2019 J2 TMJC H2 Math Prelim (with ans)

Tampines Meridian Junior College 2019 JC2 Prelim H2 Math CANDIDATE NAME CIVICS GROUP __________________________________________________________________________________ H2 MATHEMATICS 9758/01 Paper 1 18 September 2019 3 hours Candidates answer on the question paper. Additional material: List of Formulae (MF26) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and civics group 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 different level of accuracy is specified in the question. The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. 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 100. _________________________________________________________________________________ TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION For Examiners’ Use 1 2 3 4 5 6 7 8 9 10 11 Total [Turn Over www.KiasuExamPaper.com 806

2 1 Using an algebraic method, solve the inequality    2 1 1.4 21 1 x x x x   [4] Hence or otherwise, solve the inequality    2ee 1 1 e 4 21 e 1 xx xx     , leaving your answers in exact form. [2] 2 Referred to the origin O, A is a fixed point with position vector a, and d is a non-zero vector. Given that a general point R has position vector r such that rd ad ,s h o w that ,ra d where  is a real constant. Hence give a geometrical interpretation of r . [3] Let 12 2 and 1 35           ad . By writing r as x y z      , use rd ad to form three equations which represent cartesian equations of three planes. State the relationship between these three planes. [3] 3( i) The sum of the first n terms of a sequence is denoted by nS . It is known that 5 30S  , 14 168S  and 18 10 9.SS  Given that nS is a quadratic polynomial in n, find nS in terms of n.[ 4] (ii) The nth term of the sequence is denoted by nT . Find an expression for nT in terms of n. Hence find the set of values of n for which 12nT  .[ 4] 4( i) Given that f is a strictly inc reasing continuous function, explain, with the aid of a sketch, why  1 0 10 1 1f f ... f f d n xxnn n n  ½     ¾       ¿  , where n is a positive integer. [3] (ii) Hence find the least exact value of k such that 024 2 2 1 e e e ... e n nnn n kn      , where n is a positive integer. [2] www.KiasuExamPaper.com 807

3 5 It is given that  3f4 xx x . (i) On separate diagrams, sketch the graphs of fyx and fyx , showing clearly the coordinates of any axial intercept(s) and turning point(s). [4] (ii) Find the exact value of the constant k for which   2 02 f d f d k xx xx   .[ 4] 6 Show that     2cos sin sin 1 sin 1rr r   {     .[ 1] By considering the method of differences, find  1 cos n r r   where 0 2  . (You need not simplify your answer.) [3] Hence evaluate the sum 19 20 21 56 57cos cos cos cos cos ,666 66                    leaving your answer in exact form. [4] 7 The function f is defined by  for 1, where is any positive odd integer, f for 1, where is any positive even integer.2 nx nxn n x nxn x n n          (i) Show that f 1.5 0.5 and find f 2.5 .[ 2] (ii) Sketch the graph of fyx for 15 x .[ 2] (iii) Does f have an inverse for 15 x ? Justify your answer. [2] (iv) The function g is defined by 21g: , , 1 1 xxx xx   6 . For 23 x , find an expression for gf x and hence, or otherwise, find  1 2gf 3     .[ 4] [Turn Over www.KiasuExamPaper.com 808

4 8 At the start of an experiment, a particular solid substance is placed in a container filled with water. The solid substance will begin to gradually dissolv e in the water. Based on experimental data, a student researcher guesses that the mass, x grams, of the remaining solid substance at time t seconds after the start of the experiment satisfies the followi ng differential equation   d1 1d1 x xx ktk  , where k is a real constant and 3k  . (i) Show that a general solution of this differential equation is ln 1 xk tCx   , where C is an arbitrary real constant. [3] For the rest of the question, let 4k  . It is given that the initial mass of the solid substance is 3 grams. (ii) Express the particular solution of the differential equation in the fo rm fxt . [4] (iii) Find the exact time taken for the mass of the solid substance to become half of its initial value. [2] (iv) Sketch the part of the curve with the equation found in part (ii) which is relevant in this context. [2] www.KiasuExamPaper.com 809

5 9 From a point O , a particle is projected with velocity 1 msv  a t a f i x e d a n g l e o f elevation  from the horizontal, where v is a positive real constant and 0. 2  The horizontal displacement, x metres, and the vertical displacement, y metres, of the particle at time t seconds may be modelled by the parametric equations    2cos , si . n5xv t v t yt  (i) Using differentiation, find the maximum height achieved by the particle in terms of v and . (You need not show that the height is a maximum.) [3] The particle is now projected from point O situated at a height of 29 m above the horizontal ground. The particle hits the ground at A which is at a horizontal distance of 104.4 m from .O The maximum height (measured from horizontal ground) that the particle reaches is 57.8 m. The diagram above shows the path of the particle (not drawn to scale). (ii) Find the time taken for the particle to hit the ground at A and find the corresponding value of .v [5] (iii) Find the exact gradient of the tangent at A .[ 2] O , a p a rt 29 m 104.4 m Horizontal ground [Turn Over www.KiasuExamPaper.com 810

6 10 Two houses, A and B, have timber cladding on the end of their shed roofs, consisting of rectangular planks of decreasing length. (i) The first plank of the roof of house A has length 350 cm and the lengths of the planks form a geometric progression. The 20th plank has length 65 cm. Show that the total length of all the planks must be less than 4128 cm, n o matter how many planks there are. [4] House B consists of only 20 planks which are identical to the first 20 planks of house A. (ii) The total length of all the planks used for house B is L cm. Find the value of L, leaving your answer to the nearest cm. [2] (iii) Unfortunately the construction company misunderstands the instr uctions and covers the roof of house B wrongly, so that the lengths of the planks are in arithmetic progression w ith common difference d cm. If the total length of the 20 planks is still L cm and the length of the 20th plank is still 65 cm, find the value of d and the length of the longest plank. [4] It is known that house C has timber cladding on the end of its shed roof, consisting of rectangular planks of increasing length. The first plank of the roof of house C has length 65 cm and the lengths of the planks are in arithmetic progression with common difference 11 cm. The total length of the first