HL Classwork_5-4
HL MATH – Monday, 5/4/15 –3– Spring Break: Paper 1, problems 2, 3, 7 (no calculator) 2. [Maximum mark: 7] SPEC/5/MATHL/HP1/ENG/TZ0/XX 9 x 3 45 x 2 74 x 40 0 . [1 mark] , 3. ± –4– SPEC/5/MATHL/HP1/ENG/TZ0/XX [6 marks] [Maximum mark: 6] ....................................................................... ....................................................................... ....................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .[3 . . marks] . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .–. 8. .–. . . . . . . . . . . .SPEC/5/MATHL/HP1/ENG/TZ0/XX ......................... [3 marks] ....................................................................... [Maximum mark: 8] ....................................................................... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ..1.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .x. ..+.. y.. ..+.. z.. .= . ..3.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2.. ..x..+.. 3.. y.. ..+.. z.. .= x 3y z .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..!.. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . [4 . . .marks] . ....................................................................... ....................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [4 . . .marks] . ....................................................................... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....................................................................... ....................................................................... ....................................................................... ....................................................................... ....................................................................... HL MATH – Monday, 5/4/15 –8– Spring Break: Paper 2, problems 6 (with calculator) 6. [Maximum mark: 6] f is of the form f ( x) x 4 and y x+a ,x bx + c c b f has 2 ,1 3 2 –9– a , b and c . 7. SPEC/5/MATHL/HP2/ENG/TZ0/XX SPEC/5/MATHL/HP2/ENG/TZ0/XX [Maximum mark: 9] ............................................................................ 1 with . . . .a.constant . . . . . . .velocity . . . . . . .of . . . . . . . . . . . . . . . . . .1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . it moves with a constant velocity of in a direction ............................................................................ ............................................................................ – 13 – SPEC/5/MATHL/HP2/ENG/TZ0/XX[4 marks] ............................................................................ Do NOT. .write . . . .solutions . . . . . . .on . . this . . . page. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .[5 . . marks] .. 11. . . . . . . . .mark: . . . . .13] ............................................................... [Maximum ............................................................................ . . . . . . . . . . . . . .$P ......................................................... interest . . . . . rate . . . .of. .I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$R ............................................ . . $. S. n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n. th. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Let ....................................................................... ............................................................................ . . . . . . . . . . . . . . . . . . . . .S.1 .and . . .show . . . . that ......................................... ............................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . I. . .2 . . . . . . . . . . .I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S2 P 1 R 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . .100 . . . . . . . . . . . . . .100 ................................ ....................................................................... S n . Hence show that ....................................................................... n n . . . . . . . . . . . . . . . S. n. . .P. .1. . . I. . . . . 100 . . . R. . . 1. . . .I . . . . .1. .. . . . . . . . . . . . . . . .[7 . .marks] ........ 100 I 100 ....................................................................... ....................................................................... in 5 years (i.e. 60 months). ....................................................................... [6 marks] & ! "2# 3 [3 marks] HL MATH – Monday, 5/4/15 Solutions 2. (a) sum " Total [6 marks] 45 40 , product " 9 9 A1 [1 mark] (b) it follows that 3! " solving, ! " 45 40 and ! (! 2 # " 2 ) " 9 9 5 3 A1 5 $ 25 40 2% & #" !" 3" 9 # 9 1 " " ( %) 3 the other two roots are 2, A1A1 M1 A1 4 3 A1 [6 marks] Total [7 marks] –6– SPEC/5/MATHL/HP1/ENG/TZ0/XX/M HL MATH – Monday, 5/4/15 3. (a) P (no heads from n coins tossed) ! 0.5n 1 1 1 1 1 1 P (no head) ! " # " # " 3 2 3 4 3 8 7 ! 24 (A1) M1 A1 [3 marks] (b) P (2 coins and no heads) P (no heads) 1 ! 12 7 24 2 ! 7 P (2 | no heads) ! M1 A1 A1 [3 marks] Total [6 marks] 4. (a) 1 E ( X ) ! % 12 x 3 (1 $ x)dx M1 0 1 & x 4 x5 ' ! 12 ( $ ) * 4 5 +0 ! A1 3 5 A1 [3 marks] (b) f !( x) ! 12(2 x $ 3x 2 ) at the mode f !( x) ! 12(2 x $ 3 x 2 ) ! 0 therefore the mode ! 2 3 A1 M1 A1 [3 marks] Total [6 marks] 5. (a) f ($ x) ! 2cos ( $ x) # ($ x)sin ($ x) ! 2cos x # x sin x therefore f is even "! f ( x) # M1 A1 A1 [3 marks] (b) f !( x) ! $2sin x # sin x # x cos x (! $ sin x # x cos x) f !!( x) ! $ cos x # cos x $ x sin x (! $ x sin x) so f !!(0) ! 0 A1 A1 AG [2 marks] continued !! [4 marks] Total [7 marks] HL MATH – Monday, 5/4/15 7. (a) using row operations, M1 to obtain 2 equations in the same 2 variables A1A1 for example y + z " 1 2 y + 2z " ! +1 the fact that one of the left hand sides is a multiple of the other left hand side indicates that the equations do not have a unique solution, or equivalent R1AG [4 marks] (b) (i) ! "3 A1 (ii) put z " " then y " 1 # " and x " +2" or equivalent M1 A1 A1 [4 marks] Total [8 marks] –6– 5. (a) (i) 3 displacement ! " v dt 0 ! 0.703 (m) (ii) 3 total distance ! " v dt 0 ! 2.05 (m) SPEC/5/MATHL/HP2/ENG/TZ0/XX/M (M1) A1 (M1) A1 [4 marks] (b) solving the equation t 0 cos (u 2 ) du ! 1 (M1) " t ! 1.39 (s) A1 [2 marks] Total [6 marks] 6. vertical asymptote x ! #4 $ #4b % c ! 0 1 horizontal asymptote y ! #2 $ ! #2 b 1 b ! # and c ! #2 2 2 %a 1! 3 1 2 # & #2 2 3 a ! #3 M1 M1 A1A1 M1 A1 [6 marks] HL MATH – Monday, 5/4/15 –7– SPEC/5/MATHL/HP2/ENG/TZ0/XX/M 7. (a) let the interception occur at the point P, t hrs after 12:00 then, SP ! 20t and MP ! 30t using the sine rule, SP 2 sin ! ! ! MP 3 sin135 whence ! ! 28.1 A1 M1A1 A1 [4 marks] (b) using the sine rule again, MP sin135 ! MS sin (45 " 28.1255...) sin135 30t ! 10 # sin16.8745... t ! 0.81199... the interception occurs at 12:49 M1A1 M1 A1 A1 [5 marks] Total [9 marks] – 10 – SPEC/5/MATHL/HP2/ENG/TZ0/XX/M HL MATH – Monday, 5/4/15 11. (a) (i) I " ! S1 # P &1 $ !%R " 100 # A1 2 I " I " ! ! S2 # P &1 $ ! % R &1 $ !%R " 100 # " 100 # M1A1 2 I " ! ! I "" ! # P &1 $ ! % R &1 $ &1 $ !! " 100 # " " 100 # # (ii) AG extending this, n n ! ! I " I " I " ! ! Sn # P &1 $ ! % R && 1 $ & 1 $ ! $ ... $ & 1 $ ! " 100 # " 100 # " " 100 # n !! " I " R 1 $ % 1! & & ! n & " 100 # ! I " ! " # # P &1 $ ! % I " 100 # 100 n n " I " 100 R ! ! I " ! % 1 $ % 1! = P &1 $ & ! & ! ! I &" " 100 # " 100 # # 1 " !! # M1A1 M1A1 AG [7 marks] (b) (i) M1 putting S60 # 0, P # 5000, I # 1 60 60 A1 A1 5000 $ 1.01 # 100 R (1.01 % 1) R # ($)111.22 (ii) putting n # 20, P # 5000, I # 1, R # 111.22 20 20 S20 # 5000 $ 1.01 % 100 $ 111.22(1.01 % 1) # ($)3652 which is the outstanding amount M1 A1 A1 [6 marks] Total [13 marks] HL MATH – Monday, 5/4/15
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