2023 JSS AMATH P1 PRELIM QP

4049/1/GE/23 [Turn Over] O JURONG SECONDARY SCHOOL 2023 GRADUATION EXAMINATION SECONDARY 4 EXPRESS/ SECONDARY 5 NORMAL (ACADEMIC) CANDIDATE NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 PAPER 1 22 August 2023 Candidates answer on the Question Paper. 2 hours 15 minutes Additional Materials: Writing Paper (1 sheet) READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 que stion. The total number of marks for this paper is 90. For Examiner’s Use This document consists of 19 printed pages including this page. 90

2 4049/1/GE/23 1. ALGEBRA Quadratic Equation For the equation Binomial expansion , where n is a positive integer, and 2. TRIGONOMETRY Identities Formulae for 02 =++ cbxax a acbbx 2 42−−= nrrnnnnn bbar nbanbanaba ++    ++     +     +=+ −−− ......21)( 221 ! )1)...(1( )!(! ! r rnnn rnr n r n +−−=−=      1cossin 22 =+ AA 22sec 1 tanAA=+ 22cosec 1 cotAA=+ ( ) BABABA sincoscossinsin = ( ) BABABA sinsincoscoscos = ( ) BA BABA tantan1 tantantan  = AAA cossin22sin = AAAAA 2222 sincossin211cos22cos −=−=−= A AA 2tan1 tan22tan − = ABC C c B b A a sinsinsin == Abccba cos2222 −+= Abcsin2 1=

3 4049/1/GE/23 [Turn Over] 1 A curve has an equation 225y x x= − + . (a) Express 225y x x= − + in the form of ( ) 2 a x b c−+ . Hence state the coordinates of the turning point. [3] (b) The line 27yx=+ intersects the curve at points A and B. Find the distance AB. [3]

4 4049/1/GE/23 2 Express 32 32 3 10 1 3 x x x xx + + + + in partial fractions. [6]

5 4049/1/GE/23 [Turn Over] 3 A curve has equation 321 3y x x kx= + + , where k is a constant and 1k . Explain why the curve does not have a stationary point. [4]

6 4049/1/GE/23 4 The diagram shows a circle. The line PC is the tangent to the circle at P. A and B are points on the circle such that PAB is a straight line. Prove that (a) triangle BPC is similar to triangle CPA, [3] (b) 2PA PB PC= . [2]

7 4049/1/GE/23 [Turn Over] 5 The equation of a curve is ( ) 3 2 24 13y x k x k x=− − + − , where k is a constant. (a) Find the range of values of k for which y is always decreasing. [4] (b) Given that y has three distinct roots, find the range of values of k. [2]

8 4049/1/GE/23 6 A curve is such that 2 2 d 3sin 4cos 2d y xxx =− . The curve passes through ( )0,1A and ( )π,3B . Find the equation of the curve. [7]

9 4049/1/GE/23 [Turn Over] 7 For the curve 22yx= , the tangent at point P where xa= , intersect the y-axis at A. The normal to the curve at point P intersects the y-axis at B. Given that 0a , show that the area of triangle ABP is ( ) 216 1 8 aa + . [7]

10 4049/1/GE/23 8 (a) The equation of a curve is sin , 0y a bx c a= +  . The curve attains maximum and minimum values of 4 and 2 respectively, and the period is π radians. Show that 1a= , 2b= and show that 3c= . [3] (b) (i) Sketch, on the same diagram, the curves sin 2 3yx=+ and 3cosyx= for 02 πx radians. [4]