Queensway Sec Prelim 2024 Add Math P2

1 Calculator Model: ________________ NAME: CLASS: INDEX NO: QUEENSWAY SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 SECONDARY 4 EXPRESS / SECONDARY 5 NORMAL ACADEMIC ADDITIONAL MATHEMATICS 4049/02 Paper 2 27 August 2024 2 hour 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces at the top of this page. 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 questions. The number of marks is given in brackets [ ] at the end of each question or part question. 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 total of the marks for this paper is 90. 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. Gives answers in degrees to one decimal place. For π, use either your calculator value or 3.142. This document consists of 18 printed pages. Setter: Ms Chen Zhiyun [Turn over Parent’s Signature:

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0 , a acbbx 2 42 −−= Binomial expansion (𝑎 + 𝑏)𝑛 = 𝑎𝑛 + (𝑛 1) 𝑎𝑛−1𝑏 + (𝑛 2) 𝑎𝑛−2𝑏2 + ⋯ + (𝑛 𝑟) 𝑎𝑛−𝑟𝑏𝑟 + ⋯ + 𝑏𝑛, where n is a positive integer and ( ) !! ! rrn n r n −=      = ! )1)...(1( r rnnn +−− 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin(𝐴 ± 𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵 cos(𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵 tan(𝐴 ± 𝐵) = tan 𝐴 ± tan 𝐵 1 ∓ tan 𝐴 tan 𝐵 sin 2𝐴 = 2 sin 𝐴 cos 𝐴 cos 2𝐴 = 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2𝐴 = 2𝑐𝑜𝑠2𝐴 − 1 = 1 − 2𝑠𝑖𝑛2𝐴 A AA 2tan1 tan22tan −= Formulae for  ABC C c B b A a sinsinsin == a 2 = b 2 + c 2 − 2bc cos A  = 2 1 bc sin A

3 1 The Singapore government issued a savings bond in January 2024 with a yield of 2.75% per year. Mr Tan invested $15 000 in the bond. The total amount he will receive, after t years, is given by 𝐴 = 15000(1.0275)𝑡. (a) Calculate the total amount he will receive in January 2030, correct to the nearest dollar. [2] (b) In which year will the amount first exceed $22 000? [2]

4 2 (a) Without using a calculator, evaluate the value of 6𝑥 given that 22𝑥+6 × 35𝑥−1 = 27𝑥+1. [5] (b) Solve the equation 𝑙𝑜𝑔𝑥9 = 5𝑙𝑜𝑔3𝑥, giving your answers correct to 2 significant figures. [4]

5 3 (a) A curve has the equation 𝑦 = (𝑝 − 1)𝑥2 + 2(𝑝 − 3), where p is a constant. A line has the equation 𝑦 = 6𝑥 + 3. Find the range of values of p if the curve lies completely above the line. [5] (b) By expressing 𝑦 = −2𝑥2 + 10𝑥 − 5 in the form 𝑦 = 𝑎(𝑥 + 𝑏)2 + 𝑐, where a, b and c are constants, find the maximum value of y. [3]

6 4 (a) In the binomial expansion of (1 − 2 7 𝑥) 𝑛 , the sum of the coefficients of the second and third terms is zero. Calculate the value of 𝑛 and hence, find the sixth term. [4]

7 (b) Write down the general term in the binomial expansion of ( 1 𝑥3 − 2𝑥) 8 . Hence, find the value of the constant term in the expansion of (3 + 𝑥2 2 ) 2 ( 1 𝑥3 − 2𝑥) 8 . [6]

8 5 (a) The function f is defined as 𝑓(𝑥) = 𝑝 − 𝑞 sin(𝑟𝑥), for −𝜋 ≤ 𝑥 ≤ 𝜋, where p, q and r are positive integers. Given that the amplitude of the function is 6, the period is  and the maximum value is 9. (i) State the values of p, q and r. [3] (ii) Hence, sketch the graph of 𝑓(𝑥). [2]

9 (b) The acute angles A and B are such that cot(𝐴 − 𝐵) = 1 3 and cot 𝐴 = 1 5. Without using a calculator, find the exact value of cos 𝐵. [5]

10 6 A circle passes through the points (−5, 12) and (9, 14). The centre of the circle lies on the line 2𝑦 + 𝑥 = 15. (a) Find the equation of the circle. [7]