Paper G Sec 3

1 Name: __________________________________ ( ) Class: _____ HWA CHONG INSTITUTION PAPER G 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 7 printed pages, including this page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c  2 4 2 bb a cx a   Mensuration Arc Length = r , where  is in radians Sector area = 21 2 r  , where  is in radians 2. TRIGONOMETRY Formulae for ABC sin sin sin abc A BC 22 2 c2c o sab b c A 1=s i n2 ab C

3 1 (i) Express 2 32 2 42 x x x   as a single fraction in its simplest form. [3] (ii) Hence, solve 2 23 32 2 2 4 42 8 xx x xx x    . [3] 2 (a) Without using a calcula tor, given the equation 1125x x  , ( i ) find the value of 2 5 x   . [2] (ii) Hence, solve for x , leaving your answer in exact form. [2] (b) Solve the equation 15xx  . [4] (c) A rectangle has length  51 2 cm and the area is   2823 c m . Find the breadth of the rectangle in the form  3 c mab , where a and b are integers. [4] 3 (a) The equation   2 310xk x kk  has two non-zero roots. Find the range of values of k . [3] (b) The function g is defined by 4g: 3 2x x  , 2x . (i) Find 1g . [3] (ii) Hence, state the domain and range of 1g . [2] 4 The polynomial 32P( ) 3 2 7 2yyyy  . ( i ) Show that 1y is a root of P( ) 0y  . [1] (ii) Factorise 32327 2yyy  completely. [3] (iii) Hence , solve 64 22 481 42yy y  , leaving your answers in exact form. [2] 5 (i) Show that   66log 2 1 log log 2 1x x xx .   [1] (ii) Hence, solve  66log log 2 1 log 1xx xx   . [3] [Turn Over

4 6 A curve has the equation 21 42yx x  . (i) Using completing the square met hod, show that the minimum point on the curve has coordinates 114 2   , . [3] (ii) Find the coordinates of the poi nts at which the curve intersects the x-axis. [2] (iii) Hence, sketch 21 42yx x  for 26 x  , indicating clearly the x-intercepts, y-intercept and turning point. [3] (iv) Write down the range of values of k such that 21 42 x xk  has 3 solutions. [1] 7 (i) State precisely 2 successive geo metrical transformations which would transform the graph of xye  into the graph of 2 xye  . [2] (ii) Hence, sketch the graph of 2 xye  , showing clearly the x-intercept , y-intercept and asymptote if any. [3] 8 The diagram shows a circle with centre  31A , and radius 7 units. (i) Given that the equati on of the circle is 22 22 0x ya x b y c ,   find the value of a, of b and of c. [4] The points B and C lie on the circle such that AB is parallel to the y-axis and the line BC passes through the origin. (ii) Write down the coordinates of B. [1] (iii) Find the coordinates of C. [4]

5 9 (a) Given that 82 11A   and 21 32B   , find matrix C such that 22 3CB A . [3] (b) Tea D’light sells beverages at Orchard and Ang Mo Kio outlets. The number of cups of milk tea, white coffee and green tea sold at the two outlets on the first day of reopening in Phase 2 on 19th June was shown in the table below. (i) The information in the table can be represented by a 23 matrix Q. Write down the matrix Q. [1] The profit made from selling milk tea, white coffee and green tea are $3.00, $1.80 and $2.00 respectively. (ii) Write down a column matrix P to represent the above information. [1] (iii) Given that V = QP, find matrix V. [2] (iv) Using matrix multiplication, find the total amount of profit made by Tea D’light on the first day of reopening in Phase 2. [2] 10 The figure shows part of a straight line graph obtained by plotting 2x y against x . Express y in terms of x . [3] Orchard Ang Mo Kio Milk Tea 400 450 White Coffee 250 200 Green Tea 50 140 [Turn Over

6 11 Solution to this part of the question by accurate drawing will not be accepted. The diagram sho ws a trapezium ABCD with A 244555   , , C 22 18, and D 12 2, . The equation of AB is 23yx and the equation of AD is 21 6yx  . (i) Find the coordinates of B. [4] (ii) Show that 90 oABC . [2] (iii) Given that ABHD is a parallelogram, find the coordinates of H. [3] Point  16 35E , lies on AB produced. (iv) Find the value of area of triangle area of triangle AEC ABC   . [3]

7 12 In the diagram P, Q and R represent three small islands. I s l a n d P is 26 km from island Q on a bearing of 200. Island R is 30 km from island Q and on a bearing of 115. (i) Calculate the distance of island R from island P. [2] (ii) Calculate the bearing of island R from island P. [3] (iii) A boat leaves island P and travels directly to island Q. Find the shortest distance of the boat from island R. [2] 13 The diagram shows t wo circles, centres O and C with radii 8 cm and r cm respectively, which touch each other externally. PABQ is a common tangent to both circles. Given that 12AB cm, (i) show that 45r. , [2] (ii) find AOC and OCB in radians, [4] (iii) find area of the shaded region. [4] End of Paper