PHS 2024 Prelim A Math P1 (final) QP with answer key

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/01 Paper 1 19 August 2024 Monday 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 11 12 Marks Deducted Marks Setter: Ms Sabrina Tan Vetter: Mr Tan Lip Sing This question paper consists of 19 printed pages and 1 blank page. Category Accuracy Units Notations Others Question No. TOTAL MARKS 90

2 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 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 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A= ! ( 1)...( 1) ( )! ! ! n n n n n r r n r r r  − − +== − ABC

3 1 Express as the sum of 3 partial fractions. [5]

4 2 (a) State, in terms of π, (i) the principal value of , [1] (ii) the values between which the principal value of must lie. [1] (b) Given that A is a reflex angle and , find the exact value of without the use of a calculator. [3]

5 3 (a) Find the set of values of the constant k for which the curve y = – kx2 – 2x + 2k – 3 does not intersect the x-axis. [3] (b) Using the answer in part (a), explain whether it is possible for – kx2 – 2x + 2k – 3 to be positive for all x. [2]

6 4 A roller coaster is being designed such that the height, h m, of a rider above the ground in a section of the roller coaster ride is given by , where x is the horizontal distance of the rider from the starting point and 1 ≤ x ≤ 4. (a) Express h in the form a + b(x + c)2 where a, b and c are constants to be determined. [3] (b) Hence, explain why the rider cannot reach a height of 10 m. [1] (c) After testing the prototype, the roller coaster designer wants to make the ride more exciting by moving the highest point of this section of the roller coaster ride up by 0.2 m and left by 0.1 m. Write down a possible new expression for h. [1]

7 5 Water is poured into an empty inverted conical container with radius 6 cm and slant height 10 cm. After t seconds, the radius of the top surface of the water is r cm. (a) Show that the surface area, S, of water in contact with the container at any time is given by . [The curved surface area of a cone of base radius r and slant height l is .] [2] (b) Water is poured into the container such that r increases at a constant rate. Given that it takes 30 seconds to completely fill up the empty container, write down the rate at which r increases. [1] 10 cm 6 cm r cm

8 (c) Hence calculate the rate at which S increases when the container is one-eighth filled. [4]

9 6 A curve has equation , where x > 0. (a) Find the x-coordinates of the stationary points of the curve. [5] (b) Determine the nature of each of the stationary points. [3] (c) Using your answer in part (b), infer and write down the set of values of x for which y is an increasing function. [2]

10 7 The diagram shows part of the curve , where x > –8. The curve intersects the x-axis and y-axis at P and Q respectively. The tangent to the curve at R is parallel to line PQ. (a) Show that the coordinates of point R are (– 4, – 10). [5] P Q O x y