Ping Yi Prelim2021 P1 4E AM

‘Perseverance Yields Success’ Ping Yi Secondary School Preliminary Examination 2021 Sec 4 Express 4049 / 01 Additional Mathematics (Paper 1) 2 hours 15 minutes INSTRUCTIONS TO CANDIDATES Do not open this booklet until you are told to do so. Write your name, class and register 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. Write your answers and working in the spaces provided. Calculators may be used in this paper. All workings must be clearly shown. Omission of essential working may result in loss of marks. For π, use the calculator value or 3.142, unless the question requires the answer in terms of π. 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 intended marks for the question are given in the brackets [ ] at the end of each question or part question. FOR EXAMINER’S USE TOTAL 90 Expected Grade  A1  A2  B3  B4  C5  C6 Teacher’s Comment Student’s Comment Parent’s Comment and Signature This document consists of 16 printed pages including the cover page

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ,02 =++ cbxax a acbbx 2 42 −−= Binomial expansion ,......21)( 221 nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− where n is a positive integer and ! )1(...)1( )!(! ! r rnnn rnr n r n +−−=−=      2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin += += =+ BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(   = = = A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin − = −=−=−= = Formulae for ABC Cab Abccba C c B b A a sin2 1 cos2 sinsinsin 222 = −+= ==

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 3 Answer all questions. 1 A ball is thrown from a cliff overlooking the sea. The vertical height of the ball above sea level, h metres, is given by 28 36 20h t t=− + + , where t is the time in seconds after the ball is thrown. (i) Find the height of the cliff. [1] (ii) By expressing h in its completed square form, determine whether the ball can reach a height of 65 metres above sea level. [3] sea level Height of cliff

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 4 2 Without using a calculator, find the value of 10x , given that 2 4 28 25 2 5x x x x +− =  . [4] 3 It is g iven that 4sin 5A=− , where 180 270A , and that 5cos 13B= , where B is acute. Without using a calculator, find the value of (i) ( )cos AB− , [2] (ii) cosec 2A . [2]

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 5 4 A curve is such that 2 2 2 d 12 7d xxy eex =− . The curve passes through the point ( )0, 4P − and the gradient of the curve at P is 2. Find the equation of the curve. [6]

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 6 5 (a) The first 3 terms in the expansion, in ascending powers of x, of ( )13 n x− , is 21 21 x ax−+ , where a is a constant and n is a positive integer greater than 2. Find the value of n and of a. [4] (b) Using your values of n and a, find the coefficient of x2 in the expansion of ( ) ( ) 2 2 1 3 n xx+− . [2]

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 7 6 The variables x and y are related by the equation 3ln 3 xy x +=  − , where 3x . (a) Express x y d d in the form 2 9 k x − where k is a constant. [3] (b) Explain whether y is an increasing or decreasing function. [2] (c) Given that y is increasing at the rate of 8 units/s, find the rate of change of x when =x 3.5. [2]

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 8 7 (a) Using ( )sin sin cos cos sinA B A B A B+ = + , show that 13sin 75 22 += . [2] (b) Hence, express 2cosec 75 in the form 3ab+ , where a and b are integers. [4]

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 9 8 (a) Given that 3 2 2 22 5 13 1 ( 2)( 3) ( 3) ( )( 2)x x x A x x B x Cx D x− + −  − + + + + + − for all values of x, find the values of A, B, C and D. [4] (b) When a polynomial ( )f x is divided by ( )1x− , the remainder is 4. When the same polynomial is divided by ( )3x+ , the remainder is 24− . Find the remainder when ( )f x is divided by ( ) 2 23xx+− , leaving your answer in the form Ax B+ , where A and B are constants. [5]

This exam paper is the property of Ping Yi Secondary School. It must not be duplicated in part or whole. 10 9 An object is heated until it reaches a temperature of T0 C. It is then allowed to cool. Its temperature T C, when it has been cooled for t minutes, is given by the equation 0.7536 17 tTe −=+ . (a) Find the value of T0. [1] (b) Find the value of t when T = 47 C. [2] (c) Calculate the minimum number of complete minutes required for the temperature to reach 40 C. [4] (d) Explain whether the temperature of the object will reach 36 C. [1]