1 Mathematical Formulae Compound Interest Total amount = r P ( 1 ) 100 n Mensuration Curved surface area of a cone = rl Surface area of a sphere = 2 4 r Volume of a cone = 1 3 r 2 h Volume of a sphere = 4 r 3 3 Area of triangle ABC = 1 absin 2 Arc length = r, where is in radians C Sector area = 1 r 2, where is in radians 2 Trigonometry a a b c sin A sin B sin C 2 b 2 c 2 2bccos A Statistics Mean = fx f Standard Deviation = fx f 2 2 fx f
2 1 (a) Evaluate 3 2.67 + 5.47 0.43 2 3 11. Write down all the figures shown on your calculator. Answer (a)..[1] (b) Give your answer to part (a) correct to (i) 3 significant figures, Answer (b)(i).[1] (ii) 2 decimal places. Answer (b)(ii) [1] 2 Written as a product of its prime factors, 198 2 3 2 11. (a) Express 840 as the product of its prime factors. Answer (a).....[1] (b) Find the smallest positive integer that is a multiple of both 198 and 840. Answer (b).....[1] (c) Write down the smallest integer k such that 840k is a perfect cube. Answers (c) k = [1]
3 3 A map is drawn to a scale of 1: 40 000. Two buildings are 2.4 km apart. Calculate, in centimeters, their distance apart on the map. 2 5 4 Simplify 2x 3 3x Answer.. cm [2] Answer. [2] 5 8 x 1 x 5 Write 2 1 as a fraction in its simplest form. Answer. [2]
4 6 Carina bought a purse at $320 after a discount of 20%. Find the original cost of the purse. Answer $.... [2] 7 Catherine deposits $5000 in the bank. The bank pays a compound interest of 1.66% per annum. How much interest can Catherine receive after 2 years? Answer $.... [3]
5 10 8 It is estimated that by 2070 the population of the world will be1.03 10. 10 (a) 1.03 10 can be written as k billion. Find k. Answer (a).... [1] (b) 9 The population of the world in 2000 was 6.5 10. Find the estimated percentage increase in the population from 2000 to 2070. Answer (b) [2] 9 (a) Factorise 9y 2-81. Answer (a) [1] 2 (b) Factorise x 13x 36. Answer (b)... [1]
6 10 Solve9 (2k 3) 7(3k 5) 1. Answer x = [2] 11 A regular polygon has n sides. The size of each interior angle is nine times the size of each exterior angle. (a) Find the size of each exterior angle. (b) Calculate the value of n. Answer (a) [2] Answer (b) n =... [1]
7 12 Thomas has a box containing 6 red pens and 9 black pens. He selects a pen at random, without replacing the first one. What is the probability that Thomas chooses (a) two pens of the same color. Answer (a) [1] (b) one pen of each color. Answer (b) [2]
8 13 The two cups shown in the diagram are geometrically similar. The radii of their bases are 3 cm and 5 cm respectively. 25 35 3 cm 5 cm (a) Given that the height of the smaller cup is 12 cm, find the height of the larger cup. Answer (a). cm [2] (b) If both cups can hold a total volume of 380 ml of water, calculate the amount of water the smaller cup can hold. Answer (b). ml [3]
9 14 30 35 40 45 50 55 60 65 Weight (kg) The dot diagram shows the weight of 20 students in a class. (a) Find the median weight. (b) Find the interquartile range. Answer (a). kg [1] Answer (b). kg [2] (c) Two students join the class and their weights are recorded to be the same. Given that the modal weight is now 47 kg, find the weight of the two new students. Answer (b). kg [2]
10 15 Solve the simultaneous equation using any method. 2x y 6 3x 2y 26 4 Answer x =. y = [3] 2 16 By completing the square, x 6x 8 can be expressed in the form (x + p) 2 + q. (a) Find the value of p and of q. (b) Hence solve x 2-6x +8 = 0. Answer (a) p = [1] q = [1] Answer (b) x =... or.. [2]
11 17 The exchange rate between the English pound ( ) and the American dollar (US$) on a particular day was 1 to US$1.60 and the exchange rate of American dollars to Singapore dollars is US$1 = S$1.40. Calculate the amount received in Singapore dollars if 380 was exchanged on that same day. Answer S$...... [3] 18 In the quadrilateral ABCD, the diagonals AC and BD intersects at X. AX = BX and CX = DX. Show that triangle AXD and BXC are congruent. A B X D C Answer...............................................[3]
