2019 JC1 RVHS H2 Maths CT Question paper

RIVER VALLEY HIGH SCHOOL 2019 JC1 Common Test Higher 2 NAME CLASS INDEX NUMBER MATHEMATICS Paper 1 Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) 9758/01 4 July 2019 2 hours 30 minutes READ THESE INSTRUCTIONS FIRST This document consists of 25 printed pages and 3 blank 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 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. Up to 2 marks may be deducted for poor 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 85.

2 1 In the annual Best Infantry Unit Competition, 3 top units – Units A, B and C, ar e shortlisted for the eventual champion. The organizer decides to award , and x y z points for each NAPFA Gold Award, Shooting Marksmanship Badge and Army Proficiency Badge respectively and to declare the unit with the highest total points as the champion. With the emphasis of training of more marksman in the units, the organiz er decides to have the points awarded for each Shooting Marksmanship Badge to be equal to the sum of points for an NAPFA Gold Award and an Army Proficiency Badge. The number of NAPFA Gold Awards, Shooting Marksmanship Badges and Army Proficiency Badges achieved by each unit are summarized in the following table. NAPFA Gold Awards Shooting Marksmanship Badges Army Proficiency Badges Unit A 24 27 118 Unit B 29 21 124 Unit C 26 25 121 Upon computation of points, it is known that the unit with the most NA PFA Gold awards scores a total of 295 points and it falls behind the unit with the most Marksmanship Badges by 3 points. By solving a system of equations involving , and x y z , determine the eventual champion of the competition. [5]

3 [Turn over

4 2 (i) Show that 22 2 1xx−+ is always positive for all real x. [1] (ii) Hence, without using a calculator, solve ( )( ) 32 2 22 0 2 1 1 x x x xx −+  +− . [3]

5 [Turn over (iii) Using your result in part (ii), find the exact range of values of x for which ( ) ( ) ( )( ) 32 2 2 ln 2 ln ln 0 2ln 1 ln 1 x x x xx −+  +− . [3]

6 3 The functions f and g are defined below. ( ) ( ) ( ) 2 2 for 0 2,f 2 ln 2 for 2 4 g 2 1 for 0 1 xx x xx x x x x  = −  −   =− + +   (i) Sketch the graph of y = f(x). [3] (ii) State the range of g and show that the composite function fg exist. [2]

7 [Turn over (iii) Find the range of the composite function fg. [2]

8 4 The curve C has the equation 2 3ax x by xc −+= − where a, b, and c are real constants. It is given that the equations of asymptotes of C are 2x= and 1yx=− . Also, C passes through the point P 30, 2  − . (i) Explain why the value of a is 1. State the value of c and show that b = 3. [3] (ii) Sketch the curve C, indicating clearly the equations of asymp totes, axial intercepts and stationary points. [2]

9 [Turn over (iii) Find the exact equation of the normal to C at the point P. [3]

10 5 The triangle ABC is such tha t 3AB= , 4AC = and angle BAC =  radians. Given that  is sufficiently small, show that 2 2 4 1 12 1BC p q    +  + + for constants p and q to be determined. [4] By using the above result and substituting 1 4 = , find an approximate value of 7 , leaving your answer as a single fraction. [2] Without further calculations, state with a reason whe ther the approximate value obtained by the substitution 1 10 = would be a better estimate for 7 . [1]