2024 TPSS AMATH P1 PRELIM QP

This document consists of 22 printed pages TAMPINES SECONDARY SCHOOL Secondary Four Express / Five Normal Academic Preliminary Examination 2024 NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 28 August 2024 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST For Examiner’s Use Write your name, class and register 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 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. [Turn over O

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, 2 4 2 b b acx a −± −= Binomial expansion ( ) nr r nnnnn b b ar nb anb ana b a + +     + +     +     + = + −−− ......21 2 21 , where n is a positive integer and ( ) .! ) 1 )...( 1 ( ! ! ! r r n n n r r n n r n + − −=−=      2. TRIGONOMETRY Identities A AA AA A A A A A A B A B AB A B A B A B A B A B A B A AA AA A A 2 2222 22 22 22 tan1 tan22tan sin2 1 1cos2sin cos2cos cos sin2 2sin tan tan1 tan tan)tan( sin sin cos cos)cos( sin cos cos sin)sin( cot 1 cosec tan 1 sec 1 cos sin − = − = − = − = = ±= ± = ± ±= ± + = + = = +   Formulae for ABC∆ C ab A bc c b a C c B b A a sin 2 1 cos 2 sin sin sin 2 2 2 = − + = = = ∆

3 1 The equation of a curve is 22 1 xy x= + where x ≠ −1. Determine the range of values of x for which y is a decreasing function. [4]

4 2 A missile, TP-1 was launched such that its height, h1 metres above the ground was given by h1 (x) = −20x2 + 120x + 3, where x metres was the horizontal distance of the missile from the launched position. (a) Express –20x2 + 120x + 3 in the form a(x + b)2 + c, where a, b and c are constants. [2] Another missile, TP-2 was launched from the same position as TP-1. The height of TP-2, h 2 metres above the ground was given by h 2 (x) = ( ) 249 6 18310 x− −+ . (b) Find h1 (0) and h2 (0) and hence interpret the meaning of your answers. [2] (c) The missile that could reach a greater height and travel a distance further away from its launched position was acquired. Determine if missile TP-1 or TP-2 was acquired. Show your working clearly. [3]

5 3 A cuboid with volume (2x + 1)2 cm3 has a rectangular base with dimensions x2 cm and (2x – 1) cm. Find an expression for the height of the cuboid, leaving your answer in partial fractions. [6]

6 4 (a) (i) Find in ascending powers of x, the simplified first three terms in the expansion of (2 + qx)6. [3] (ii) Given that the first two non-zero terms in the expansion of (2 + px)(2 + qx) 6, where q > 0, are 128 and −168x2, find the values of p and q. [4]

7 (b) Find the term independent of x in 12 3 2x x  − . [3]

8 5 A function f(x) is defined for all real values of x such that f ′′(x) = 18 e−3x. The gradient of the curve y = f(x) at x = 0 is 2 and the curve passes through 3 21,e   . (a) Find the exact value of the x-coordinate of the stationary point of the curve and determine its nature. [6] (b) Find the equation of the curve. [2]

9 6 A graph has the equation y = x2 + (6 – 2m)x + m + 5 where m is a constant. (a) Given that the graph cuts the x-axis at A and B and the point (4, 0) is the midpoint of AB, find the value of m. [3] (b) If m = 8, find the range of values of p such that the graph of y = x 2 + (6 – 2m)x + m + 5 + p lies above the x-axis. [2]

10 7 The number of insects present in a colony t weeks after observation began, can be modelled by the equation 3 t m ab= . Measurements of m and t are shown in the table below. t 2 4 6 8 10 m 920 1108 1333 1605 1930 (a) Plot lg m against t and draw a straight line graph to illustrate the information. [2] (b) Use your graph to estimate the values of a and b. [3] The number of insects present in another colony t hours after observation began, can be modelled by the equation lg m 40 = t + 120. (c) By drawing a suitable straight line on your graph estimate the time taken for the two colonies to have the same number of insects. [2]