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Slide 1: Name Reg. No. DUNMAN HIGH SCHOOL Class 4 ( ) PRELIMINARY EXAMINATION 2009 SECONDARY FOUR __________________________________________________________________________ 4016/02 MATHEMATICS PAPER 2 15 SEPTEMBER 2009 100 MARKS 2 hours 30 minutes TUESDAY Additional Materials: Writing Paper Graph paper (1 sheet) String ___________________________________________________________________________ INSTRUCTIONS TO CANDIDATES Do not open this booklet until you are told to do so. Write your name, Register Number and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid / tape. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π , use either your calculator value or 3.142, unless the question requires the answer in terms of π . At the end of the examination, fasten all you work securely together. The number of marks is given in brackets [ The total of the marks for this paper is 100. ________________________________________________________________________ This question paper consists of 14 printed pages. [Turn over ] at the end of each question or part question.
Slide 2: 2 Mathematical Formulae Compound interest r⎞ ⎛ Total amount = P ⎜1 + ⎟ ⎝ 100 ⎠ Mensuration n Curved surface area of a cone = πrl Surface area of a sphere = 4πr 2 Volume of a cone = 12 πr h 3 43 πr 3 Volume of a sphere = Area of triangle ABC = 1 ab sin C 2 Arc length = rθ , where θ is in radians Sector area = 12 r θ , where θ is in radians 2 Trigonometry a b c = = sin A sin B sin C a2 = b2 + c2 – 2bc cos A. Statistics Mean = ∑ fx ∑f Standard deviation = ∑ fx ∑f 2 ⎛ ∑ fx ⎞ −⎜ ⎜∑f ⎟ ⎟ ⎝ ⎠ 2
Slide 3: 3 p2 − q p = , express q in terms of p. 2 q Express as a fraction in its lowest terms, 1 (a) (b) Given that [3] 3 − 2x x . − x − 5x + 6 3 − x 2 [3] ________________________________________________________________________________ 2 1st pattern 2nd pattern 3rd pattern In the diagram above, each pattern is made up of dots, lines and small triangles. In the 1st pattern, there are 9 dots, 15 lines and 7 small triangles. How many small triangles are there in the (i) (ii) (b) (c) (a) 4th pattern, n th pattern? [2] [1] [2] How many lines are there in the n th pattern? If there are d dots, l lines and T triangles in one of these patterns, write down an equation connecting d, l and T. ________________________________________________________________________________ [Turn over
Slide 4: 4 3 A cylindrical container which has an internal diameter of 60 cm and an internal height of 1.05 m weighs 7 kg when empty. (You may assume that π = (a) 22 .) 7 Find the weight of the container when it is full of oil, if the density of oil is 7 g/cm3 . 9 (b) How many times will the oil in the container fill a hemispherical bowl of internal diameter of 7 cm? [5] [2] Find the internal surface area of the hemispherical bowl in contact with the oil. (c) ________________________________________________________________________________ In May 2007, the Credit Bureau Singapore released the following data on Singaporeans’ home loans/ mortgages for the period from March 2005 to March 2007. No of Singaporeans with: March 2005 March 2006 March 2007 4 2 or more home loans 2 or more home loans valued at a total of more than S$1 million More than S$1 million 19901 1416 2381 25977 1962 2381 41078 2925 4291 in home loans The information for those Singaporeans with 2 or more home loans over this period of ⎛ 19901 ⎞ comparison can be represented by the matrix P = ⎜ 25977 ⎟ . ⎜ ⎟ ⎜ 41078 ⎟ ⎝ ⎠ The information for those Singaporeans with 2 or more home loans valued at a total of more than S$1 million over this period of comparison is represented by a matrix Q. (i) (ii) Write down the matrix Q. Calculate the matrix ( P − Q ) . [1] [1]
Slide 5: 5 (iii) Describe what is represented by the elements of ( P − Q ) . [1] The information for those Singaporeans with home loans in 2005 is represented by the matrix A = (19901 1416 2381) . Information for those Singaporeans with home loans in 2007 is represented by the matrix B. (iv) (v) Write down the matrix B. Show that the matrix C, in terms of A and/ or B, which has its elements showing the increase of each category over the period of 2005 to 2007 is [1] ( 21177 1509 1910 ) . [1] ⎞ 0⎟ ⎟ 0 ⎟ . Evaluate (100CD ) , rounding ⎟ ⎟ 1⎟ ⎟ 2381 ⎠ (vi) ⎛1 0 ⎜ 19901 ⎜ 1 A matrix D is given by ⎜ 0 ⎜ 1416 ⎜ ⎜0 0 ⎜ ⎝ off each element to the nearest whole number. (vii) [1] [2] Describe what is represented by the elements of the matrix (100CD ) . ________________________________________________________________________________ [Turn over
