华中中三E Math Practice B

1 Hwa Chong Institution Secondary 3 Examination (Practice B) CANDIDATE NAME CLASS REGISTER NUMBER Mathematics 2 hours Additional materials: Writing papers READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work that you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Answer all 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. The use of an approved scientific calculator is expected, where appropriate. For  , use either your calculator value or 3.142, unless the question requires the answer in terms of . You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 80. This document consists of 7 printed pages.

2 Mathematical Formulae Compound interest Total amount = 1 100 nrP Mensuration Curved surface area of a cone = rl Surface area of a sphere = 24 r Volume of a cone = 21 3 rh Volume of a sphere = 34 3 r Area of triangle ABC = 1 sin2 ab C Arc Length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Trigonometry sin sin sin abc A BC 22 2 2c o sab c b c A Statistics Mean = fx f   Standard deviation = 2 2 fxf x f f   

3 1 (a) Factorise completely 236 4 41x xy y y   . [ 3 ] (b) Express as a single fracti on in its simplest form 15 1227xx xx      . [3] (c) It is given that 22 2 azx y z  where 0z . Express a in terms of , and x yz . [3] 2 (a) Solve the equation 13 221x x  . [ 3 ] ( b ) Solve the simultaneous equations 2 6xy x yx    . [4] 3 A factory produces two types of toys – handmade and machine-made. The following table shows the cost of the components used in producing each toy. Labour (hours) Material (boxes) Electricity (units) Handmade 8 5 0.7 Machine-made 3 5 0.4 (a) Represent the information given in the table as a 23 matrix P. [1] (b) The labour costs $7 per hour, the material used costs $1.20 per box and electricity costs 60 cents per unit. (i) Represent this information as a column matrix C. [1] (ii) Evaluate the matrix T = PC. [ 2 ] (iii) State what the elements in T represent. [1] 4 The function f is defined by 2f: 1 5 8 2x xx . (i) Express  f x in the form 2()ax b c  where a, b and c are constants. [2] (ii) Hence, solve 215 8 2 0xx  . Leave your answers in exact form. [2] (iii) State the equation of the line of symmetry of  f x . [ 1 ] [Turn Over

4 5 The diagram below shows a straight line obtained by plotting 3 y x against x. (i) Express y in terms of x. [3] (ii) Amelia claims that the point (5, 215) lies on the straight line. Is she correct? Justify your answer. [1] 6 The diagram shows the cross section of a cylindrical barrel floating in water. The radius of the barrel is 30 cm and its highest point P is 40 cm above the water level AB. O is the centre of the circular face. Calculate (i) the angle AOB i n r a d i a n s , [ 2 ] (ii) the major arc length APB, [ 2 ] (iii) the percentage of the area of th e barrel that is below the water level. [3] O A P B 40 cm 30 cm (10,75) (1,13) x

5 7 (a) T h e curve  1yx x k  cuts the x-axis at A and B, and y-axis at C. It also passes through the point D (20, 21). Find (i) the value of k, [ 2 ] (ii) the area of triangle ABC, [ 2 ] (iii) the equation of the straight line parallel to CD and passing through A. [2] y x (b) Find the equation of the tangent at the point 24,255A  on the circle 22 244xy xy . [4] 8 A straight line passing through the point 1, 4 intersects the curve 22 67 0yxx y at P 1 ,32   . Find (i) the coordinates of the point Q at which the line meets the curve again, [4] (ii) the equation of the perpendicular bisector of PQ. [2] 9 (i) Given that f( ) lnxx and g( ) 4ln 3xx  , describe the transformation of the graph f( )x i n t w o s u c c e s s i v e s t e p s . [ 3 ] (ii) Sketch the graph of 4l n 3yx  , indicate clearly the asymptote, y-intercept and x-intercept. [3] A B C D (20, 21) [Turn Over

6 10 The diagram shows points A, B, C and D on level ground, with A due North of B. Given that angle BAD = 32, angle ADB = 46, BD = 6 km, BC = 14.8 km and CD = 10 km, calculate (i) angle DBC, [ 2 ] (ii) bearing of B from C, [ 2 ] (iii) shortest distance from D to BC. [ 2 ] TD represents a vertical tower. The angle of elevation of T from B is 47.5, find (iv) the height of the tower (correct to the nearest metres), [2] (v) the maximum angle of elevation of the top of the tower T when observed along BC. [2] A B C D 6 km 10 km 32 46 14.8 km N

7 11 In the figure, PQRS is a rhombus, with vertices  3,14P  and 9,13Q .The diagonals of the rhombus intersect at  1, 9T  . (i) Find the coordinates of S and of R. [4] (ii) Hence, find the area of rhombus PQRS. [3] 12 The graphs of () ( )yx xp x q  , where p and q are positive integers, and 41yx are drawn on the same axes as shown in the diagram below. (a) State the values of p and of q. [ 2 ] (b) The points of intersection of the curve and the straight line give the solutions of the cubic equation. Using your answer in (a), find the cubic equation, express your answer in the form 3240xb xc x d   . [ 2 ] End of Paper