2022 CGS AMath Prelims P1

Name: Register No.: Class: CRESCENT GIRLS’ SCHOOL SECONDARY FOUR 2022 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS 4049/01 Paper 1 23 August 2022 Candidates answer on the Question Paper. No Additional Materials are required. 2 hours 15 min READ THESE INSTRUCTIONS FIRST Write your name, register number and class in the spaces at the top of the 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 the questions. Write your answers in the answer spaces provided. 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 of the marks for this paper is 90. For Examiners’ Use Question 1 2 3 4 5 6 7 8 9 10 11 12 13 Marks Table of Penalties Question Number Presentation −1 Accuracy / Units −1 Parent’s/Guardian’s Signature This document consists of 21 printed pages. 90

Crescent Girls’ School 2022 Sec 4 A Math Prelims P1 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= Binomial expansion 1 2 2( ) ... ... 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ! ( )...( 1) !( )! ! n n n n r n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+=  = 1 sin2 bc A

Crescent Girls’ School [Turn Over 3 1 A trapezium of area ( ) 212 7 1 cm+ has parallel sides of length ( )2 7 m1 c− and ( )7 m6 c+ . Without using a calculator, obtain an expression for the height, h, of the trapezium in the form 7ab + , where a and b are integers. [4] 2 Integrate 2sin 1x x+ with respect to x. [3]

Crescent Girls’ School 2022 Sec 4 A Math Prelims P1 4 3 Solve the simultaneous equations. 1 11 2 5 89 2 5 11 xy xy + −− += −= [4]

Crescent Girls’ School [Turn Over 5 4 Express 2 2 7 23 6 ( 2)( 4) xx xx −+ −− in partial fractions. [5]

Crescent Girls’ School 2022 Sec 4 A Math Prelims P1 6 5 The gradient function of a curve is 2 4 (1 2 ) p x + − . It is given that the gradient of the curve at the point (1, –3) is 7. (a) Find the value of p. [2] (b) Find the coordinates of the point where the curve meets the y-axis. [4]

Crescent Girls’ School [Turn Over 7 6 (i) Express 32 2 4log 16 n n − in the form an b+ , where a and b are integers. [3] (ii) Solve the equation 2 4 2log ( 4) lo 1)g (7 4xx+ =+ + . [4]

Crescent Girls’ School 2022 Sec 4 A Math Prelims P1 8 7 A container is in the shape of an inverted right circular cone where its base radius is equal to its height. Alcohol in the container evaporates at a constant rate of 10 cm 3/s. The volume of alcohol, V, in the container is 31 3 x when the height of the alcohol is cmx . (i) Calculate the rate of change of the height of alcohol when the height of alcohol is 5 cm, leaving your answer in exact form. [3] x

Crescent Girls’ School [Turn Over 9 (ii) Calculate the rate of change of the exposed surface area of the alcohol when the depth of alcohol is 5 cm. [3]

Crescent Girls’ School 2022 Sec 4 A Math Prelims P1 10 8 The vertical displacement of the waves, y m, at a port can be modelled by the equation 53cos 8.kty=− + , where k is a constant and t is the number of hours after midnight. The time between two consecutive high tides is 6 hours. (i) Show that the value of k is 3  . [1] (ii) Sketch the graph of 33c 8.os 5y t +=− for 0 24t . [3] y 0 t