2023 BPGHS Prelim P2

Name of Candidate: _____________________ ( ) Class: ______ Calculator Model: This paper consists of 23 printed pages BUKIT PANJANG GOVERNMENT HIGH SCHOOL PRELIMINARY EXAMINATION 2023 SECONDARY 4/5 GCE 'O' LEVEL SYLLABUS ADDITIONAL MATHEMATICS Paper 2 4049 / 02 Date: 24 August, 2023 Duration: 2h 15 min Time: 0750 – 1005 h 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, glue or correction fluid. 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 number of marks for this paper is 90. [Turn over] No Additional Materials are required.

2 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0, a acbbx 2 42 −−= . Binomial Theorem 1 2 2( ) ... ... , 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and ( ) ! !! n n r r n r  = − = ( 1)......( 1) ! n n n 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 BABABA sincoscossin)sin( = cos(𝐴 ± 𝐵) = cos 𝐴 cos𝐵 ∓ sin 𝐴 sin 𝐵 tan tantan( ) 1 tan tan ABAB AB = sin2A = 2 sin A cos A AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for  ABC C c B b A a sinsinsin == a 2 = b 2 + c 2 − 2bc cos A  = 1 sin2 bc A

3 1 (a) Solve the equation ( ) 5 2 5 2log 4 log 2 5 3.x x+ − + =− [5]

4 (b) Given that 132 2 16 ,x x x+++= find the value of x. [4]

5 2 The equation of a curve is 3 21ln . 25 xy x  +=  − (i) Explain why the curve does not have a stationary point. [4] (ii) If y decreases at the rate of 0.4 units/s, find the rate of change of x when x = 2. [2]

6 3 (i) Express 2 32 4 2 1 2 xx xx −+ + in partial fractions. [5]

7 (ii) Hence, show that 24 322 4 2 1 9 1 6ln 4ln 2 .2 5 4 xx dxxx −+ = − ++ [4]

8 4 Solution to this question by accurate drawing will not be accepted. The parallelogram ABCD is such that A is (−2, 6) and C is (4, 3). Given that point B lies on the x-axis and BC is perpendicular to the line 2 12 20,yx+= find (a) (i) the coordinates of B and of D, [4]

9 (ii) the area of par