3E NV AM EOY 2021

NORTH VISTA SECONDARY SCHOOL END-OF-YEAR EXAMINATION 2021 90 SUBJECT: ADDITIONAL MATHEMATICS DATE: 8 OCTOBER LEVEL/ STREAM: SECONDARY 3 EXPRESS CODE : 4049 TIME : 2 HOURS 15 MINUTES READ THESE INSTRUCTIONS FIRST Write your name, register number and class 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 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 number of marks for this paper is 90. NAME: _____________________________( ) CLASS: ____________ For Examiner’s Use Category Question No. Accuracy Brackets Fractions Units Others Marks Deducted This question paper consists of 21 printed pages. [Turn over

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0,ax bx c   2 4 2 b b acx a    Binomial expansion 1 2 2( ) ... ... , 12 n n n n n r r nn n na b a a b a b a b b r                          where n is a positive integer and ! ( 1)...( 1) ( )! ! ! n n n n n r r n r r r      2. TRIGONOMETRY Identities 22sin cos 1AA 22sec 1 tanAA 22cosec 1 cotAA sin( ) sin cos cos sinA B A B A B   cos( ) cos cos sin sinA B A B A B tan tantan( ) 1 tan tan ABAB AB  sin 2 2sin cosA A A 2 2 2 2cos2 cos sin 2cos 1 1 2sinA A A A A      2 2tantan 2 1 tan AA A   Formulae for ABC sin sin sin a b c A B C 2 2 2 2 cosa b c bc A   1 sin2 ab C

3 [Turn over 1 Express 2 2 53 5 7 (3 1)( 10) xx xx   in partial fractions. [6]

4 2 (a) Variables x and y are connected by the equation 23y ax by , where a and b are constants. Explain clearly how a and b can be calculated when a graph of 2x y against 2y is drawn. [3] (b) The mass, m grams, of a radioactive substance decreases with time, t years. The observation started on 1 January 2021. Measured values of m and t are shown in the table below. m (grams) 53.6 35.9 24.1 16.2 t (years) 10 20 30 40 It is known that m and t are related by the equation 0 ktm m e  , where 0m and k are constants. (i) Draw the graph of ln m plotted against t, using a scale of 4 cm for 10 units on the t-axis and a scale of 2 cm for 1 unit on the ln m -axis. [3]

5 [Turn over

6 Use your graph to estimate 2 (b) (ii) the value of k, [2] (iii) the mass of the substance when the observations began, [2] (iv) the year for the substance to reach half its original mass. [3]

7 [Turn over 3 (a) In the expansion of 9 2 3 52x x   , determine whether the term independent of of x exists. [4]

8 3 (b) (i) Find the first 3 terms in the expansion, in ascending powers of x, of 8 3 2 x  . Give the terms in their simplest form. [3] (ii) Hence, find the coefficient of 3x in the expansion of   8 23 2 7 92 x xx    . [2]

9 [Turn over 4 The points (5,17)A and (12,10)B lie on a circle. The centre of the circle lies on the y-axis. (i) Find the equation of the perpendicular bisector of AB. [5] (ii) Find the equation of the circle. [3]

10 4 (iii) Does the point ( 7,4)C  lie on the circle, inside the circle or outside the circle? Explain clearly by showing all your working. [2] The circle gets reflected about a vertical line that passes through the point ( 9,6) . (iv) Find the equation of the reflected circle. [2]