2022 XMS AMATH P1 PRELIM QP

This document consists of 22 printed pages. [Turn over XINMIN SECONDARY SCHOOL SEKOLAH MENENGAH XINMIN Preliminary Examinations 2022 CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 1 Secondary 4 Express/ 5 Normal Academic Setter : Mr Johnson Chua Vetter : Mrs Loh Si Lan Moderator : Ms Pang Hui Chin Candidates answer on the Question Paper. No Additional Materials are required. 4049/01 29 August 2022 2 hour 15 minutes READ THESE INSTRUCTIONS FIRST Write your name, register number and class 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 the 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. At the end of the paper, fasten all your work securely together. 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. Errors Qn No. Errors Qn No. Accuracy Simplification Brackets Units Geometry Marks Awarded Presentation Marks Penalised For Examiner’s Use 90 Parent’s/Guardian’s Signature:

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0, 2 4 2 b b acx a −  −= Binomial Expansion ( ) 1 2 2 12 n n n n n r r nn n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ( ) ! ( 1)...( 1) ! ! ! n n n n n r r n r r r  − − +== − 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 tan 1 tan tan AB AB  sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2 sin2 A tan 2A = 2 2 tan 1 tan A A− Formulae for  ABC sin sin sin a b c A B C== a 2 = b 2 + c 2 − 2bc cos A  = 2 1 bc sin A

3 [Turn over 1 Solve the equation 22 5 3 2 0x x x− + − = . [3] 2 (a) Express 23 24 53xx−+ in the form 2()a x b c++ where a, b and c are constants. (b) Hence explain why there is no intersection between the graph 23 24 53y x x= − + and the line y = 4. [2] [1]

4 3 Given that 6f '( ) 35x x= + and that f ( )x passes through the origin, find the equation of f ( )x . [3]

5 [Turn over Turn over for Question 4

6 4 The diagram shows part of the graph ( ) 2 2 3 1y x x= − − . Find the range of values of x for which y is increasing. [5] x y ( ) 2 2 3 1y x x= − − O

7 [Turn over 5 (a) Solve 2 1 234xx−−= and show that lg 48 2lg 6x= . [3] (b) In order to obtain a graphical solution of the equation 2 7ln 5 xx −=  , a suitable straight line can be drawn on the same axes as the graph 254 x ye=− . Find the equation of this line. [2]

8 6 Solve the equation 3sin cosec 5cotA A A=− for 180 180A−    . [5]

9 [Turn over 7 Express ( ) 32 2 3 13 8 4 x x x xx + + + + in partial fractions. [6]

10 8 The diagram below shows the base of a hexagonal pyramid. The base is a regular hexagon, made up of six equilateral triangles of sides ( )13+ cm. (a) Show that the area of the hexagon is ( )6 3 9+ cm2. [3] (b) Given that the volume of the hexagonal pyramid is ( )9 4 3+ cm3, find the exact height of the hexagonal pyramid. (Volume of pyramid = 1 base area height3 ) [3] ( )13+ cm