[FFM] [2023] 4E AM 4049 Prelims P1 QP

NAME: ____________________________ ( ) CLASS: _______ ADDITIONAL MATHEMATICS 4049/01 Paper 1 Date: 24 August 2023 Duration: 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. ____________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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. For Examiner’s Use Table of Penalties Question Number 90 Presentation  1  2 Rounding off  1 Parent’s/Guardian’s Signature Setter: Mdm Haliza This question paper consists of 21 printed pages including the cover page. FAIRFIELD METHODIST SCHOOL (SECONDARY) PRELIMINARY EXAMINATION 2023 SECONDARY 4 EXPRESS

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 2 Additional Mathematics Paper 1 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= Binomial expansion ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21 221 , where n is a positive integer and ! )1)...(1( )!(! ! r rnnn rnr n r n +−−=−=      2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += cosec 2 A = 1 + cot 2 A BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= Cabsin2 1=

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 3 Additional Mathematics Paper 1 Answer all the questions. 1 Without using a calculator, find the exact value of (i) cosec , given that  is acute and 3cos 4 = , [1] (ii) cos30 (tan 45 sin 60 ) +  . [2]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 4 Additional Mathematics Paper 1 2 Find the set of values of a for which the curve 227yx=+ lies entirely above the line 3y ax=− . [4]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 5 Additional Mathematics Paper 1 3 The equation of a curve is 11 22xx y Ae Be − =+ , and that 11 223 252 xxdy y e edx − + = − . Find the value of each of the constants A and B. [5]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 6 Additional Mathematics Paper 1 4 Water is leaking from a container at a rate of 8 cm3/s. If the volume of water, V cm3, in the container is given by ( ) 23 82V h h=+ where h is the depth of the water , in cm , remaining in the container , find the rate of change of h when the volume of water is 13.5 cm3. [5]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 7 Additional Mathematics Paper 1 5 A certain radioactive material, radium -226, decomposes according to the formula 0 ktA A e= where A is the remaining mass in grams, after decomposition, A0 is the original mass in grams, t is the time in years and k is a constant. A radioactive substance is often described in terms of its half-life, which is the time required for half the material to decompose. (i) Given that after 400 years, a sample of radium-226 has decayed to 84.1% of its original mass, show that 0.000433k =− , rounded off to 3 significant figures . [2] (ii) Hence, find the half-life of radium-226, to the nearest whole number. [2] (iii) If a sample of radium -226 has an initial mass of 100 grams, what is the remaining mass after 3200 years? [2]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 8 Additional Mathematics Paper 1 6 In the diagram, A, B, C and D are points on the circle. The tangent at A meets CD produced at R. The chords AC and BD intersect at S. The line BSD bisects angle ABC. Prove that (i) DAR CAD = , [3] (ii) RAD is similar to RCA , [2] (iii) 2RA RC RD= . [1] A R B C D S

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 9 Additional Mathematics Paper 1 7 (i) Explain why there is no constant term in the expansion of 8 2 3 1      − x x . [2] (ii) Show that the coefficient of 6x − in the expansion of ( ) 8 35 2 1 1xx x  −+ is 20. [4]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 10 Additional Mathematics Paper 1 8 The equation of a curve is 22 4 9y x x= − + . (i) By expressing 22 4 9xx−+ in the form 2()a x b c++ , where a, b and c are constants, find the coordinates of the stationary point on the curve. [2]