswered. Award (R) for a correct comparison of either their p-value 2 2 to the test level or their χ statistic to the χ critical value, award (A) for the correct result from that comparison. Total [0 marks]

2 3 M5/5/MATSD/SP2/ENG/TZ/XX/M 2. (a) ( p q) r (A)(A)(A) [3 marks] Notes: Award (A) for conjunction seen, award (A) for implication seen, award (A) for correct simple propositions in correct order (the parentheses are required). Accept r ( p q). (b) p q r ( p q) ( p q) r T T T T T T T F T F T F T F T T F F F T F T T F T F T F F T F F T F T F F F F T (A)(ft)(A)(ft) Notes: Award (A)(ft) for each correct column, follow through to the final column from their ( p q) column. For the second (A)(ft) to be awarded there must be an implication in part (a). Follow through from part (a). (c) The argument is not valid since not all entries in the final column are T. (A)(ft)(R) Notes: Do not award (A)(ft)(R0). Follow through from part (b). Accept The argument is not valid since ( p q) r is not a tautology. (d) (i) ( p q) r (A)(ft)(A)(ft) ( p q) r (A)(ft)(A)(ft) Notes: Award (A)(ft) for the negation of their antecedent and the negation of their consequent, (A)(ft) for their fully correct answer. Follow through from part (a). Accept r ( p q) or r ( p q). Follow through from part (a). continued

3 4 M5/5/MATSD/SP2/ENG/TZ/XX/M Question 2 continued (ii) if it is not the case that the land has been purchased and the building permit has been obtained then the land can not be used for residential purposes. (A)(A)(ft) if (either) the land has not been purchased or the building permit has not been obtained then the land can not be used for residential purposes. (A)(A)(ft) [4 marks] Notes: Award (A) for if then seen, (A)(ft) for correct statements in correct order. Follow through from part (d)(i). Total [ marks]

4 5 M5/5/MATSD/SP2/ENG/TZ/XX/M 3. (a) 0 ( km h ) (b) 36 (c) 4.5 (A) (G2) (G) [ mark] [ mark] (d) (M) = 9 ( ± ) (A)(ft)(G2) Notes: Award (M) for quartiles seen. Follow through from part (c). (e) 20 0 = 0 Note: Award (M) for 0 seen. (M) (A)(G2) (f) p = 4 q = 0 (A)(ft)(A)(ft) Note: Follow through from part (e). (g) (i) 30 < s 40 (A) (ii) 35 Note: Follow through from part (g)(i). (A)(ft) (h) (i) 36.8 (km h ) ( ) (G2)(ft) Notes: Follow through from part (f). (ii) 8.85 ( ) (G)(ft) [3 marks] Note: Follow through from part (f), irrespective of working seen. (i) (M) Note: Award (M) for seen = 2.7 (%) , 2, 3 3 (A)(G2) Total [7 marks]

5 6 M5/5/MATSD/SP2/ENG/TZ/XX/M 4. (a) AC = cos0 (M)(A) AC = (A)(G2) length of course = 2920 (m) ( m) (A) [4 marks] Notes: Award (M) for substitution into cosine rule formula, (A) for correct substitution, (A) for correct answer. Award (G3) for 2920 ( ) seen without working. The final (A) is awarded for adding 900 and 700 to their AC irrespective of working seen. (b) (M) Note: Award (M) for their length of course divided by.5. Follow through from part (a). = (seconds) (A)(ft) = 32 (minutes) (A)(ft)(G2) [3 marks] Notes: Award the final (A) for correct conversion of their answer in seconds to minutes, correct to the nearest minute. Follow through from part (a). (c) = (M)(A)(ft) sin ACB sin cos ACB = ACB = 30.0 ( ) (M)(A)(ft) (A)(ft)(G2) [3 marks] Notes: Award (M) for substitution into sine rule or cosine rule formula, (A) for their correct substitution, (A) for correct answer. Accept 29.9 for sine rule and 29.8 for cosine rule from use of correct three significant figure values. Follow through from their answer to (a). continued

6 7 M5/5/MATSD/SP2/ENG/TZ/XX/M Question 4 continued (d) sin0 2 (M)(A) Note: Accept AC 900 sin ( ACB). 2 their their Follow through from parts (a) and (c). 2 2 = m (296003m ) (A)(G2) [3 marks] Notes: Award (M) for substitution into area of triangle formula, (A) for correct substitution, (A) for correct answer. Award (G) if is seen without units or working. (e) sin = distance (M) 900 ( distance = ) 450 (m)( ) (A)(ft)(G2) Note: Follow through from part (c) distance = (M) ( distance = ) 450 (m)( ) (A)(ft)(G2) Note: Follow through from part (a) and part (d). 450 is greater than 375, thus the course complies with the safety regulations Notes: A comparison of their area from (d) and the area resulting from the use of 375 as the perpendicular distance is a valid approach and should be given full credit. Similarly a comparison of angle 375 ACB and sin should be given full credit. 900 Award (R0) for correct answer without any working seen. Award (R)(ft) for a justified reason consistent with their working. Do not award (M0)(A0)(R). (R) [3 marks] continued

