AISS 2023 4E Add Math Paper 1 Prelim

_____________________________________________________________________________ This document consists of 19 printed pages. AISS PRELIM/4E/4049/P1/2023 [Turn over AHMAD IBRAHIM SECONDARY SCHOOL GCE O-LEVEL PRELIMINARY EXAMINATION 2023 SECONDARY 4 EXPRESS Name: Class: Register No.: ADDITIONAL MATHEMATICS Paper 1 Candidates answer on the Question Paper. 4049/01 7 August 2023 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. 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 questions. Give non-exact numerical answers 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 examination, 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. For Examiner’s Use /90

2 AISS PRELIM/4E/4049/P1/2023 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 2 4 2 b b acx a −  −= 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) ( 1) !( )! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A=

3 AISS PRELIM/4E/4049/01/2023 [Turn over 1 The variables x and y are related by the equation 2 hy xk= − . The diagram below shows the graph of 1 y against x. Calculate the value of h and of k. [4] ( )0, 4− x

4 AISS PRELIM/4E/4049/01/2023 2 The Richter scale measures the intensity of an earthquake using the formula 0 lg IM I =   , where M is the magnitude of the earthquake, I is the intensity of the earthquake, and 0I is the intensity of the smallest earthquake that can be measured. (a) Calculate the magnitude of an earthquake if its intensity is 1000 times the intensity of the smallest earthquake that can be measured. [1] (b) In February 2011, an earthquake with magnitude 6.2 was recorded in Christchurch, New Zealand. Few weeks later, an earthquake with magnitude 9.0 was detected in Fukushima, Japan. How many times stronger in intensity was the Japan’s earthquake as compared to the New Zealand’s earthquake? Give your answer to 2 decimal places. [3]

5 AISS PRELIM/4E/4049/01/2023 [Turn over 3 The diagram shows a hemispherical bowl of radius 12 cm. Water is poured into the bowl and at any time t seconds, the height of the water level from the lowest point of the hemisphere is h cm. The rate of change of the height of the water level is 0.4 cm/s. (a) Show that the area of the water surface, A, is given by ( )24A h h=− . [2] (b) Find the rate of change of A when h = 5 cm. Leave your answer in terms of  . [3]

6 AISS PRELIM/4E/4049/01/2023 4 (a) Explain why there is only one solution to the equation ( ) ( )5 5log 13 4 log 2xx− = − . [5]

7 AISS PRELIM/4E/4049/01/2023 [Turn over (b) Solve the simultaneous equations ( ) 34 32 2 , 9 3 10 . x x y xy ++ = += [7]

8 AISS PRELIM/4E/4049/01/2023 5 (a) Prove the identity 11cot 2 tan2 tan 2xx x=− . [2] (b) Hence solve the equation ( )tan 3 4cot 2 3xx −= for 0 360x   . [5] (c) Without further solving, e xplain why there are 6 roots to the equation ( )tan 3 4cot 32 x x−= for 360 720x−    . [2]

9 AISS PRELIM/4E/4049/01/2023 [Turn over 6 The curve 2 13xy e x=− intersects the y-axis at the point P. The tangent and the normal to the curve at P meet the x-axis at A and B respectively. Find the exact area of triangle PAB. [7]

10 AISS PRELIM/4E/4049/01/2023 7 In the diagram, CE is a tangent that touches the circle of centre O at D. AD is the diameter of the circle, EA cuts the circle at points G and A, and EB cuts the circle at points F and B. (a) Given that ABC is a straight line, show that triangle ABD and triangle DBC are similar. [3]