CBSS Prelim 2024 4E AM P2 MS

CANBERRA SECONDARY SCHOOL 2024 Preliminary Examination Secondary Four Express ADDITIONAL MATHEMATICS 26 Aug 2024 4049/02 2 hours 15 minutes 1130h — 1345h Name: ______________________________________ ( ) Class: ________ READ THESE INSTRUCTIONS FIRST Write your full name, class and index number on all work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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. The use of an approved scientific calculator is expected, where appropriate. 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 90. _________________________________________________________________________________ This question paper consists of 22 printed pages including the cover page. Setter: Mr Muhamad Lathif FOR MARKER’S USE Marks Awarded Max Marks Total 90 O

2 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ,0,02 =++ acbxax .2 42 a acbbx −−= Binomial expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21)( 221 , where n is a positive integer and ! )1)...(1( !!)( ! r rnnn rrn n r n +−−=−=      . 2. TRIGONOMETRY Identities sin² A + cos² A = 1 sec² A = 1 + tan² A cosec² A = 1 + cot² A sin (A  B) = sin A cos B  cos A sin B cos (A  B) = cos A cos B  sin A sin B BA BABA tantan1 tantan)tan(  = sin 2A = 2sin Acos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A tan 2A = A A 2tan1 tan2 − Formulae for ABC C c B b A a sinsinsin == . a² = b² + c² − 2bc cos A.  = Cab sin2 1

3 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express Answer all the questions 1 A curve has an equation 2 2 1.y x x=+ (a) Show that 21 dy ax b dx x += + , where a and b are positive integers. [2] 2ux= ( ) 1 22 1 2 1v x x= + = + 2du dx = ( ) 1 2 112 1 (2)2 21 dv xdx x − = + = + 2 2 2 1 21 dy x xdx x = + + + M1 2 4 2 6 2 2 1 2 1 x x x xx + + +== ++ A1 (b) Hence, find 31 . 21 x dx x + + [2] 62 2 2 1 21 x dx x x c x + = + + + M1 312 2 2 1 21 x dx x x c x + = + + + 31 2 1 21 x dx x x c x + = + + + A1 [Turn Over

4 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express 2 A calculator must not be used in this question. (a) Show that 62sin15 4 o −= . [4] ( )sin15 sin 45 30o o o=− sin 45 cos30 cos 45 sin 30o o o o=− M1 1 3 1 1 2222  =−    M1 31 22 −= 3 1 2 2 2 2 −= M1 62 4 −= A1

5 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express (b) Hence, find 2sin 15 o , giving your answer in the form 3a b − , where a and b are positive integers. [2] 2 2 62sin 15 4 o  −= M1 6 2 6 2 44   −−=       6 2 12 2 16 −+= 8 2 12 16 −= 23 4 −= A1 (c) By using part (b), show that 2 23cos 15 . 4 o += [2] 22sin 15 cos 15 1oo+= M1 223 cos 15 14 o− += 2 23cos 15 1 4 o −=− 4 2 3 2 3 4 4 4 −+= − = A1 [Turn Over

6 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express 3 (a) Prove that cos 2 cot 2 cotec x x x+= . [4] cos 2 cot 2LHS ec x x=+ 1 cos 2 sin 2 sin 2 x xx=+ M1 1 cos 2 sin 2 x x += 21 2cos 1 2sin cos x xx +−= M1 cos sin x x= M1 cot x RHS== (Proved) A1

7 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express (b) Hence, solve cos cot 3ecx x+= where 20  x . [3] cos cot 3ecx x+= 1cot 32 x= M1 11tan 2 3 x= Basic angle 1 1tan 3 − =  6 = M1 Tangent is positive, 1st and 3rd quadrant. 17 ,2 6 6x = 3x = A1 [Turn Over

8 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express 4 (a) Solve the equation 23 3 8xx −−= . [4] 23 3 8xx −−= 2338 3 x x−= M1 23 8(3 ) 9 0xx− − = Let 3x X= 2 8 9 0XX− − = ( )( )1 9 0XX+ − = M1 31x =− (NA) or 39x = M1 2x= A1 (b) The curve ( )5log 2 5yx=+ intersects the x-axis at A and the y-axis at B. (i) Find the coordinates of A and B. [3] Cuts x-axis, y-coordinate = 0 ( )5log 2 5 0x+= 2 5 1x+= M1 24x=− 2x=− A(-2 , 0) A1 Cuts y-axis, x-coordinate = 0 ( )5log 2(0) 5y=+ 5log 5 1y== B(0 , 1) B1

9 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express (ii) Explain why the graph does not exist for all values of 5 2x− . [2] ( )5log 2 5yx=+ does not exist for 2 5 0x+ M1 25x− 5 2x− A1

10 Canberra Secondary School Additional Mathematics 4049/02 2024 Preliminary Examination Secondary 4 Express 5 A compound produced in the laboratory has a growth equation given by 0.5 600 10 30 tw e−= + where w is the mass in grams and t is the time in hours after the compound was first produced. Find (ai) the initial mass of the compound, [2] 0.5 600 10 30 tw e−= + Initial mass when t = 0 M1 0.5(0) 600 600 1510 30 10 30wg e−= = =++ A1 (aii) the mass of the compound after 5 hours 15 minutes, [2] 5h 15 min = 5.25 h M1 0.5(5.