7 8 M5/5/MATSD/SP2/ENG/TZ/XX/M Question 4 continued (f) AH tan5 = 700 (M) Note: Award (M) for correct substitution into trig formula. AH = 88 (m) ( ) (A)(ft)(G2) (g) HC = (M)(A) Note: Award (M) for substitution into Pythagoras, (A) for their and their correctly substituted in formula. HC = 330 (m)( ) (A)(ft)(G2) [3 marks] Note: Follow through from their answer to parts (a) and (f). Total [2 marks]

8 9 M5/5/MATSD/SP2/ENG/TZ/XX/M 5. (a) k (A)(A)(A) [3 marks] x Note: Award (A) for 92, (A) for 3 x, (A) for k (only). (b) at local minimum f ( x) = 0 (M) Note: Award (M) for seeing f ( x) = 0 (may be implicit in their working) k = 0 (A) 3 4 k = 3 (AG) Note: Award (A) for substituting x = 4 in their f ( x), provided it leads to k = 3. The conclusion k = 3 must be seen for the (A) to be awarded. (c) (2) 2 + (M) Note: Award (M) for substituting x = 2 and k = 3 in f( x ). = 30 (A)(G2) (d) (M) Note: Award (M) for substituting x = 2 and k = 3 in their f ( x). = 2 (A)(ft)(G2) Note: Follow through from part (a). continued

9 20 M5/5/MATSD/SP2/ENG/TZ/XX/M Question 5 continued (e) y 30 = ( x 2) (A)(ft)(M) 2 Notes: Award (A)(ft) for their 2 seen, (M) for the correct substitution of their point and their normal gradient in equation of a line. Follow through from part (c) and part (d). gradient of normal = 2 (A)(ft) 30 = 2 + c 2 (M) 9 c = y = x+ 29 ( y = x ) 2 2 x 2y+ 628 = 0 (A)(ft)(G2) [3 marks] Notes: Accept equivalent answers. (f) (A)(A)(A)(A) [4 marks] Notes: Award (A) for correct window (at least one value, other than zero, labelled on each axis), the axes must also be labelled; (A) for a smooth curve with the correct shape (graph should not touch y-axis and should not curve away from the y-axis), on the given domain; (A) for axis intercept in approximately the correct position (nearer 5 than zero); (A) for local minimum in approximately the correct position (first quadrant, nearer the y-axis than x = 0 ). If there is no scale, award a maximum of (A0)(A)(A0)(A) the final (A) being awarded for the zero and local minimum in approximately correct positions relative to each other. continued

10 2 M5/5/MATSD/SP2/ENG/TZ/XX/M Question 5 continued (g) ( 3.7, 0) (( , 0) ) (G)(G) Notes: If parentheses are omitted award (G0)(G)(ft). Accept x= 3.7, y = 0. Award (G) for 3.7 seen. (h) 0< x 4 or 0< x < 4 (A)(A) Notes: Award (A) for correct end points of interval, (A) for correct notation (note: lower inequality must be strict). Award a maximum of (A)(A0) if y or f( x ) used in place of x. Total [20 marks]

11 22 M5/5/MATSD/SP2/ENG/TZ/XX/M 6. (a) the temperature of the water cannot fall below room temperature (R) m an (informal) explanation that as m, k 0 (R) recognition that there is a horizontal asymptote at y Note: Award (R) for a contextual reason involving room temperature. Award (R) for a mathematical reason similar to one of the two alternatives. = a (R) (b) 0 00 = 20 + bk ( ) (M) Note: Award (M) for substituting 00, 20 and 0. b = 80 (A)(G2) Note: The (A) is awarded only if all working seen is consistent with the final answer of 80. (c) 84 = k (M) Note: Substituting k =.25 at any stage is an invalid method and is awarded (M0)(M0). Award (M) for correctly substituting 84, 20 and their = k k =.25 (M) (AG) Note: Award (M) for correct rearrangement that isolates k ; k =.25 must be consistent with their working and the conclusion k =.25 must be seen. (d) T 3 = (.25 ) (M) Note: Award (M) for their correct substitutions into T. Follow through from part (b) and k =.25. T = 6.0 (60.96) (A)(ft)(G2) m (e) 35 = (.25 ) (M) Note: Award (M) for their correct substitutions into T. Follow through from part (b). Accept graphical solutions. Award (M) for sketch of function. ( m = ) 7.50( minutes ) ( ) (A)(ft)(G2) 7 minutes and 30 seconds (A) [3 marks] Note: Award the final (A) for correct conversion of their m in minutes to minutes and seconds, but only if answer in minutes is explicitly shown. Total [ marks]

Markscheme May 2015 Mathematical studies Standard level Paper 2

Markscheme May 2015 Mathematical studies Standard level Paper 2 M15/5/MATSD/SP/ENG/TZ1/XX/M Markscheme May 015 Mathematical studies Standard level Paper pages M15/5/MATSD/SP/ENG/TZ1/XX/M This markscheme is the property of the International Baccalaureate and must not