VJC 9758 2024 Promo Qns

2024/VJC/Math Dept VICTORIA JUNIOR COLLEGE JC1 PROMOTIONAL EXAMINATION H2 MATHEMATICS 9758 QUESTION PAPER 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 an approved graphing calculator. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. 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 5 printed pages and 3 blank pages.

2024/VJC/Math Dept 1 A small garment shop produces dresses, blouses and skirts. The production process includes cutting, sewing and packaging. The table below shows the time in mi nutes required to produce each piece of garment. The total time spent on cutting, sewing and packaging are 3150, 4300 and 480 minutes per week respectively. Garment Time required (min) Cutting Sewing Packaging Dress 45 70 8 Blouse 50 60 6 Skirt 20 25 3 Write down and solve equations to find the number of pieces of each garment the shop produces each week. [4] 2 (a) Without using a calculator, solve the inequality 2 2 23 1 0 2xx xx   . [3] (b) Hence, find the set of values of x for which 2 2 2e 3e 10 2e+ e xx xx   . [2] 3 The diagram shows the curve f( )yx which cuts the axes at 0,Aa  and  ,0Bb , where a and b are positive constants. (a) State, if it is possible to do so, the coordinates of the points where the following curves cut the axes. (i) 1f( )yx  [1] (ii) f( 2 )yx [1] (iii) f( 3 )yx [1] (b) Sketch the curve 1 f( )y x , stating the equations of any asymptot es and the coordinates of the point where 1 f( )y x crosses the axes. [3] x y O

2024/VJC/Math Dept 4 (a) The sum, nS , of the first n terms of a sequence 123, , , . . .uuu is given by 1 2 375 5 n n nS   . Show that 1 3 5 n nuk    , where k is a constant to be determined. Find S . [4] (b) Another sequence of numbers 123, , , . . .vvv is such that 1 2 21 n i i nv n   . Find the exact value of 11 i i v    . [3] 5 A curve C has equation 223241xx y y . (a) Show that d3 .d4 yx y x yx   [2] (b) Find the acute angle between the tangent to C at the point (1, 1) and the x-axis. [2] (c) Show that there is no point on C where the tangent to the curve is parallel to the x-axis. [3] 6 (a) An infinite sequence is such that 1up= , where p is a constant, and 1 2412n n u u   , where 1n . (i) Showing your working clearly, explain why p cannot be 2.4. [2] (ii) A constant sequence is a sequence in wh ich every term is the same. Given that p > 5, find the value of p for which the sequence is a constant sequence. [2] (b) An arithmetic series has a positive first term 1v and common difference d. Given that 10 1935vv  , find the largest possible value of the sum of the first n terms, leaving your answer in terms of d. [4] 7 (a) (i) Showing your working, find the complex numbers v and w which satisfy the following simultaneous equations. 25vw 3 *15 ivw  [4] (ii) Points O, V and W on an Argand diagram represent the complex numbers 0, v and w respectively. Plot these points on an Argand diagram and state the transformation that maps the point W onto V. [2] (b) The roots of the equation 43 2 23 0zzz a z b   , where a and b are real numbers, are 1z , 2z , 3z and 4z . It is given that 2222 1234 0zzzz . Explain why at most two of 1z , 2z , 3z and 4z are real. [2]

2024/VJC/Math Dept 8 The curve 1C has equation 22(1 )4 4xy  . The curve C2 has parametric equations secx  and 2ta ny  . (a) Show that the cartesian equation of C2 is 22 22 1xy ab , where a and b are positive integers to be determined. [2] (b) Sketch 1C and 2C on the same diagram, indicating clearly the equations of any asymptotes and coordinates of vertices. [4] (c) Given that k > 0, find the range of values of k such that 22(1 )4 4kx y  cuts C2 at most twice. [2] (d) Describe a sequence of transformations that maps C1 to a unit circle with centre (0, 0). [2] 9 (a) Using standard series from the List of Formula (MF27), expand 22 cosax x a as far as the term in 4x , where a is a positive constant. Give your answer in the form 24 12 3cc x c x++ where 12, cc and 3c are in terms of a. [6] (b) Find the range of values of x, in terms of a, for which the expansion in part (a) is valid. [1] (c) Using your answer in part (a) and a suitable value of a, show that 1 2 0 cos 2 119 d49 6 0 x xx  . [3] 10 (a) (i) State the derivative of tane x . [1] (ii) Hence, find   tan 3es e cs i n dx x xx . [2] (b) Write down constants A and B such that, for all values of x, 14 ( 38 )x Ax B  . Hence, find 2 14 d 14 3 x x xx   . [5] (c) Use the substitution 1ux to find the exact value of  2 3 3 1 d 1 x x x . [4]

2024/VJC/Math Dept 11 The functions f and g are defined by 2 1f 51 3: xx x   for , 1xx  , g: l n 1xx  - for 2x . (a) Sketch the graph of fyx , stating the equations of any asympt otes, the coordinates of any turning points and points of intersections with the axes. [4] (b) Explain why gf does not exist. [1] (c) Show that g has an inverse and find  1g x . [4] (d) Find the range of 1fg . [2] 12 [The volume of a square-based pyramid is 1 base area height3 .] A manufacturer makes open containers in the shape of an inverted square pyramid with square base of side length l cm, vertical height h cm and fixed slant edge k cm, as shown above. He decides that the volume of each container, V cm3, should be as large as possible. (a) Show that 24 6 2 91 8 kl lV  . [2] (b) Use differentiation to find, in terms of k, the exact maximum value of V, proving that it is a maximum. [6] (c) Water is poured into an empty contai ner with the dimensions found in part (b) at a constant rate of 1 3 cm3 per second. It is given that at time t seconds, the volume of water in the container is W cm3 and the water level is p cm. Show that 34 3Wp and hence, find the rate of increase of p at the instant when 4 kp . [4] l k h

2024/VJC/Math Dept BLANK PAGE

2024/VJC/Math Dept BLANK PAGE

2024/VJC/Math Dept BLANK PAGE