SMSS 2024 AMP2 Preliminary Exam updated

This document consists of 19 printed pages and 1 blank page. ST. MARGARET’S SCHOOL (SECONDARY) Preliminary Examinations 2024 CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS Paper 2 Secondary 4 Express / 5 Normal (Academic) Candidates answer on the Question Paper. 4049/02 20 August 2024 2 hours 15 minutes Additional Materials: NIL READ THESE INSTRUCTIONS FIRST Write your name, registration number and class on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the 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 π, use either your calculator or 3.142, unless the question requires the answer in terms of π. 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.

2 SMS(S) Prelims 2024 [Turn Over . Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion nr r nnnn n b b ar nb anb ana b a + +     + +     +     + = + −−− ......21) ( 2 21 , where n is a positive integer and ! ) 1 ( ... ) 1 ( )! ( ! ! r r n n n r n r n r n + − −=−=      2. TRIGONOMETRY Identities 1cos sin2 2 = +A A AA 22 tan1sec + = AA 22 cot1cosec + = B A B A B Asin co s cos sin)sin( ±= ± B A B A B Asin sin cos cos)cos( = ± B A B AB A tan tan1 tan tan)tan(  ±= ± A A Acos sin2 2sin = AA A A A 222 2 sin2 1 1cos2sin cos2cos − = − = − = A AA 2tan1 tan22tan − = Formulae for ∆ABC, C c B b A a sin sin sin= = A bc c b acos22 2 2− + = C absin2 1= ∆

3 SMS(S) Prelims 2024 [Turn Over 1 The growth in population of bacteria in an experiment is modelled by the equation 80 mtPe= , where P is the population of the bacteria t hours after the start of the experiment. (a) Find the initial number of bacteria when the researcher started the experiment. [1] (b) The population of the bacteria grew to 1600 after 12 hours. Show that m = 0.24964, correct to 5 significant figures. [2] (c) Find the amount of bacteria after 1 day. Round off answer to the nearest whole number. [2] (d) The experiment will end when the bacteria population reaches 1.5 million. A researcher claims that he will complete his experiment in 2 days. Is his claim true? Show your working clearly. [3]

4 SMS(S) Prelims 2024 [Turn Over 2 It is given that 32f ( ) 3 18x x ax bx= + +− , where a and b are constants. 2x− is a factor and leaves a remainder of 51− when divided by 1x+ . (a) Find the values of a and b, [4]

5 SMS(S) Prelims 2024 [Turn Over (b) Explain why f( ) 0x = has only one real root. [3]

6 SMS(S) Prelims 2024 [Turn Over 3 A calculator must not be used in this question. (a) Use the identity for cos 2A to show that 2 23sin 15 4 −°= . [3] (b) Prove that 22sin 15 sin 75 1°+ °= . [5]

7 SMS(S) Prelims 2024 [Turn Over 4 (a) Prove the identity 11 sin 22 tan 2cot 4 θθθ =+ . [4] (b) Hence solve the equation 1 sin2 tan 2cot θθθ =+ , for 02 θπ≤≤ . Explain why there is no real solution for equation. [3]

8 SMS(S) Prelims 2024 [Turn Over 5 (a) Show that the solution of the equation 1264 3xx x −+×= is lg36 lg8x= . [3] (b) Explain why there is no real solution for the equation 22log (4 3) log (2 1) 2xx−− −= . [3]

9 SMS(S) Prelims 2024 [Turn Over (c) Solve the equation 8 1log 2log 16 12 xx = + . [5]

10 SMS(S) Prelims 2024 [Turn Over 6 The equation of a curve is 5 23 xy x = − , xa≠ . (a) State the value of a. [1] (b) Find dy dx . [2]