TJC 8865 2022 Prelim Solutions vetted

1 TEMASEK JUNIOR COLLEGE 2022 JC2 Preliminary Examination Higher 1 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 8865/01 Paper 1 26 August 2022 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. Give non -exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. This document consists of 21 printed pages and 1 blank page. [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 12 Total Marks

2 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2022 8865 Section A: Pure Mathematics [40 marks] Qn 1 Solution For ( ) 22 4 1 0kx x k− + +  for all real values of x, 20k  and Discriminant < 0 0k  and ( )( )16 4 2 1 0kk− +  ( )16 8 1 0kk− +  216 8 8 0kk− −  2 20kk+ −  ( )( )1 2 0kk− +  0k  and 1 or 2kk − Combining, 1k  1 k

3 @TJC 2022 8865 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN Qn 2 Solution (a) 311 2233 1 23 1 23 2 3 2 3 11 2233 ddln 9 2 3 ln 9 2dd 1d3 9 2 d 92 3 3 4 9 12 or 9 2 9 2 x x x xxx xxx xx xx xx x x x x − −     − = −           =−   −  − −= −− (b) ( ) 2 2 3 1 1 2 2 2 5 3 1 2 2 2 23 4 12 9dd 4 12 9 d 8 8 185 x xxxx xx x x x x x x x C − − −+= = − + = − + +  

4 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2022 8865 Qn 3 Solution (i) 222e 1 3exx −−= ( )( ) 2 2 2 2 2 2 2 32e 1 0 e 3Let e 2 1 0 2 3 0 2 3 1 0 3e or e 1 (reject e 0)2 13ln22 x x x x x x uu u uu uu x − − = =  − − = − − = − + = = =−  =  (ii) (iii) 222e 3e 1 0xx −− −  222e 1 3exx −− From (i) and (ii), 13ln22x   (iv) Replace x by 2 p− 13ln2 2 2 32ln ln23 p p −     − =       y x y = y = 0

5 @TJC 2022 8865 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN Qn 4 Solution (i) 2 d d ( 5) ya xx= − At 2x= , equation of tangent is 2 16 33yx=− + . (gradient is 2 3− ) 2 2 6(2 5) 3 a a=−  =−− ( ) 2 6 5 dy x x − =− − ( ) 1165 1 xC −=− − +− 6 5 Cx=+ − At 2x= , 4y= 64 3 C = +−  6C= Equation of curve C: 61 6 6 155y xx = + = + −−  . (Shown) (ii) (iii) Equation of line which passes through origin and P: 2yx= Sketch line (see ii) (iv) Exact area of the curve = Area of triangle + Area under C 1 242=   + 4 2 6 6 d5 xx  + − 4 2 4 6ln 5 6xx= + − +   4 6ln1 24 6ln 3 12= + + − − 16 6ln3=− 2yx= y x = x y = 6 C

6 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2022 8865 Qn 5 Solution (i) Let n represent cost price of a nineth scale War-Robot Let s represent cost price of a sixth scale War-Robot Let q represent cost price of a quarter scale War-Robot 20 4 30 ..............(1)ns−= 30 16 6 8517 ..............(2)n s q+ + = 1.2 0.8 661.5 ..........(3)n s q+ + = By GC, Cost price for a nineth scale War-Robot is $57.00 Cost price for a sixth scale War-Robot is $277.50 Cost price for a quarter scale War-Robot is $394.50 (ii) ln(3 60) 1120 wCw= − − + d 1 3 1 1 d 20 3 60 20 20 C w w w= − = − −− d 1 1For 0d 20 20 11 20 20 20 20 40 C wW w w w = − = − = − −= = Method 1 ( ) 2 22 d1 0 for 20d 20 C ww w =   − Method 2 w 39 40 41 d d C w −0.00263<0 0 0.00238>0 Shape of graph C is minimum when w = 40 (iii) Estimated minimum cost is $8905.66 (iv) 10.42 0.42 ln(3 60) 11 20 CwPw ww = − = − − − +  From GC, the manufacturer should only accept order with at least 24 sixth scale War-Robots. (40, 8.91) w C w=20

7 @TJC 2022 8865 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN Section B: Probability and Statistics [60 marks] Qn 6 Solution (i) Number of different committees can be formed 14 9 5 5 5 5 1875      = − −           = (ii) Number of ways 33! 2! 2 36 =   =

8 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2022 8865 Qn 7 Solution (i) ( )P AB represents the probability that B occurs and A does not occur. ( ) ( )PP 0.3 P( ) P( ) 0.3 P( ) 0.3 0.4 0.12 ABAB B A B B ==  =  =  = P( ) P( ) P( ) 0.4 0.12 0.28A B B A B = −  = − = (ii) P( ) P( ) P( ) 0.8 0.28 0.52 A A B A B =  −  =− = Or P( ) P( ) P( ) P( ) 0.8 P( ) 0.4 0.12 P( ) 0.52 A B A B A B A A  = + −   = + − = (iii) (iv) Since P(A) ∙P(B) = 0.52  0.4 = 0.208 ≠ 0.12 = P(AB), A and B are not independent. 0.12 0.28 0.52 − 0.12 = 0.4 A B 0.2

9 @TJC 2022 8865 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN Qn 8 Solution (i) ~ B(60, )Xp ( ) ( ) 0 60 1 59 60 59 P( 1) = P 0 P 1 0.61 60 60 (1 ) (1 ) 0.6101 (1 ) 60 (1 ) 0.61 X X X p p p p p p p  = + = =     − + − =       − + − = Using GC, 0.0224p= (3 s.f.) (ii) ~ B(60,0