Queensway Sec Prelim 2024 Add Math P1

Calculator Model: _____________ NAME: CLASS: INDEX NO: QUEENSWAY SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 SECONDARY 4 EXP / 5NA ADDITIONAL MATHEMATICS 4049/01 Paper 1 21 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 tape. Answer all the questions. Give non-exact numerical values 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 bracket [ ] at the end of each question or part question. The total number of marks for this paper is 90. This document consists of 15 printed pages and 1 blank page. Setter: Mrs Sheryl Soh [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 Given that 𝑓(𝑥) = 3𝑥5 − 11𝑥3 + 30𝑥2 + 39 = (𝑥 − 1)(𝑥 + 3)𝑄(𝑥) + 𝑎𝑥 + 𝑏 for all values of 𝑥 and that 𝑄(𝑥) is a polynomial, find (a) the values of 𝑎 and of 𝑏. [5] (b) the remainder when 𝑓(𝑥) − 3 is divided by 𝑥2 + 2𝑥 − 3. [2]

4 2 (a) Water is poured into a jar at a rate of 35 cm3/s. The volume of water in the jar is 𝑉 cm3, where 𝑉 = 3 ( ℎ2 4 + 8𝜋 ℎ3) and ℎ is the height of water in the jar. Find the rate at which the height of the water is increasing when ℎ = 4 𝑐𝑚. [3] (b) (i) Find the set of values of 𝑥 for which 𝑦 = 2−5𝑥 𝑒𝑥 is a decreasing function of 𝑥. [3] (ii) Find the gradient of the curve 𝑦 = 2−5𝑥 𝑒𝑥 at the point where it cuts the 𝑥- axis. [2]

5 3 (a) Express 1−3𝑥−3𝑥2 𝑥(𝑥+1)2 in partial fractions. [5]

6 3 (b) Hence, find ∫ 1−3𝑥−3𝑥2 2𝑥(𝑥+1)2 𝑑𝑥. [4] 4 The diagram shows a kite 𝐴𝐵𝐶𝐷 in which 𝐴𝐵 = 𝐴𝐷 and 𝐵𝐶 = 𝐶𝐷, Point 𝐴 and 𝐵 lie on the y-axis and 𝐷 lies on the x-axis. The coordinates of 𝐶 is (4, 8), 𝐵 is (0, ℎ) and 𝐷 is (𝑘, 0), where ℎ and 𝑘 are positive constants. (a) Show that ℎ2 − 𝑘2 = 16ℎ − 8𝑘. [2] 𝐴 𝑦 𝑥 𝐵(0, ℎ) 𝐶 (4, 8) 𝐷 (𝑘, 0) 𝑂

7 (b) It is now given that ℎ = 1. (i) Find the coordinates of 𝐴. [4] (ii) Find the area of the kite. [2]

8 5 In the diagram, angle 𝐴𝐵𝐹 = 𝜃, angle 𝐹𝐵𝐷 = 90°. 𝐴𝐵𝐶 is a straight line and BE is perpendicular to AC. 𝐴𝐵 = 𝐵𝐹 = 7 cm, 𝐵𝐶 = 𝐵𝐷 = 8 cm and 𝐵𝐸 = 5.6 cm. (a) Show that the area 𝑄 cm2 of the quadrilateral 𝐴𝐶𝐷𝐹 is given by 𝑄 = 51.6 cos 𝜃 + 46.9 sin 𝜃. [3] (b) Express 𝑄 = 51.6 cos 𝜃 + 46.9 sin 𝜃 in the form 𝑅 cos(𝜃 − 𝛼), where 𝑅 is a positive constant and 𝛼 is acute. [2] (c) State the maximum value of 𝑄 and find the corresponding value of 𝜃. [2] (d) State the maximum value of 1 𝑄2+3. [1] A B C D E 7 cm 8 cm F 𝜃 8 cm 5.6 cm 7 cm

9 6 An insect leaves its nest and flies along in a straight path. The velocity, 𝑣 m/s, that it flies in time, 𝑡 s, after it leaves the nest is given by 𝑣 = 4𝑒−𝑡 − 1 2 𝑒2𝑡. (a) Find the acceleration when 𝑡 = 0.5. [2] (b) Find the value of 𝑡 when the acceleration is maximum. [3] (c) Show that the insect is instantaneously at rest when 𝑡 = ln 𝑘 where 𝑘 is an integer. [3]

10 (d) Find the total distance travelled by the insect between 𝑡 = 0 and 𝑡 = 3. [4] 7 (a) Prove the identity cot 𝐴−tan 𝐴 cot 𝐴+tan 𝐴 = 2𝑐𝑜𝑠2𝐴 − 1. [4]