ACJC_8865_2022_Prelim_Solution_vetted

2022 JC2 H1 Math Prelim Marking Scheme Qn Solution 1 Let the amount invested in Banks A, B and C be x, y and z respectively. We have 25000 (1) 0.06 0.07 0.08 1620 (2) 6000 (3) x y z x y z yz + + = −−− + + = −−−− − = −− By GC, 15000, 8000, 2000x y z= = = .  the amount invested in Banks A, B and C are $ 15000, $ 8000 and $ 2000 respectively. 2(a) 2 31 22 5 2 31 d 2 31 d 22 3 5 x x x x x x x x c − + =+ = + +   where c is an arbitrary constant 2(b) ( ) ( ) 2 2 2 d 4 3lnd 1 d1 ln 4 3 ln 1d2 22 43 1 x x x xxx x x x  − − = − − − =− − − 3i ( )( ) 2 2 2 2 2 41 410 41 4 40 2 2 0 2 or 2 (rejected since 0) dy dx x x x x x xx xx x =− −   − − +  −  

y=x2+5 3ii 3iii ( ) 32 22 2 2 54 54 since is positive, 4 5 Add 5 to , intersection pt is (0.842,5.6 8) 0 x 0.824 x x x x x x x xx x y x C +  + +  + +  + =+    4(i) 1.5x=− y (0,5) (-0.75,0) x 4(ii) ( )( ) ( ) 2 1510 5 23 20 30 15 523 20 15 5 2 3 2 3 10 0 mxx x mxx x mx x mx m x − = ++ +− =++ + = + + + − = discriminant > 0 ( ) 2 3 10 0 10, ,0 3 m mm −  y=x x y o o 1510 23y x=− + 10y=

4(iii) 0 3 4 0 3 4 15 1 510 d 52 3 2 2 15 2510 ln | 2 3 |24 15 15 15 3 250 ln 3 ln2 2 2 2 4 15 15 15 15 25ln 3 ln 3 ln 22 2 2 2 4 15 55ln 224 1 55 30ln 24 55, 30 xx xx pq 5i 220 5 0.01 500C x x x= − + + 5ii ( ), (0, 420) and (45,60) 420 60 80 45 sub (0, 420) and 8 into straight line: 420 8(0) 420 8 420 xS m m S mx c c c Sx = −= =−− =− =+ =− + =  =− + 5iii Revenue = ( )8 420xx−+ Profit = revenue - cost 22 2 = 8 420 20 5 0.01 500 = 8.01 400 500 5 500 x x x x x x x x a − + − + − − − + − +  =− 5iv 32 2 2 d5 400 16.02d 2 d 0d 5 400 16.02 0 2 using GC graph, find -intercept 25 8(25) 420 220 million dollars =$220000,000 Method 1: d5 16.02 16.03 0 is maximum.d4 P xx x P x x x xx S P xPx − = + − = + − = =  =− + = =− − =−   Method 2:

x 24.99 25 25.01 d d P x 0.160 0 -0.160 shape is maximum.P 5v when 40, d 240.40 = 240 million dollars/plane d When the production level increased from 40 to 41 planes, the profit decreases by approximately $240 million dollars. The production level should be x P x = =− − decreased in this case. 6(i) The probability of an employee is working from home is constant at 0.37. Whether an employee is working from home is independent of any other employee. (ii) Let X be the number of employees from the 30 who are working from home. ( )30,0.37XB ( ) ( )P 18 1 P 18 = − XX = 0.0030511= 0.00305 (3sf) (iii) ( )30,0.37XB Mean = 30 0.37 11.1 (exact)= Variance = ( )30 0.37 1 0.37 6.993 (exact)  − = 6(iv) Method 1: Probability of a company getting the monetary incentive = ( )P 18 0.0030511=X Answer: 0.0030511  (1 – 0.003511)  2 = 0.00608 Method 2: Let Y be the no.