Slide 6: 6 5 In Singapore, the rate for the usage of water for the month of July in 2009 is as follows: Water used Water borne fee Sanitary Appliance fee Water Conservation tax : $1.17 per m3 : $0.28 per m3 : $2.80 per fitting : 30% of the amount payable for water used (before GST) Goods and Services tax (GST): 7% of all the above fees/ tax In July, the GST payable for water used only by a Pasir Ris 5-room household is $3.11. Calculate the amount, excluding GST, paid for water used in July by this household. (ii) (i) [2] [1] [2] Show that the amount of water used by this household in July, is approximately 38.0 m3. Hence, by using the result found in (ii), find the overall water bill if this household has 2 sanitary fittings. If the national average of water usage per month for a typical 5-room HDB flat in Singapore is 19.1 m3, (a) (iii) (iv) how many percent above average is the water usage for this household? [2] [1] what is the average water usage per day for a typical 5-room HDB flat in Singapore for the month of July? (b) ________________________________________________________________________________
Slide 7: 7 6 C P D B 42° R H A The points D, H, R and P lie on the circumference of a circle. DR is a diameter of the circle, DA is a tangent to the circle at D, RDC, RHA and CBH are straight lines and ˆ DRH = 42° . (a) Find, with reason, (i) (iii) ˆ DHR , ˆ DAR , (ii) (iv) ˆ RDH , ˆ RPH . [4] (b) ˆ Given also that DBH = 107° , find (i) ˆ RCH , (ii) ˆ DHC . [2] [2] (c) Show that the triangles DHR and AHD are similar. ________________________________________________________________________________ [Turn over
Slide 8: 8 Q 7 P R S A 18 cm B C The diagram shows three semicircles each of radius 18 cm with centres at A, B and C in a straight lines shown above. A fourth circle centre at P and with radius r cm is drawn to touch the other three semicircles. Given that BPQ is a straight line which is tangential to the two semicircles with centres A and C at point B, (a) (b) (c) show that r = 4.5 cm. ˆ Find the value of PAC in radians. [3] [2] [3] Calculate the area of the shaded region. ________________________________________________________________________________
Slide 9: 9 8 P Q 108.3° 0.874 km 1.3 km R T 26.3° S North In the diagram, ST represents the northward-bound MRT line. The quadrilateral PQRS formed the fence that boarded a carnival for the F1 Night Race in September. The point P is due west of S and PS is parallel to QR. Given that PRT is a straight line, ˆ ˆ ˆ QR = 0.874 km, PS = 1.3 km, PQR = 108.3° RST = 26.3° and SRT = 90° . Find (i) (ii) the bearing of R from T, the length of PR, Hence, show that PQ = 0.54 km, [1] [1] [2] [1] (iii) (iv) ˆ QPR . The base of the Singapore Flyer is at point Q. If the angle of depression of P from the highest point of the flyer is 8° , find the height, in metres, of the entire flyer. [1] (v) A man walked from P along PS and reached a point X such that the angle of elevation of the highest point of the flyer is a maximum. Find this maximum angle of elevation. (You may ignore the height of the man.) [3] ________________________________________________________________________________ [Turn over
Slide 10: 10 9 According to the Straits Times, a check on a random selection of basic goods at several supermarkets in Singapore revealed an increase in the prices since the beginning of the year. In particular, a pack of fresh chicken (between 1 to 1.3 kg) now cost 70 cents more than its original cost at the beginning of the year. In 2008, Yusof budgeted $234 for fresh chicken to be used during his wedding reception in January 2009. (i) If x represents the number of packs of fresh chicken (between 1 to 1.3 kg) which Yusof could buy at the beginning of 2009, write down an expression, in terms of x, for the original cost of a pack of fresh chicken (between 1 to 1.3 kg). [1] Yusof found that he would get 7 packs of fresh chicken (between 1 to 1.3 kg) less than that at the beginning of the year if he decided to delay the wedding reception till September 2009. Write down an expression, in terms of x, for the current cost of a pack of fresh chicken (between 1 to 1.3 kg). [1] [3] [2] [2] Write down an equation in x, and show that it reduces to x 2 − 7 x − 2340 = 0 . Solve the equation x 2 − 7 x − 2340 = 0 . Calculate the percentage increase in the price of a pack of fresh chicken (between 1 to 1.3 kg). (ii) (iii) (iv) (v) _____________________________________________________________________________