12 19 In the diagram, ABC 90, BC = 24 cm, CD = 6 cm and AD = 34 cm. A 34 cm B 24 cm C 6 cm D E (a) Find the length of AB. (b) Find the area of triangle ACD. Answer (a).. cm [1] Answer (b).. cm 2 [2] (c) Write down the value of (i) sin ADB. (ii) cos ADE. Answer (c)(i) sinðadb=.. [1] (ii) cosðade=.. [1]
13 20 y is inversely proportional to (x + 2) and when x = 5, y = 3. (a) write down an equation connecting x and y, (b) find the value of x when y 2. Answer (a) [2] Answer (b) x =... [1] 21 (a) Solve the inequality11 6x 7 50, showing the solution in the number line below. Answer (a)... [3] (b) Hence, write down the possible integer values of x which satisfy 11 6x 7 50. Answer (b) x =..... [1]
14 22 The diagram below shows a solid which is composed of a pyramid VABCD and a cuboid ABCDEFGH. Vertex V is vertically above point D and the height VD of the pyramid is 6 cm. EFGH is a square of side 8 cm. The height CG of the cuboid, is 12 cm. 6 cm V D C A B 12 cm H G E 8 cm F (a) Calculate the volume of the solid. Answer (a) cm³ [2] (b) Calculate the surface area of the solid. Answer (b) cm² [2]
15 23 (a) Sketch the graph of y 2x(5 x) in the axes provided. Answer (a) y 0 x (b) Write down the equation of the line of symmetry. [2] Answer (b) [1] 24 The first four terms in a sequence of numbers, T 1, T 2, T 3, T 4, are given below. T 1 = 1 2 + 2 2 = 5 T 2 = 2 2 + 3 2 = 13 T 3 = 3 2 + 4 2 = 25 T 4 = 4 2 + 5 2 = 41 (a) Write down the next two terms of the sequence. Answer (a).,. [1] (b) Find the expression, in its simplest form, for the nth term. Answer (b) [1]
16 25 In a quadrilateral ABCD, AB = AD = 10 cm, BC = 5 cm and CD = 7 cm. (a) Measure the angle ABC. [1] Using ruler and compasses only, (b) complete the quadrilateral ABCD, [1] (c) construct (i) the perpendicular bisector of BC, [1] (ii) the angle bisector of angle ABC. [1] Answer (b) and (c) C A B (d) The perpendicular bisector of BC meets the angle bisector of ABC at X. Mark the point X and measure AX. Answer (a) ABC.. [1] (d) AX =... cm [1]
17 Section A (44 marks) Answer all the questions in this section. 1 (a) By rounding each number to 1 significant figure, estimate the value of [2] 3375 7.25 435.2-293.8. (b) Einstein s equation, E = mc 2, is used to determine the amount of energy E for a given mass, m, which is expressed in grams. The variable c represents the speed of light in m/s. It is given that c = 300 million. (i) Express the value of c in standard form. [1] 2 (ii) Calculate the value of E when m 1.8 10. Give your answer in standard form. [2] 12 8 2 (a) Simplify. 2 [2] ( x 3) x 3 (b) Make c the subject of the formula b = 2ac. [2] a 3 (a) Janice bought an antique necklace for $620 and sold it some years later for $1085. Calculate the percentage profit. (b) The price of a washing machine is $700. Mrs. Wong decides to pay by hirepurchase. She pays a 15% deposit and then 24 equal payments of $28. Calculate the amount she pays in total by using hire-purchase. [2] [2] 4 The diagram shows the speed-time graph of a car s journey for 60 seconds. Speed (m/s) 60 0 24 40 60 Time (s) Calculate (i) the acceleration of the car when t 10 s, [1] (ii) the speed of the car at the 45 th second, [2] (iii) average speed of the car. [3]
