VJC 2023 H2 MATH CT QP

VICTORIA JUNIOR COLLEGE JC2 COMMON TEST 2023 CANDIDATE NAME CLASS INDEX NUMBER H2 MATHEMATICS 9758/01 Paper 1 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) Writing paper READ THESE INSTRUCTIONS FIRST Write your class 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 angl es 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. This document consists of 19 printed pages and 1 blank page. For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 Total Marks

2 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN Section A: Pure Mathematics (60 marks) 1 Express 2 7 132 x xx − −−+ as a single simplified fraction. [1] Hence, without using a calculator, solve the inequality 2 7 132 x xx − −+ . [4]

3 [ Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 2 The curve C is given by the equation e 1 3y x a −=+ , where a is a real constant. (a) Show that d ln ,d 1 3 ax yk x −= + where k is a constant to be determined. [3] (b) Find the value of a such that T, the tangent to C at x = 0, makes an angle of 45 with the positive x-axis and hence find the equation of T. [4]

4 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 3 Using the standard series from the List of Formulae (MF26), show that the series expansion for ( )ln sec x in ascending powers of x, up to and including the term in 4,x is 24 .2 12 xx+ [4]

5 [ Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 3 [Continued] By putting 4x = , show that an approximate value for ln 2 is given by 24 mn  + , where m and n are integers to be found. [2] Deduce the series expansion for tan ,x up to and including the term in 3.x [2]

6 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 4 The diagram below shows a sketch of the curve ( )fyx= . The curve cuts the x-axis at the origin and ( )6, 0 . It has turning points at ( )7, 10−− and ( )2, 2− , and an asymptote with the equation 2.x=− (a) Sketch including the points of intersection with the axes, turning points and equations of asymptotes, if any, the following graphs (i) ( )f3yx=− , [3] (6, 0) ( )fyx=

7 [ Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 4 [Continued] (ii) ( ) 1 fy x= . [3] (b) It is given that ( ) ( ) 2 2 12 f a x a x − = −  , where a is a positive constant. By sketching a suitable graph on the diagram in part (a)(ii), find the range of values of a such that the equation has 2 real distinct roots. [3]

8 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 5 Plane 1 contains the line with equation 21 , 3 3 zxy −+ = = and the point A with position vector 3−+ij . (a) Show that 1 is perpendicular to 3 2 1  −− and find the cartesian equation of 1 . [3] (b) The perpendicular to 1 from the point B with position vector 5 6 10−+i j k meets 1 at the point N. Find the position vector of N. [3]

9 [ Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 5 [Continued] (c) Find a vector equation of the line which is a reflection of the line BA in the line BN. [3]

10 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 6 The function f is defined by 2 19f: 2 xxx x −+→ + , , 2xx − . (a) Sketch the graph of ( )fyx= , stating the equation of any asymptotes and the coordinates of any points where ( )fyx= crosses the axes and of any turning points. [4]