Practice Paper B Sec 3 IP Math

1 Name: __________________________________ ( ) Class: _____ HWA CHONG INSTITUTION Practice Paper B INTEGRATED MATHEMATICS Level : Secondary Three Duration : 2 hours 30 minutes Do not open this booklet until you are told to do so. INSTRUCTIONS TO CANDIDATES Write your name, class and index number on all the work you hand in. 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. INFORMATION FOR CANDIDATES 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. You are expected to use an electronic calculator to evaluate explicit numerical expressions. The use of a graphing calculator is not permitted. You are reminded of the need for clear presentation in your answers. Omission of essential working will result in loss of marks. This question paper consists 6 printed pages, including this page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2. TRIGONOMETRY Identities Formulae for 2 0ax bx c   2 4 2 b b acx a    22sin cos 1AA 22sec 1 tanAA 22cosec 1 cotAA ABC sin sin sin a b c A B C 2 2 2 c 2 cosa b bc A   1 = sin2 ab C

3 1 (a) Factorise completely . [3] (b) Simplify . [3] 2 The area of a rectangle ABCD is given by m2, and the length of side AB is m. Find , without using a calculator, the length of side BC in the form of m, where a and b are real numbers. [3] 3 Solve the equations (a) , [3] (b) 1ln 2 ln 0, where 02x x x    . [4] 4 Two outlets of a local bakery sold the following types of cakes on a particular day. Pandan cake Chocolate cake Outlet A 100 120 Outlet B 70 85 The costs of baking a pandan cake and a chocolate cake are $5 and $6 respectively. In both outlets, each pandan cake was sold at $x, and each chocolate cake was sold at $y. It is given that 100 120 70 85  P , 5 6  Q and x y  R . (i) Write down an expression connecting P, Q and R, such that it represents the profit earned by each outlet on that particular day. [1] (ii) If the profits earned by Outlet A and Outlet B on that particular day were $1710 and $1205 respectively, find the selling price for each type of cake using the inverse matrix method. [4] 5 Find the range of values of m for which 2 ( 1)y mx m x m    is always positive for all real values of x. [4] 33( 3) ( 1)xx   1 2 12 xx x x    1 3 2  28 ( 2)ab 2 ( 1)log (3 7 2) 2x xx   

4 6 It is given that the equation 2 2 1 ( 2)x k k x    has two real roots and  . (i) Find the range of values of k. [3] (ii) If 22 11 , determine the value(s) of k. [3] 7 (a) Solve the equation 322 5 2x x x   . [4] (b) Find the remainder when 200 100( 3) ( 2)xx   is divided by 2 56xx . [4] 8 S is the sum of two parts. One part varies directly as the square of x and the other part varies inversely as y. When 2 and 3, 1x y S   . When 14 and 6, 7 2x y S   . Express S in terms of x and y. [3] 9 (a) Solve the trigonometric equation 26 4cos 30 9sin 3022 xx               for 0 720x   . [4] (b) Prove that 2 tan cot2sin 1 tan cot xxx xx   . [3] 10 Answer the whole of this question on a sheet of graph paper. The table shows experimental values of two variables x and y. It is known that x and y are related by an equation e bxya  , and that one of these values of y given in the table is subject to an abnormally large error. x 1.5 2.5 3.5 4.5 5.0 6.0 y 9.9 6.0 3.6 2.2 1.7 1.4 (i) Draw a straight line graph of ln y against x, using a scale of 2 cm to 1 unit on the x-axis and 1 cm to 0.2 units on the ln y -axis. [3] (ii) Use your graph to estimate the value of a and of b. [3] (iii) Identify the abnormal reading of y and estimate its correct value. [2]

5 11 A function f is defined, for 0x , by f : 1 ax x , where a is a real constant. Given that 12f( 1) f (2) 4   , calculate the value of a. [4] 12 The function f is given by f : 2 3cos 2xx  for the domain 0 180x   . Sketch the graph f( )yx . Hence, state the range of the function for its given domain. [4] 13 (a) Sketch the graphs of 10xy and 1 (2 1)( 2)2y x x   on the same axes, showing clearly the x-intercepts, y-intercepts and turning points, if any. Hence, state the range of values of x such that 110 (2 1)( 2)2 x xx   . [5] (b) Describe the sequence of transformations to obtain a graph of 2ln( 1)yx from the graph of e1xy . [2] 14 Three points, S, Y and T are on level ground. Y is the foot of a vertical flagpole XY, and S is 30 m due south of Y. (i) The angle of elevation of X from S is 25 . Calculate the height of the flagpole. [2] (ii) The bearing of T from S and the bearing of T from Y are 042 and 075 respectively, calculate the distance ST. [3] (iii) A man is standing at the midpoint of ST. Calculate the angle of elevation of X from the man. [4] T N X S 30 Y

6 15 Given that  is an angle that lies in the 4th quadrant, simplify the expression 2 2 1 2cot 1cos 1 tan 1sin       . [3] 16 The diagram shows two intersecting circles 12 and CC with centres P and Q respectively. A straight line L passes through M and N, which are the points of intersections of the two circles. (i) Given that the equation of the circle 1C is 22 6 4 21 0x y x y     , find the coordinates of P and the radius of 1C . [3] (ii) It is also given that the equation of the line L is 46yx , find the coordinates of M and of N. [4] (iii) Find the equation of the perpendicular bisector of MN. [3] (iv) Given that the radius of circle 2C is 1 2212 units, find the coordinates of Q. [4] (v) Calculate the area of the quadrilateral PMQN. [2] End of Paper x y C1 C2 L N Q M P O