SRJC H2 MATH P1

1 [TURN OVER] SERANGOON JUNIOR COLLEGE 2018 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9758/1 11 Sept 2018 3 hours Additional materials: Writing paper List of Formulae (MF 26) TIME : 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on the cover page and on all the work you hand in. Write in blue or black pen on both sides of the paper. You may use a soft 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. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calcul ator are not allowed in a question, you are required to present the mathematical steps us ing mathematical notatio ns 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. At the end of the examination, fasten all your work securely together. Total marks for this paper is 100 marks. This question paper consists of 6 printed pages (inclusive of this page) and 2 blank pages.

2 Answer all questions [100 marks]. 1 Find 2 2 d25 x xxx++ . [3] 2 The complex numbers z and w satisfy the equations *2 1 5 izw z+= and 23 1 1wz+= . Find the complex numbers z and w. [6] 3 Without the use of a graphing calculator, find the range of values of x for which 218 492 xxx ≥++− . [3] Hence find the exact range of values of x for which 218 e4 e 92e xx x ≥+ +− . [3] (i) The curve with equation 2 2 19 y x−= undergoes a two-step transformations to become curve C with equation () 22 1 194 xy −−= . State the two transformations involved for the curve 2 2 19 y x−= . [2] (ii) Draw a sketch of the curve C, labelling clearly the equation(s) of its asymptote(s), intersection with the axes and the coordinates of any turning points. [3] (iii) Show that the point () 1, 3−− lies on the line 3ym x m=+ − for all real values of m. [1] (iv) Hence using the diagram drawn in (ii), find the range of values of k such that the equation () () 22 31 194 kx k x+− − −= has 2 negative real roots. [2] 4

3 [TURN OVER] 5 (i) Find ()ln 1 dxx+ for 1x >− . Show your working clearly. [2] (ii) The curve C is defined by the parametric equations () ()22 l n 12 , 22 l n 11xt t y t t=− + + = − − + + where 1t >− . Another curve L is defined by the equation () 2 21 6xy=+ − . The graphs of C and L intersect at the point A()2,1 as shown in the diagram below. Find the area of the shaded region bounded by C, L and the line 1l n 4y =− − , express your answer in the form 3 262 164233 AAA+− − , where A is an exact real constant. [6] 6 (a) (i) The twelfth, eighth and fifth terms of an arithmetic progression are three consecutive terms of a converging geometric progression of positive terms with common ratio r. Find the value of r. [3] (ii) For this part, take the value of r to be 3 4 . If the difference between the sum of the first n terms of the geometric progression and its sum to infinity is less than 0.3% of the sum to infinity, find the least value of n.[ 3 ] (b) A convergent geometric sequence of positive terms, G has a non-zero first term a and common ratio r. (i) The sum of the first n odd-numbered terms of G is equal to the sum of all terms after the ()21n − th term of G. Show that 22 121 0nnrr −+− = . [2] (ii) In another sequence H, each term is the reciprocal of the corresponding term of G. If the nth term of G and H is denoted by nu and nv respectively, show that a new sequence whose nth term is ln n n u v    , is an arithmetic progression. [2]

4 7 (a) The function f is defined by 2f : 2 8, , .x xx x x k−− ∈ >a¡ (i) State the least value of k such that 1f − exists and find 1f − in a similar form. [3] (ii) Using the value of k found in (i), state the set of values of x such that 11f f () f f ()x x−− = . [1] (b) The functions g and h are defined by g : 41 , 41, ,xx a x a ++ ≥ − ∈a¡ 2h : 10 16, , 7xx x x x+− ∈ < −a¡ . (i) Find the exact value of x for which 1h( ) h ( )x x− = . [3] (ii) Explain clearly why the composite function gh exists. [1] (iii) Find gh in the form bx + c, where b is a real constant and c is in terms of a. Explain your answers clearly. [2] (iv) State the exact range of gh in terms of a. [1] A sequence 012,,. . .uuu is such that 1 ! ru r= and 2 2 1 ,! rr rruu r − +−=+ when 2n ≥ . (i) Show that 2 2 11 2!! n r rr n rn= −− + =− . [3] (ii) Hence find () 25 8 31 1! n r rr r + = −+ − in terms of n. [3] Limit Comparison test states that for two series of the form n r rk a =  and n r rk b =  with ,0nnab ≥ for all n, if lim 0n n n a b→∞ > , then both r rk a ∞ =  and r rk b ∞ =  converges or both diverges. (iii) Given that 2 2 1 !r rr r ∞ = −−  is convergent, using the test, explain why ()2 2 1!r r r ∞ = − − is convergent. [2] (iv) Show that 1234e2 . . .22! 3! 4! 5!−< + + + + < . [3] 8

5 [TURN OVER] 9 The position vectors of points A, B, C with respect to the origin O are given by , and ab c respectively. The non-zero vectors a, b and c satisfy the equation ++=abc0 . (i) By considering the plane OAB or otherwise, explain clearly why O, A, B and C lies on the same plane. Show that ×=×= ×ab bc ca . [4] (ii) Show that the area of triangle ABC is given by k ×ab where k is a constant to be determined. [3] (iii) If b is a unit vector and a is perpendicular tob , find the length of projection of AC → onto OB → . Given that the magnitude of AC → is 2 units, deduce the angle between AC → andOA → . [5] 10 The diagram below shows a rectangular container of variable height h cm, inscribed in a right circular cone with fixed height b cm and a radius of 20 cm. The four corners of the rectangular container’s upper surface ABCD is always in contact with the conical surface. The point O is at the centre of the rectangle ABCD. The rectangular container is made with material of negligible thickness. (i) If the fixed angle BOC is ,θ show that the storage volume of the rectangular container in the shape of a cuboid is given by () 22800 sinVb b h h θ−=− . [3] (ii) Find the value of h such that the rectangular container has a maximum storage volume, leaving your answer in terms of b. [5] 20 cm h cm A D C B b cm O

6 The rectangular container is opened at its upper surface ABCD and completely filled with a type of liquid perfume. A fragrance chemist placed the container in a room and allowed the liquid perfume to evaporate. It is known that the heat energy of the liquid perfume in the container, E joules, is related to the height of the container by the equation 1 23Eh h − =− . (iii) Given that the perfume evaporates at a rate of 0.08 cm3 every hour and that the value of θ is ,6 π calculate the rate of change of heat energy of the perfume when the height of the rectangular container is 5 b cm, leaving your answer in the form 3 25pq b +  , where p and q are exact constants to be determined. [4] 11 A charged particle is