XMSS 2024 AM Prelim (P1) (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 1 Secondary 4 Express Setter : Mr Johnson Chua Vetter : Ms Low Yan Jin Moderator : Ms Pang Hui Chin Candidates answer on the Question Paper. No Additional Materials are required. 4049/01 23 August 2024 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. 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 A triangle has a base of length ( )6 2 7+ cm and an area of ( )17 7 7+ cm2. Find, without using a calculator, the perpendicular height to the base of the triangle, in cm, in the form ( )7ab+ , where a and b are integers. [3] 2 Solve the equation 51 xx− + = . [4]

4 3 The equation of a curve is 42 1 xy x += + , where 1x− . (a) Find d d y x , leaving your answer in the form ( )1 n ax b x + + , where a, b and n are constants. [2] (b) Explain why the curve is a increasing function. [2]

5 [Turn over 4 The line 2 3 12xy+= intersects the curve 2 48yx=− at points A and B. Find the value of p and q for which the length of AB can be expressed as pq . [6]

6 5 The height, h m, of a baseball above ground t seconds after it has been hit is given by 224 4h c t t= + − , where c is a constant. (a) If c = 1.65, express h in the form 2()h p q t r= + + where p, q and r are constants to be determined. Hence, state the maximum height attained by the baseball and the time at which this occurs. [4] (b) Find the range of values of c if the baseball did not reach a height of 40 m. [2]

7 [Turn over 6 A curve is such that 2 63 2 d 12 15d xxy eex −=+ . The point ( )0, 2P − lies on the curve and the normal to the curve at P is parallel to the y-axis. Find the equation of the curve. [6]

8 7 The equation of a curve is 2y kx kx p= + + , where p and k are constants. (a) Show that 4 kp for which the curve lies completely above the x-axis. [3] (b) In the case where k = 2 and p = 4, find the values of m for which the line 4y mx=− is a tangent to the curve. [4]

9 [Turn over 8 (a) Divide 322 5 5 9x x x+ + + by 3 3xx+ . [1] (b) Express 32 3 2 5 5 9 3 x x x xx + + + + in partial fractions. [5]

10 9 The value, $V, of a watch is related to t, the number of years since 1980. The table below gives the value of the watch in 1990, 2000, 2010, 2020. Year 1990 2000 2010 2020 t (years) 10 20 30 40 V ($) 7200 9600 12 800 17 200 (a) Plot lnV against t and draw a straight line graph to illustrate the information. [2] | | | | 0 10 20 30 40 8.4 8.6 – 8.8 – 9.0 – 9.2 – 9.4 – 9.6 – 9.8 – 10.0 – t lnV