JVSS 4E A Math Prelim 2021 P1 QP (Final edited)

This document consists of 16 printed pages. [Turn over JURONGVILLE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2021 Secondary 4 Express STUDENT NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 27 AUGUST 2021 2 hours 15 Minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class and index number 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. DO NOT WRITE ON ANY BARCODES. Answer ALL 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 marks for this paper is 90. DO NOT OPEN THE BOOKLET UNTIL YOU ARE TOLD TO DO SO Setter: Mrs Adeline Pang For Examiner’s Use 90

2 Jurongville Secondary School 4E PRELIM / 4049_01 / 2021 1. ALGEBRA Quadratic Equation For the equation ,02 =++ cbxax a acbbx 2 42 −−= Binomial expansion ,......21)( 221 nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− where n is a positive integer and ! )1(...)1( )!(! ! r rnnn rnr n r n +−−=−=      2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin += += =+ BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(   = = = A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin − = −=−=−= = Formulae for ABC Cab Abccba C c B b A a sin2 1 cos2 sinsinsin 222 = −+= ==

3 Jurongville Secondary School 4E PRELIM/ 4049_01 / 2021 [Turn Over Answer all the questions. 1 A particle moves along the curve 𝑦 = 𝑥 cos 3𝑥 in such a way that the y-coordinate decreases at a constant rate of 0.2 radian/sec at the instant when 𝑥 = 𝜋. Find the rate of change of the x-coordinate at this instant. [3] 2 Given that 𝑦 = 𝑥3 + 𝑚𝑥2 + 𝑛𝑥 + 15 where m and n are integers, is a decreasing function of x for −2 < 𝑥 < 4 3 . Find the values of m and of n. [4]

4 Jurongville Secondary School 4E PRELIM / 4049_01 / 2021 3 (a) Simplify 71−5𝑥×49𝑥 3(343𝑥)+73𝑥 and express it as 𝑎𝑏−6𝑥 where a and b are non-zero constants. (b) Given that 71−5𝑥×49𝑥 3(343𝑥)+73𝑥 = 64𝑥. Hence, find the exact value of 146𝑥. [2] [2]

5 Jurongville Secondary School 4E PRELIM/ 4049_01 / 2021 [Turn Over 4 A factory purchased a new machine for its operation. After t months, the value of the machine $A is given by 𝐴 = 5600𝑒−𝑘𝑡, where k is a constant. (a) Find the initial value of the machine when it is newly purchased. (b) The value of the machine after 6 months is expected to be $4800. Find the value of k. (c) Calculate the age of the machine, to the nearest month, when the expected value is $1200. [1] [2] [2]

6 Jurongville Secondary School 4E PRELIM / 4049_01 / 2021 5 Find the stationary point of the curve that is given by 𝑦 = 𝑥𝑒1−2𝑥 and determine the nature of the stationary point. [5]

7 Jurongville Secondary School 4E PRELIM/ 4049_01 / 2021 [Turn Over 6 (a) Find the range of values of x such that 2𝑥2 + 𝑥 − 15 is always positive. (b) Given the line 𝑦 = 𝑥 + 2𝑝 is a tangent to the curve 𝑦2 = 𝑞𝑥 , where p and q are positive constants, show that q is a multiple of p. [2] [3]

8 Jurongville Secondary School 4E PRELIM / 4049_01 / 2021 7 Given that sin 𝜃 = − 3 5 and tan 𝜃 > 0, find the exact value of (a) tan 2𝜃, (b) cos 𝜃 2 . [2] [3]

9 Jurongville Secondary School 4E PRELIM/ 4049_01 / 2021 [Turn Over 8 (a) Given that cos(𝐴+𝐵) cos(𝐴−𝐵) = 5 6 , show that cot 𝐴 = 11 tan 𝐵 . (b) Hence, find the value of tan 𝐴 when tan 2𝐵 = 0.18 and B is acute. [3] [3]

10 Jurongville Secondary School 4E PRELIM / 4049_01 / 2021 9 The diagram shows the graph of the eq uation 𝑦 = tan 𝑎𝑥 + 𝑏 where a and b are constants. (a) Write down the value of a and of b. (b) Given that the line whose equation is 𝑦 = 𝑘, intersects the curve 𝑦 = tan 𝑎𝑥 + 𝑏 at 2 points for 0 ≤ 𝑥 ≤ 3𝜋 ,find the range of value of 𝑘. (c) Using the values of a and b, solve tan 𝑎𝑥 + 𝑏 = 1 where 0 ≤ 𝑥 ≤ 3𝜋. [2] [1] [3] y x 3𝜋 2 3𝜋 0 3𝜋 4 9𝜋 4 0.5