Canberra AM 2024 Prelim P1 QP

CANBERRA SECONDARY SCHOOL 2024 Preliminary Examination Secondary Four Express ADDITIONAL MATHEMATICS 22 August 2024 4049/01 2 hours 15 minutes 1130h – 1345h Name: ______________________________________ ( ) Class: ________ READ THESE INSTRUCTIONS FIRST Write your full name, class and index number on all work you hand in. Write in dark blue or black pen. You may use a 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. 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. This question paper consists of 19 printed pages including the cover page. Setter: Mr Tan Qi Guang FOR MARKER’S USE Marks Awarded Max Marks 90

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ,0,02 =++ acbxax .2 42 a acbbx −−= Binomial expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21)( 221 , where n is a positive integer and ! )1)...(1( !!)( ! r rnnn rrn n r n +−−=−=      . 2. TRIGONOMETRY Identities sin² A + cos² A = 1 sec² A = 1 + tan² A cosec² A = 1 + cot² A sin (A  B) = sin A cos B  cos A sin B cos (A  B) = cos A cos B  sin A sin B BA BABA tantan1 tantan)tan(  = sin 2A = 2sin Acos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A tan 2A = A A 2tan1 tan2 − Formulae for ABC C c B b A a sinsinsin == a² = b² + c² − 2bc cos A  = 1 sin2 bc A

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 3 [Turn Over Answer all the questions 1 Express 2 2 3 12 1 (3 1)( 1) xx xx − + − +− in partial fractions. [5]

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 4 2 (a) Find the amplitude and period of (i) 3cos x , [1] (ii) 1 4sin 2 x− [2] (b) Sketch, on the same diagram, the curves 3cosyx= and 1 4sin 2yx=− for 0 360x . [3] (c) State the number of solution(s) for which the equation 4sin 2 1 3cosxx=− has. [1]

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 5 [Turn Over 3 (a) (i) Find the first three terms in the expansion, in ascending powers of x, of 7 2 3 x − . [2] (ii) Hence find the value of p, where p is an integer, such that the coefficient of 2x in the expansion of ( ) 7 2 2 3 xpx +−  is 232 3 p− . [4]

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 6 3 (b) By considering the general term in the binomial expansion of 17 2 1 2x x  − , explain why there is no independent term in this expansion. [4]

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 7 [Turn Over 4 The equation of a curve is 2 4 3 7y ax x a= + + − , where a is a constant. The line 25y ax=− is a tangent to the curve as point Q. (i) Find the possible values of a. [4] (ii) Given that the gradient of the line 25y ax=− is positive, find the coordinates of Q. [2]

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 8 5 Mr Lu invested a sum of money in a bank in 1980. The value of the investment, V in thousands dollars can be modelled by an equation of the form 0 tV V k= , where 0V and k are constants and t is the number of years since Mr Lu invested the sum of money. The table below gives some values of V and t. t (years) 5 10 15 20 25 V (in thousands $) 192 370 714 1374 2642 (i) On the grid provided in the next page, plot lgV against t and draw a straight line graph to illustrate the information. The vertical lgV-axis should start at 1.8. [3] Use your graph to estimate (ii) the initial value of the investment, [2] (iii) the value of k, [3] (iv) the time for the initial value of the investment to increase by 50%. [2]

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 9 [Turn Over

____________________________________________________________________________________________________ Canberra Secondary School Additional Mathematics 4049/01 2024 Preliminary Examination Secondary 4 Express 10 6 (a) Find the range of values of x for which 2 ( 3) 3 5x x x−  − . [3] (b) Find the greatest value of k such that the line 2y k x+= does not intersect the curve 62yx x=+ , where k is an integer. [4]