18 5 The diagram shows a sector ABCDE with centre B and radius 13 cm. Given that the arc length is 7.28 cm and ACB 90. 13 cm (i) Show that θ = 0.56 radian. [1] (ii) Find the area of the shaded region. [3] 6 The diagram shows a right-angled triangle ABC. C x +1 cm x cm A x 7 cm B (i) Show that x 2 16x + 48 = 0. [2] (ii) Solve the equation x 2 16x + 48 = 0 and find the value of x. [3] 7 A is the point (2, 5) and B is the point ( 7, 3). (i) Calculate the length of the line AB. [2] (ii) Find the equation of the line AB. [4] 8 The diagram shows the positions of four lighthouses P, Q, R and S. Q, P and S are on a straight line and R is due East of Q. PR = 50 km, PS = 60 km, QPR= 116 and PQR = 30. Q 30 P 116 60 km 50 km Calculate (i) QR, [2] (ii) RS. [3] (iii) the bearing of S from Q. [1] R S
19 9 A solid metal hemisphere has a diameter of 42 cm. (a) Find its volume. [2] (b) The metal is melted and moulded into a solid cone with the same diameter as shown below. Find the height of the cone. [2] 42 cm
20 Section B (16 marks) Answer any two questions from this section. 10 (a) The time taken, in seconds, by 20 Formula 1 drivers to complete 1 lap of the race track on a sunny day and a rainy day is represented by the box-and-whisker plot below. Rainy Day Sunny Day Time in Seconds 70 75 80 85 90 95 100 (i) State the median time taken during the rainy day. [1] (ii) Find the interquartile range for the sunny day. [2] (iii) Compare the time taken on both days. [1] (b) The waiting time, in minutes, for 150 patients at two hospitals were given as follows. Hospital A Time (minutes) Number of patients 13 44 64 15 14 Hospital B Mean = 29.2 Standard Deviation = 3.5 (i) For Hospital A, calculate (a) the mean waiting time, [1] (b) the standard deviation. [2] (ii) Compare the waiting time of the two hospitals. [1]
21 11 A, B, P are points on the circle with centre O. TA and TB are tangents to the circle. Angle APB = 65. A T O 65 P B (a) Give a reason why TBO = 90. [1] (b) Find, giving reasons, (i) reflex angle AOB. [2] (ii) angle ABO, [1] (iii) angle ATB. [1] (c) Given that the radius is 4 cm. Find the area of the quadrilateral TAOB. [3] 12 Answer the whole of this question on a single sheet of graph paper. 2 The table of values for the graph of y x 3x 2. x 1 0 1 2 3 4 y 6 2 0 p 2 6 (a) Find the value of p. [1] (b) Use a scale of 2 cm to represent 1 unit on the x-axis and 2 cm to represent 1 unit 2 on the y-axis, draw and label the graph of y x 3x 2 for 1 x 4. [3] (c) Use your graph to find (i) the value of y when x = 3.5, [1] (ii) the value of x when y = 3. [1] (d) By drawing a tangent, find the gradient of the graph at the point where at. [2]
22 Answer Key Paper 1 1 (a) 67.34396265 (b) 67.3 (c) 67.34 2 (a) 2 3 3 5 7 (b) 27720 14 (a) 41 kg (b) 7 kg (c) 47 kg 15 x = 8 y =10 (c) 11025 3 6 16 (a) p = -3 (b) 4, 2 q = -1 4 24x 17 S$851.20 5 13-8x (x -1) 2 19 (a) 16 cm (b) 48 cm 2 (c)(i) 8 17 (c)(ii) - 15 17 6 $400 20 (a) y = 21 x + 2 (b) -12.5 7 $167.38 21 (a) 3 < x 9 1 2 (b) 4, 5, 6, 7, 8, 9 8 (a) 10.3 (b) 58.5 % 22 (a) 896 cm 3 (b) 576 cm 2 9 (a) 9(y-3)(y+3) 23 (b) x = 2.5 (b) (x - 9)(x - 4) 10 k = 2 24 (a) 61, 85 (b) 2n 2 + 2n +1 11 (a) (b) 20 25 (a) (d) AX = 7.6 cm (± 0.1) 12 (a) 17 35 (b) 18 35 13 (a) 20 (b) 67.5
23 Paper 2 Section A 1 (a) 210 (b)(i) 3.0 10 8 (b)(ii) 1.62 10 19 2 36-8x (a) (x - 3) 2 (b) c = b2 2a 3 3 (a) 75 % (b) $777 4 (i) 2.5 m/s 2 (ii) 45 m/s (iii) 38 m/s 5 (ii) 9.29 cm 2 6 (ii) x =12 (rej. x = 4 ) 7 (i) 12.0 (ii) y = 8 5 x - 41 5 8 (a) 89.9 km (b) 58.9 km (c) 9 (a) 19400 cm 3 (b) 42.0 cm Section B 10 (a) (i) 90 s (a) (ii) 10 s (a) (iii) Rainy day has longer time taken based on the median of 90 s as compared to sunny day s median of 83 s. (b) (i) (a) 29.28 min (b) 4.16 (c) Their waiting times are the same as both means are the same but Hospital A is more consistent from since it has a lower standard deviation. 11 (a) Tgt perpendicular to radius (b) (i) (b) (ii) (b) (iii) (c) 34.3 cm 2 12 (a) p = 0 (c) (i) y = 3.75(±0.1) (c) (ii) x = -0.3 and 3.3(±0.1) (d) Gradient = 1
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