PHS 2024 A Math Prelim P2 QP w Ans

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/02 Paper 2 20 August 2024 Tuesday 2 hours 15 min PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2024 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) PRELIMINARY EXAMINATIONS DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. INSTRUCTIONS TO CANDIDATES Write your name, index number and class in the spaces provided above. 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 spaces provided below 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. 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 Qn 1 2 3 4 5 6 7 8 9 10 Marks Deducted Marks Setter: Mr Gregory Quek Vetter: Mr Tan Lip Sing Category Accuracy Units Symbols Others Question No. TOTAL MARKS 90

2 This question paper consists of 21 printed pages and 1 blank page.

3 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0,ax bx c+ + = 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 22sin cos 1AA+= 22sec 1 tanAA=+ 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 2cos2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2tantan 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=

4 1 (a) Write down, and simplify, the first three terms in the expansion of 5 23 x  − in descending powers of x. [2] (b) Given that there is no term independent of x in the expansion of ( ) 5 2 25 3 ,ax x +−  hence find the value of the constant a. [3]

5 2 In the figure, ABCD is a rectangle inscribed within a semicircle of radius 4 cm and centre O. It is given that AB = x cm and BC = y cm. (a) Show that the area of the rectangle, A cm, is given by 21 64 .2A x x=− [2] O A B C D x cm y cm

6 (b) Find the exact value of x for which A has a stationary value. Give your answer in the form 2,k where k is an integer. [4]

7 3 The diagram shows a triangle ABC inscribed in the circle with centre O. BD is a tangent to the circle at B and AB is parallel to CD. Point M is the midpoint of BC. (a) Prove that triangles ABC and BCD are similar. [3] (b) Prove that ABMO is a trapezium. [2] A B C D O M

8 (c) Prove that 2 2 BCOM CD= . [3]

9 4 Milk is poured into an empty cup and heated. The temperature, C,mT  of the milk in the cup, t minutes after it is heated, is modelled by the formula, ( )5 2 20. t mT =+ (a) State the initial temperature of the milk. [1] Coffee is poured into another empty cup. The temperature, C,cT  of the coffee in the cup, t minutes after it is poured, is modelled by the formula, ( )60 2 25. t cT − =+ (b) Find the time taken for the temperature of the coffee to drop to 35 C. [3]

10 (c) Find the time taken for the milk and the coffee to reach the same temperature. [4]