Slide 11: 11 10 Answer the whole of this question on a sheet of graph paper. In a recent Olympic diving event, a male participant stood on a platform and performed a dive into the water. During the dive, the horizontal distance of the participant away from the platform, x m, and the corresponding vertical distance of the participant above the platform, y m, are related by the equation y= 13 x2 x− . 10 2 Some corresponding values of x and y are given in the table below. 0 0 1 0.8 2 0.6 3 −0.6 4 −2.8 5 −6 6 p [1] x y (a) (b) Find the value of p. Using a scale of 2 cm to 1 unit, draw a horizontal x-axis for 0 ≤ x ≤ 6 . Using a scale of 2 cm to 1 unit, draw a vertical y-axis for − 11 ≤ y ≤ 1 . On your axes, plot the points given in the table and join them with a smooth curve. [3] [2] [1] [3] (c) Use your graph to find the distance(s) the participant was from the platform when he was 0.5 m above the platform. Use your graph to find the maximum height above the platform reached by the participant. By drawing a tangent, find the gradient of the curve at the point (3, −0.6). What can be said about the movement of the participant at this instant? The participant entered the water when he was 4.4 m away from the platform horizontally. Use your graph to determine the height of the platform above the (d) (e) (f) water for this participant. (g) [1] [1] Is the graph useful in finding the position of the participant beyond a horizontal distance of 4.4 m? Justify your answer. ________________________________________________________________________________ [Turn over
Slide 12: 12 11 A bag holds some coloured balls. There are 15 red, 3 blue and 2 white balls. Two balls are picked from the bag at random and the colours are noted. The tree diagram below shows the possible outcomes and some of their probabilities. Second Pick b Red Blue White 15 19 2 19 Red Blue White Red Blue White First Pick Red 3 4 3 19 2 19 3 20 Blue c a 15 19 White d 1 19 (a) State, leaving your answers as fractions in lowest terms, the values of a, b, c and d. [2] Expressing your answers as a fraction in its lowest terms, find the probability that (i) (ii) (b) both balls are white, at least one ball is red. [1] [2] ________________________________________________________________________________
Slide 13: 13 12 In a bid to make our society more environmentally friendly, a survey was conducted and the cumulative frequency curve shown on page 14 illustrates the number of plastic bags used, by 200 Singaporeans in a week. (a) Use the graph to find (i) (ii) (iii) the median number of plastic bags used, the lower quartile, the interquartile range, [1] [1] [1] (b) A person is considered to be a ‘reddie’ if he uses more than 18 plastic bags in a week. A Singaporean is chosen at random. Calculate, leaving your answer as a fraction in its lowest term, the probability of getting a ‘reddie’. [2] Given that 19.5% of Singaporean surveyed are ‘green crusaders’, use the graph to find the minimum number of plastic bags used by a Singaporean who is not a green crusader. [2] [2] The frequency table for this set of data is given below. Showing your method clearly, prove that the values are as shown in the table. Number of plastic bags used per week 0< x≤4 4< x≤8 (c) (d) Number of Singaporeans surveyed 10 29 52 75 30 4 [3] [2] 8 < x ≤ 12 12 < x ≤ 16 16 < x ≤ 20 20 < x ≤ 24 (e) Calculate, (i) (ii) the mean, the standard deviation. (f) A similar survey was also conducted in Hong Kong and the table below shows the results of the processed data. Mean Standard Deviation 11.96 2.90 Compare, briefly, the results for the two countries. [1] [Turn over
Slide 14: Cumulative Frequency 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 14 Cumulative Frequency Curve showing the distribution of number of plastic bags used by 200 Singaporeans in a week ________________________________________________________________________________ Number of plastic bags used in a week 200 0 5 10 15 20 25 --- End of Paper 2 ---