2019 J2 JPJC H2 Math Prelim (with ans)

JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2019 MATHEMATICS 9758/01 Higher 2 18 September 2019 Paper 1 3 hours 1( i) The first four terms of a sequence are given by 1 13u  , 2 12.8u  , 3 1.8u  and 4 38u  . Given that mu is a cubic polynomial in m, find mu in terms of m. [3] (ii) Find the range of values of m for which mu is greater than 2000. [2] 2( a) Express 31 2 xy x   i n t h e f o r m 2 ByA x  , where A and B are constants to be found. Hence, state a sequence of transformations that transforms the graph of 1y x to the graph of 31 2 xy x   .[ 4] (b) It is given that 2g( ) 2 2xx x . Sketch the graph of g( )yx , stating clearly the coordinates of any turning point s and axial intercepts. Find numerically, the volume of revolution when the region bounded by the curve g( )yx and the line 5y  is rotated completely about the x-axis. [5] 3 Referred to an origin O, the points A and B are such that OA  a  and OB  b  . The point C is such that OACB is a parallelogram. The point D is on BC such that BD BC     and the point E is on AC such that AE AC     , where  and  are positive constants. The area of triangle ODE is k times the area of triangle OCE. (i) By finding the area of triangle ODE and OCE in terms of a and b, find k in terms of  and  .[ 6] (ii) The point F is on OC and ED such that :6 :1OF FC  and :3 : 4DF FE  . By finding the values of  and  , calculate the value of k.[ 3] 4 The curve C has equation 2 5 2 xy x   . (i) Prove, using an algebraic method, that C cannot lie between two values to be determined. [4] (ii) Sketch C, showing clearly the equations o f any asymptotes and coordinat es of any turning points and axial intercepts. [4] (iii) By adding a suitable graph to your sketch in (ii), deduce the range of values of h for which the equation      2 22 222 51 2 2xx x h x     has at least one positive real root. [3] [Turn over www.KiasuExamPaper.com 342

2 5 Do not use a graphing calculato r in answering this question. (a) (i) It is given that 1 35 iw   . Find the value of 3 1w , showing clearly how you obtain your answer. [2] (ii) Given that 35 i is a root of the equation 3241 4 0wp wq w   , using your result in (i) , find the values of the real numbers p and q. [3] (iii) For these values of p and q, find the other two roots of the equation in part(ii). [3] (b) It is given that 13 iz   . Find the set of values of n for which * n z z is purely imaginary. [4] 6 The function f is defined for all real x by  22fe 9 exxx  . (i) Show that f ' 0x  for all x. [2] (ii) Show that the set of values of x for which the graph fyx is concave upward is the same as the set of values of x for which f0 x  , and find this set of values of x, in the form of kln3, where k is a constant to be found. [3] (iii) Sketch the graph of fyx , showing clearly any points of intersections with the axes. [2] (iv) Hence, find the exact value of 2 22 0 e9 e dxx x . [4] 7 It is given that   224 , for 0 2 ,f 2 2 , for 2 4 , ax x ax ax a a x a        and that   ff 4xx a for all real values of x, where a is a positive real constant. (i) Evaluate f 2019a in terms of a. [1] (ii) Sketch the graph of fyx for 35ax a  . [3] The function g is defined by  22g: 4 2 , 2 4 .xa x a a x a  6 (iii) Determine whether the composite function gf exists, justifying your answer. [1] (iv) Give, in terms of a, a definition of fg. [2] (v) Given that  1 7fg (27) 2 a   , find the exact value of a. [2] www.KiasuExamPaper.com 343

3 8 Fig.1 Fig.2 Fig.3 Fig. 1 shows a metal sheet, ABC, in the form of an equilateral triangle of side 4 a cm. A kite shape is cut from each corner, to give the shape as shown in Fig. 2. The remaining metal sheet shown in Fig. 2 is bent along the dotted lines, to form an open triangular prism of height p cm shown in Fig. 3. (i) Show that the volume of the prism is given by  2 32 3Vp a p cm 3 . [3] (ii) Without using a calculat or, find in terms of a, the exact value of p t h a t g i v e s a stationary value of V, and explain why there is only one answer. [6] (iii) The prism is used by a housewife as a mould for making a dessert. To make the dessert, The housewife has to fill up 3 4 of the mould with coconut milk. The cost of coconut milk is 0.4 cents per cm 3 . What is the exact maximum cost in terms of a she needs to pay for the coconut milk? [3] 9 Find (a) 1 2 e d x xx , [2] (b) cos cos 2 dkx k x x  , where k is a positive constant, [2] (c)  1tan 3 dxx x  . [6] 10 (a) The Deep Space spacecraft launched in October 1998 used an ion engine to travel from Earth to the Comet Borrelly. The average speed of the spacecraft in October 1998 was 44 000 km/hr. The monthly average speed, nv of the spacecraft in month n based on its first 5 months of operation was given by: Month, n 12 3 4 5 Average speed, nv 44 000 44335 44 670 45 005 45 340 Assume that nv follows the same increment for the rest of its flight. (i) State a general formula for nv in terms of n. [1] (ii) In which month and year did the average speed of the spacecraft first exceed 53 500 km/hr ? [3] C A B p 4a p p p pp p [Turn over www.KiasuExamPaper.com 344

4 (iii) Assume that there are 30 days per month. It is known that the t otal distance travelled by the spacecraft from Earth is given by  1 n r r vT   where T is the time taken, in hours, by the spacecraf t to travel in one month. Give n that the spacecraft travelled from Earth continuously for 3 years to rea ch Comet Borrelly, find the total distance that it travelled. [3] (b) Dermontt’s Law is an empirical formula for the orbital period of major satellites orbiting planets in the solar system. It is represented by the equation 0 n nTT C , where nT is the orbital period, in days, of the (n + 1)th satellite and C is a constant associated with the satellite system in question. It is known that the planet Jupiter has 67 satellites. The orbital period of its first satellite is 0.44 days and C = 2.03. (i) Find the longest orbital period of a satellite of Jupiter. [2] (ii) Find the largest value of n for which the total orbita l periods of the first n satellites of Jupiter is within 5  106 days of the orbital period of the 20th satellite of Jupiter. [3] www.KiasuExamPaper.com 345

JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2019 MATHEMATICS 9758/02 Section A : Pure Mathematics [40 Marks] 1 A sequence 0a , 1a , 2a , … is given by 0 3 5a  and 1 3n nnaa n  for 0n  . By considering  1 1 0 n rr r aa     , find a formula for na in terms of n. [5] 2 In this question, you may use expansions from the List of Formula (MF26). (a) (i) Find the Maclaurin expansion of ln(cos 3 x) in ascending powers of x, up to and including the term in 6x . [5] (ii) Hence, state the Maclaurin expansion of tan 3x, up to and including the term