2022 NCHS Prelim P1

Name: Register Number: Class: PRELIMINARY EXAMINATION 2022 SECONDARY FOUR EXPRESS ADDITIONAL MATHEMATICS Paper 1 Candidates answer on the Question Paper. No Additional Materials are required. 4049/01 24 August 2022, Wednesday 2 hours 15 minutes 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 a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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. This document consists of 22 printed pages including the cover page. For Marker’s Use 90 Parents’ signature: ________________

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 0 2 =++ cbxax , a acbbx 2 4 2 −−= Binomial expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−−  221 21)( , where n is a positive integer and ! )1)...(1( !)!( ! r rnnn rrn n r n +−−=−=      2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2 tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2 222 −+= Abc sin2 1=

3 1 In triangle 𝐴𝐵𝐶, the lengths of 𝐴𝐵 and 𝐵𝐶 are (√3 + √5) cm and (4√3 − 2√5) cm respectively. Given that angle 𝐴𝐵𝐶 is 120°, without using a calculator, find the value of 𝐴𝐶2. Leave your answer in the form of (𝑎 + 𝑏√15), where a and b are integers. [3]

4 2 A section of roller coaster track is parabolic in shape. The height, ℎ m, of the first carriage above the ground at time 𝑡 seconds is given by ℎ = 2𝑡2 − 36𝑡 + 164. (a) Explain why this carriage cannot go below the height of 2 m. [2] (b) Find the total duration for the carriage to be below 20 m for this section of the ride. [2]

5 3 (a) Without using a calculator, show that cot 75° = 2 − √3 . [3] (b) Hence, show that cosec2 75° = 4 cot 75°. [2]

6 4 It is given that f(𝑥) is the equation of the curve such that f ′(𝑥) = 5 𝑥2 + cos2 𝑥 + tan2 2𝑥. Given that f ( 𝜋 2) = − 𝜋 4, find the equation of the curve f(𝑥). [5]

7 5 Solve the equation (log5 𝑦)2 + log5 1 𝑦3 = 28 . [4]

8 6 The diagram shows the graph of f ′(𝑥) of a function 𝑦 = f(𝑥). f ′(𝑥) = acos 𝑏𝑥 + 𝑐 for 0 ≤ 𝑥 ≤ 𝜋 radians. The graph has maximum points at (0, 5 2) and (𝜋, 5 2) and a minimum point at ( 𝜋 2 , − 5 2) . (a) Explain why 𝑐 = 0. [2] (b) Explain why 𝑏 = 2. [2] (c) Without finding the function f(𝑥) and using the diagram above, state and explain at which x-coordinate does the maximum point of f(𝑥) occurs. [2] f ′(𝑥) 𝑥 ൬0 , 5 2൰ 𝑂 ൬𝜋 , 5 2൰ ൬ 𝜋 2 , − 5 2൰ 𝜋 2 𝜋

9 7 The diagram shows a triangular field PQR, in which PQ is x m, QR is y m and angle PQR = 90° . Points M and N lie on PQ and PR respectively, such that MN = MQ = 6 m and angle MNP = 90°. (a) Show that the area of the triangular field PQR, A m2, is given by 𝐴 = 3𝑥2 √𝑥2 − 12𝑥 . [3] P Q R M N 𝑦 m 𝑥 m 6 m

10 (b) Given that 𝑥 can vary, find the stationary value of A and determine its nature. [5]