SPEC/5/MATME/SP2/ENG/TZ0/XX/M MARKSCHEME SPECIMEN MATHEMATICS Standard Level Paper 2 15 pages В±5В± SPEC/5/MATME/SP2/ENG/TZ0/XX/M SE C T I O N A 1. (a) attempt to substitute into sum formula for AP 20 20 e.g. S20 2(7) 19d , (7 u20 ) 2 2 M1 setting up correct equation using sum formula A1 e.g. d (b) 20 2(7) 19d 620 2 4 correct substitution 7 77(4) u78 301 A1 A1 A1 N2 [3 marks] N2 [2 marks] Total [5 marks] В±6В± 2. (a) SPEC/5/MATME/SP2/ENG/TZ0/XX/M M1 intercepts when f ( x) 0 A1A1 (0.827, 0) (4.78, 0) (accept x 0.827 x 4.78) N3 [3 marks] (b) A1A1A1 Note: Award A1 for maximum point in circle, A1 for x-intercepts in circles, A1 for correct shape (y approximately greater than 3.14 ). (c) gradient is 1.28 N3 [3 marks] A1 N1 [1 mark] Total [7 marks] В±7В± 3. SPEC/5/MATME/SP2/ENG/TZ0/XX/M (a) A1A1A1 N3 Note: Award A1 for 450 , A1 for a to the right of the mean, A1 for area 0.27 . [3 marks] (b) valid approach e.g. P(X a) 1 P(X ! a), 0.73 M1 a 462.256} A1 a 462 A1 N3 [3 marks] Total [6 marks] 4. (a) L1 A1 N1 [1 mark] (b) В§1В· ВЁ Вё ВЁ 2Вё ВЁ 3Вё В© В№ A1 N1 [1 mark] (c) choosing correct direction vectors В§ 3 В· В§ 1 В· e.g. ВЁВЁ 2 ВёВё , ВЁВЁ 2 ВёВё ВЁ 1 Вё ВЁ a Вё В© В№ В© В№ A1A1 recognizing that a b 0 correct substitution e.g. 3 4 a 0 M1 A1 a 7 A1 N3 [5 marks] Total [7 marks] В±8В± 5. (a) (i) (ii) (b) SPEC/5/MATME/SP2/ENG/TZ0/XX/M recognizing binomial with n 600, p 0.4 E( X ) 240 M1 A1 correct substitution into formula for variance or standard deviation e.g. 144, 600 u 0.4 u 0.6 A1 sd 12 A1 attempt to find range of values e.g. 240 r 12 228 X 252 M1 evidence of correct approach e.g. P( X d 251) P( X d 228) A1 P 228 X 252 0.662 A1 N2 N1 [4 marks] N2 [3 marks] Total [7 marks] В±9В± 6. SPEC/5/MATME/SP2/ENG/TZ0/XX/M METHOD 1 appropriate approach e.g. completed diagram M1 attempt at set up e.g. correct placement of one angle A1 h h 1.6 , tan 35 x x attempt to set up equation e.g. isolate x A1A1 tan 30 M1 correct equation h h 1.6 e.g. tan 30 tan 35 A1 h 7.52 A1 N3 [7 marks] METHOD 2 sin 30 h l in triangle ATB, A 5 , T choosing sine rule correct substitution h / sin 30 1.6 e.g. sin 55 sin 5 1.6 u sin 30 u sin 55 sin 5 h 7.52 h A1 55 A1A1 M1 A1 A1 A1 N3 [7 marks] В± 10 В± 7. (a) (b) SPEC/5/MATME/SP2/ENG/TZ0/XX/M substitution into formula for area of triangle 1 e.g. r u r sin T 2 A1 evidence of subtraction M1 correct expression 1 1 1 e.g. r 2T r 2 sin T , r 2 (T sin T ) 2 2 2 A1 evidence of recognizing that shaded area is e.g. N2 [3 marks] 1 of area of circle 8 M1 1 seen anywhere 8 A1 setting up correct equation 1 1 2 e.g. r 2 (T sin T ) Sr 2 8 M1 eliminating 1 variable 1 1 e.g. (T sin T ) S , T sin T 2 8 S 4 M1 attempt to solve e.g. a sketch, writing sin x x T 1.77 (do not accept degrees) S 4 0 A1 N1 [5 marks] Total [8 marks] В± 11 В± SPEC/5/MATME/SP2/ENG/TZ0/XX/M SE C T I O N B 8. A1A1 N2 [2 marks] additional cost per box (unit cost) A1 N1 fixed costs A1 N1 [2 marks] (a) y 10.7 x 121 (b) (i) (ii) (c) (d) attempt to substitute into regression equation e.g. y 10.7 u 60 121, y 760.12} M1 cost A1 $760 (accept $763 from 3 s.f. values) setting up inequality (accept equation) e.g. 19.99 x ! 10.7 x 121 M1 x ! 12.94} A1 13 boxes (accept 14 from x ! 13.02, using 3 s.f. values) A1 Note: Exception to the F T rule: if working shown, award the final A1 for a correct integer solution for their value of x. (e) N2 [2 marks] N2 [3 marks] (i) this would be extrapolation, not appropriate R1R1 N2 (ii) this regression line cannot predict x from y, not appropriate R1R1 N2 [4 marks] Total [13 marks] В± 12 В± 9. (a) y 2x 1 x 1 interchanging x and y (seen anywhere) 2 y 1 e.g. x y 1 M1 correct working e.g . xy x 2 y 1 A1 collecting terms e.g. x 1 2 y xy , x 1 y (2 x) A1 x 1 2 x h1 ( x) (b) SPEC/5/MATME/SP2/ENG/TZ0/XX/M A1 N2 [4 marks] (i) A1A1A1A1 N4 A1A1 N2 A1 N1 Note: Award A1 for approximately correct intercepts, A1 for correct shape, A1 for asymptotes, A1 for approximately correct domain and range. (ii) x 1, y 2 (iii) 1 2 [7 marks] FRQWLQXHGВ« В± 13 В± SPEC/5/MATME/SP2/ENG/TZ0/XX/M Question 9 continued (c) A2 (i) area 2.06 (ii) attempt to substitute into volume formula do not accept SВі y 2dx a M1 2 В§ 2x 1 В· volume SВі 1 ВЁ Вё dx 2 В© x 1 В№ 3 b N2 A2 N3 [5 marks] Total [16 marks] В± 14 В± 10. (a) (i) 106 m (ii) substitute t h 44.9 m (iii) (b) 4.5 SPEC/5/MATME/SP2/ENG/TZ0/XX/M A1 N1 M1 A1 N2 set up suitable equation e.g. f (t ) 30 M1 t A1 4.91 recognising that height is 0 set up suitable equation e.g. g (t ) 0 A1 M1 t 5.39secs A1 N1 [5 marks] N2 [3 marks] FRQWLQXHGВ« В± 15 В± SPEC/5/MATME/SP2/ENG/TZ0/XX/M Question 10 continued (c) (i) A1A2 N3 Note: Award A1 for correct scales on axes, A2 for 5 correct points, A1 for 3 or 4 correct points. (ii) -DQHВ¶VIXQFWLRQZLWK2 valid reasons A1R1R1 e.g. -DQHВ¶VSDVVHVYHU\close to all the points, .HYLQВ¶VKDVWKHUock clearly going up initially В± not possible if rock falls Note: $OWKRXJK-DQHВ¶VDOVRJRHVXSLQLWLDOO\LWRQO\JRHVXSYHU\VOLJKWO\DQG so is the better model. N3 [6 marks] Total [14 marks]