2019 J2 RVHS H2 Math Prelim (with ans)

1 ©RIVER VALLEY HIGH SCHOOL 9758/01/2019 RIVER VALLEY HIGH SCHOOL 2019 JC2 Preliminary Examination Higher 2 NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) 9758/01 19 September2019 3 hours READ THESE INSTRUCTIONS FIRST This document consists of 5 printed pages. For examiner’s use only Question number Mark 1 2 3 4 5 6 7 8 9 10 Total Calculator Model: Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. 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. Write your answers in the spaces provided in the question paper. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a diff erent level of accuracy is specified in the question. The use of an approved graphing calculator is expected, where appropriate. You are required to present the mathematical steps using mathematical notations and not calculator comma nds. 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 numbe r of marks for this paper is 100. www.KiasuExamPaper.com 628

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2019 1 Find the range of values of x that satisfy 21 51 x xx . [4] Hence deduce the range of values of x that satisfy 2 1 051 x xx . [2] 2. A curve f( )yx passes through the point (0 , 2) and dc os d2 yx xy . (i) Obtain the Maclaurin series of y in ascending powers of x, up to and including the term in 2x . Write down the equation of the tangent to the curve f( )yx at 0x . [4] (ii) Show that 4s i nyx . [2] (iii) Using your results in part (ii), and assuming x to be sufficiently small for terms in 3x and higher powers to be ignored, obtain the binomial series of y. [3] 3. The curve C has equation 2 2 2 xay xb , where a and b are positive integers. C passes through the point 9 ,02 §· ¨¸ ©¹ and the equation of an asymptote of C is 3x . (i) Show that 9a and 18b . [2] (ii) Sketch C, stating clearly the equations of asymptotes, the x-coordinates of turning points and axial intercepts. [4] (iii) Hence, giving your reasons, deduce the range of values o f h such that the graph of

2 2 2 8 1100 x y h , where h is a positive integer, intersects C at exac tly 6 distinct points. [2] 4. A function is said to be self-inverse when f = 1f for all x in the domain of f. Given that the function f is defined by f: ax bx cx a

ax b cx a , for , axx cz , ax, czx, , where a, b and c are positive constants. (i) Show that f is self-inverse. [2] (ii) Using the result of part (i), deduce 2f() x and state the range of 2f. [2] (iii) Solve the equation 1f( )xx , leaving your answers in the exact form. [3] For the rest of the question, let a = 2, b = 5 and c = 3. The function g is defined by g: e 2 xx e2x for x . (iv) Show that the composite function fg exists, justifying yo ur answer clearly. [1] (v) By considering the graph of f, or otherwise, find the exac t range of fg. [2] www.KiasuExamPaper.com 629

3 ©RIVER VALLEY HIGH SCHOOL 9758/01/2019 5 Consider the following definitions: eecosh , 2 eesinh , 2 sinhtanh , cosh 11sech and cosech .cosh sinh xx xx x x xx x xx xx

They are known as hyberbolic functions. They are used in modeling suspended bridges and equations of motion related to skydiving. (i) Find an expression for tanh x in terms of 2e x . [1] (ii) Given that 11f( +1) f ( ) sinh sech sech 22nnx n x n x ªº ªº§· §· ¨¸ ¨¸«» «»©¹ ©¹¬¼ ¬¼ , where 1f( ) t a n h 2nn x§· ¨¸ ©¹ and 0xz , find an expression for 1 11sech sech22 N N n Sn x n x ªº ªº§· §· ¨¸ ¨¸«» «»©¹ ©¹¬¼ ¬¼¦ in the form 1cosech tanh tanh 2xA x x§· ¨¸ ©¹ where A is a constant to be determined. [3] (iii) Explain why Sf exists. Deduce an expression for Sf in the form cosech tanhxP Q x , where P and Q are constants to be determined. [3] 6 (a) Find the roots of the equation 2(1 i) (2 2i) 0zz , giving your answers in the form ixy , where x and y are exact real numbers. [4] (b) Given that 43 2f( ) 2zz zza z b , where a , b  , and that 1 + 2i satisfies the equation f(z) = 0, find the values of a and b and the other roots. [5] Hence solve the equation 43 22i 8i 20 0ww ww , showing your workings clearly. [2] www.KiasuExamPaper.com 630

4 ©RIVER VALLEY HIGH SCHOOL 9758/01/2019 7 The area A of the region bounded by 6sinyx , the x-axis, x = 0 and x = π 2 is to be found. (i) The area A ca n be approximated by dividing the region into 5 vertical strips o f rectangles of equal width. Show that 0.10625π 0.20625π.A [3] (ii) It is given that π 22 0 sin dn nIx x ³ , where 0nt . By writing 22 1sin sin sinnnxx x , show by integration by parts that 1 21 , for 1.2 nn nII n n

t Deduce that 2 3 16I S . Using the value of 2I , obtain the exact value of A. [7] 8. Two planes have equations given by 1 2 :5 4 4 , :4 . px y z px y z (i) Explain why 1p and 2p intersect in a line and determine the cartesian equation of . [3] (ii) The plane 3p contains the line and the point 5, 3, 6Q , find a c artesian equation of 3p . Describe the geometrical relationship between these 3 planes and . [5] (iii) The line m passes through the points 0, 2, 0S and 4, 0, 0T . Find the position vector of the foot of perpendicular of S on 2p . Hence find a vector equati on of the line of reflection of m in 2p . [5] 9 On 1 January 2019, Mr Tan started a savings account with a bank with an initial deposit of $5000. The bank offered compound interest of 1% computed based on the amount of money in this account at the end of each month. Due to personal financial needs, Mr Tan withdrew $100 from the savings account at the beginning of each month starting from the second month, i.e. 1 February 2019. (i) Find the amount of money in Mr Tan’s account at the end of March 2019. [3] (ii) Deduce that the am ount of money in Mr Tan’s account at the end of the thn month is given by $ 100 101 50 1.01n . [3] (iii) Determine the month and yea r Mr Tan will deplete the savings in his account if he continues to withdraw money from this account. [2] On 1 January 2019, Mrs Tan also started a savings account with the same bank with an initial deposit of $3000. The ba nk offered her interest of $10 for the first month and increment of $5 for each subsequent month. For example, in the first 3 months, her interest was $10 for January, $15 for February and $20 for March. In add ition, Mrs Tan further deposits $50 into her savings account every mid-month starting from 15 January 2019. (iv) Determine the month and year Mrs Tan will first have more money in her account than that of Mr Tan by the end of month. [5] www.KiasuExamPaper.com 631

5 ©RIVER VALLEY HIGH SCHOOL 9758/01/2019 END OF PAPER 10 Since the 1990s, a group of scientists has started conservation work to prevent the extinction of a rare species of flying fox in the wild Western Australia forests. The number of such species, in thousands, observed in the forests at time t years after the start of the conservation is denoted by N. It is known that the death rate of the flying fox es is proportional to the nu