XMSS 2024 AM Prelim P2 QP

This document consists of 19 printed pages and 1 blank page. [Turn over XINMIN SECONDARY SCHOOL SEKOLAH MENENGAH XINMIN Preliminary Examination 2024 CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS Paper 2 Secondary 4 Express Setter: Ms Joanne Kong Vetter: Ms Low Yan Jin Moderator: Ms Pang Hui Chin 4049/02 26 August 2024 2 hour 15 minutes Candidates answer on the Question Paper READ THESE INSTRUCTIONS FIRST Write your name, index 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 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 of the 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 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0, a acbbx 2 42 −−= Binomial Expansion ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− 221 21 , where n is a positive integer and ( ) ! )1)...(1( !! ! r rnnn rrn n r n +−−=−=      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 A company purchased a colour copier machine at a cost of $8500. The value of this machine decreases with time such that its value, $ V, after t months of usage is given by 8500 ktVe −= , where k is a constant. (a) The value of the copier machine is expected to fall to $6400 after 8 months of usage. Estimate the value, to the nearest dollar, of the machine after 2 years of usage. [4] (b) Copier machines are to be replaced when its value reaches 1 7 of its initial value. The company’s manager, Mrs Lee, claims that the machine will last for at least 5 years before a replacement is due . Showing all necessary working, explain whether you agree with Mrs Lee. [2]

4 2 (a) Differentiate 2 sin 2 xx . [2] (b) Use the result in part (a) to evaluate π 0 3 cos 2 xx dx , leaving your answer as an exact value in the form πab− , where a and b are constants. [4]

5 [Turn over 3 It is given that 32f ( ) 2 3x x px qx= + + + , where p and q are constants, has a factor of 21x− and leaves a remainder of 75− when divided by 2x+ . (a) Show that 15p=− and 1q= . [4] (b) Solve the equation f ( ) 0x = . [4] (c) Hence, solve the equation 2 3 0k k pk q k+ + + = . [2]

6 4 The diagram shows a vertical cross section of a tent in which AB = 2 m, BC = 3 m and angle angle BAD BCD == . The tent is symmetrical about its vertical height AD and it is set up on horizontal ground. (a) Show that 3sin 2cosAD =+ . [2] (b) Express AD in the form ( )cosR − , where R > 0 and 0 90   . [3] A B C D 2 m 3 m

7 [Turn over (c) Given that the vertical height of the tent is 3.45 m, calculate the value of  . [2] (d) Find the value of  for which AD is a maximum. [2]

8 5 (a) Given that 2log 10p A = and log 2p B= , find the value of log A pB . [3] (b) Solve ( )3 6 5 3xx −=− . [4]

9 [Turn over 80 m 37 m P 6 The diagram shows a wind turbine with propeller-like blades that have a length of 37 m each. Wind turns the blades that spin around a rotor in the centre to generate electricity. The height, h m, of the tip of each blade above the ground, t seconds after leaving a particular point P, can be modelled by 37 cosh a bt=− , where a and b are constants. The centre of the wind turbine’s rotor is 80 m from the ground and on average, the blades rotate in an anti-clockwise direction at a rate of 1 revolution every 8π seconds. (a) Show that a = 80 and 1 4b= . [2] (b) Find the time taken, in seconds, for the blade to first reach a height of 89 m above ground after leaving P. [3]

10 7 The equation of a curve is 5lnyx= . The tangent to the curve at 2xe= intersects the x-axis at A. (a) Show that the coordinates of A are ( ) 2, 0e− . [5] (b) Find the area bounded by the tangent, the line 2xe= and the x-axis. [2]