2022 CGS AMath Prelims P2

Name: Name: Register No.: Class: CRESCENT GIRLS’ SCHOOL SECONDARY FOUR 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS 4049/02 Paper 2 30 August 2022 Candidates answer on the Question Paper. 2 hours 15 mins No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142, unless the question requires the answer in terms of  . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. Question 1 2 3 4 5 6 7 8 9 10 Marks Table of Penalties Qn. No. Presentation –1 Significant Figures/ Units –1 Parent’s/Guardian’s Signature For Examiner’s Use This document consists of 19 printed pages. [Turn over 90

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= Binomial expansion 1 2 2( ) ... ... 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ! ( )...( 1) !( )! ! n n n n r n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+=  = Cabsin2 1

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 3 1 Show that 1x=− is a solution of the equation 34 7 3 0xx− − = and hence solve the equation completely. [5] 3Let f ( ) 4 7 3x x x= − − f ( 1) 4 7 3 0− =− + − = [M1 – apply factor thm] Hence, by the factor Theorem, 31 is a solution of 4 7 3 0.x x x=− − − = [A1] ( )( ) 2f ( ) 1x x ax bx c = + + + Comparing coefficient of 3 :4xa = Comparing constant term :3c=− Comparing coefficient of 2 :4xb =− [M1, A1] Hence ( )( ) 21 4 4 3 0x x x+ − − = ( )( )( )1 2 1 2 3 0xxx+ + − = 131, , 22x=− − [A1] [Turn over

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 4 2 Show that the equation 1 6 2 3xxe e e+−− = − has only one solution and find its exact value. [5] 1 6 2 3xxe e e+−− = − Let xye= 6. 2 3e y e y− = − [M1, M1 – apply laws of indices] ( ) 2. 6 2 3e y e y− = − ( ) 2. 3 2 6 0e y e y+ − − = [A1 – form correct quad eqn] ( )( )3 2 0ey y+ − = [A1 – correct factorization] 32 orxxee e = = − (NA) Hence x = ln 2 is the only solution. [ A1]

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 5 3(a) Given that 2 31 xy x = + , show that ( ) 3 d 3 2 d 31 yx x x += + . [3] ( ) ( ) 33 1.(2) 2 .d 2 3 1 d 3 1 xxy x xx +− += + [M1, M1 – Quotient rule and chain rule] ( ) ( ) 2 3 1 3 31 31 xx x x +− += + [M1 – simplifying numerator] ( ) 3 32 31 x x += + 3(b) Hence find the value of ( ) 5 1 3 63 d 31 x x x + +  . [5] ( ) ( ) 55 11 33 6 3 3 2 3 1dd 3 1 3 1 x x x xx xx + + + += ++  = ( ) 5 1 3 32 d 31 x x x + +  + ( ) 5 1 3 31 d 31 x x x + +  [M1] = 5 1 2 31 x x   + + ( ) 15 2 1 3 1 dxx − + [M1, M1] = 10 2 42  − + ( ) 5 1 2 1 31 1 (3)2 x   +  [M1] = ( )121 4 223+− = 17 5or 266 [A1] 4 [Turn over A D E

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 6 In the diagram, DFGE is a rectangle and triangle ABC is an isosceles with AB = AC = 5 m. ADB and AEC are straight lines. DE is parallel to BC and AB = 4AD. Angle ABC =  where 0 90   . (a) Show that the perimeter, P metres, of rectangle DFGE is given by ( )5 2cos 3sin .2P =+ [3] By similar triangles, ABC = ADE =  3 15 15 m, sin m.4 4 4BD AB DF = =  = [B1] ( ) 152 5cos m, cos m. 42BC DE BC =  = = [B1] ( )2P DE DF=+ 5 152 cos sin24 =+  ( )5 2cos 3sin2 =+ [A1]

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 7 (b) Express P in the form ( )5 cos2 R − , where R > 0 and write down the largest possible value of P. [3] 222 3 13R= + = [M1] 1 3tan 56.3102 − = =  [M1] ( )( ) 5 13 cos 56.32P = −  [–1 if final answer not given] Largest possible P = 5 13 2 [A1] (c) Find the value of  for which P = 6. [3] ( )( ) 5 13 cos 56.310 62 −  = ( ) 2.4cos 56.310 13  −  = [M1] Basic angle = 1 2.4cos 48.269 13 − = [M1] 56.310 48.269 − =−  8.041 8.0 =   [A1] [Turn over

Crescent Girls’ School 2022 Prelim Sec 4 A Math P2 8 5(a) A student is asked to solve the quadratic inequality ( )( )2 1 3 0xx− −  . The first step