2022 AMKSS AMath Prelims P2

[Turn Over Class Index Number Name ANG MO KIO SECONDARY SCHOOL PRELIMINARY EXAMINATION 2022 SECONDARY FOUR EXPRESS/5 NORMAL ACADEMIC ADDITIONAL MATHEMATICS 4049/02 Paper 2 Setter: Mrs D Sharav Wednesday 14 September 2022 2 hour 15 minutes Answer on the question paper. No additional materials are required. 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 on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use 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. At the end of the examination, fasten all your work securely together. 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. For Examiner’s Use 90 This document consists of 20 printed pages.

2 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over Mathematical Formulae 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 22 22 22 sin cos 1 sec 1 tan cosec 1 cot AA AA AA += =+ =+ BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(   = = = A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin − = −=−=−= = Formulae for ABC a2 = b2 + c2 – 2bc cos A Cabsin2 1= C c B b A a sinsinsin ==

3 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over 1 (a) Prove that cossec tan1 sin  −=+ . [4]

4 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over (b) Hence solve the equation cossec 2sin1 sin  −=+ for  20  . [4]

5 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over 2 A circle with centre C at ( )7 ,5 touches the y-axis at point P. (i) State the radius of the circle and find its equation. [2] (ii) The line 1634 =+ yx is a tangent to the circle at the point Q. Find the equation of the line CQ. [3] (iii) Hence find the coordinates of the point Q. [2]

6 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over (iv) Given that line kyx =+ 34 is another tangent to the circle, find the value of k. [3]

7 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over 3 (i) Show that 2x – 3 is a factor of 3232 23 −+− xxx . [1] (ii) Hence factorise 3232 23 −+− xxx . [2] )

8 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over (iii) Using results from (ii), express 3232 78 23 2 −+− −+ xxx xx in partial fractions. [5]

9 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over 4 (a) Solve the equation ( ) ( ) 11log23log 2 12 =−−+ xx . [5] (b) It is given that abba 722 =+ . Show that baba lglg)(3 1lg2 +=+ . [4]

10 AMKSS 4E/5NA PRELIM 4049/02/2022 [Turn Over 5 (i) Find all the angles between 0 and 360 which satisfy the equation 29 tan 24sec 25xx=− . [4]