(b) Write down the largest integer, y, which satisfies -7 < 2y + 4 < 12.

Size: px

Start display at page:

Download "(b) Write down the largest integer, y, which satisfies -7 < 2y + 4 < 12."

Earl Flynn
5 years ago
Views:

1 4E/N Prelim 010 Paper 1 1. (.1 Evaluate, giving your answer in standard form and correct to significant figures. ( Simplify p (p 1). Answer ( [] ( [1]. ( Simplify x. ( Given that 64 1 k, find the value of k. Answer ( [1] ( []. ( Factorise completely 16s s. ( Write down the largest integer, y, which satisfies -7 < y + 4 < 1. Answer ( [] ( [] 4

2 4E/N Prelim 010 Paper 1 4. Angie and Bob went for a holiday to Japan. On a particular day, the rate of exchange between Singapore dollars (S$) and Japanese yen ( ) was S$1.40 = 100. ( Angie exchanged S$ 00 into yen. How much did she receive? ( At a restaurant in Japan, Angie was billed a 10 % service charge and an 8 % government tax on goods and services. The cost of the food items was 14 00, excluding service charge and government tax. What was the total cost of the meal? ( (d) Bob had leftover from his holiday and exchanged it for Singapore dollars. Calculate the amount in Singapore dollars he received. Bob then decided to visit Bangkok. The rate of exchange between Japanese yen ( ) and Thai baht was 00 = 1 000baht. Calculate the rate of exchange between Singapore dollars (S$) and Thai baht, expressing your answer in the form S$ m = baht, where m is an integer. Answer ( [1] ( [] ( S$ [1] (d) []

3 4E/N Prelim 010 Paper 1. The force, F between two particles is inversely proportional to the square of the distance between them. The force is 48 units when the distance between the particles is r metres. Find the force when the distance is r metres. Answer units [] 6. Molly bought m candies at n cents per dozen. She sold them for p cents each. Find an expression, in terms of m, n and p, for the profit, in cents, that she made. Answer cents [] 7. The container, shown in the diagram, is initially full of water. There is a hole at the bottom and water is leaking through the hole at a constant rate. It takes 1 minute for the water level to drop by a depth of cm and minutes to drop another 8 cm. Finally, it takes minutes for the water level to drop by a depth of 1 cm. On the axes in the answer space, sketch the graph showing how the depth of the water, h cm, in the container varies over the 6 minutes. cm 8 cm Answer 1 cm h (cm) time (se [] 6

4 4E/N Prelim 010 Paper 1 8. A regular polygon has t number of sides. The size of each interior angle is 8 times the size of each exterior angle. Calculate the size of each exterior angle. Answer o [] 9. The diagram shows a straight line y = mx + c cutting the x-axis at A(8,0), and the y-axis at B(0,-6). A curve y = (x + k)(8 - x) meets the x-axis at A(8,0) and the y-axis at C(0,16). ( Find the values of m and c. ( Find the value of k. ( Write down the equation of the line of symmetry of the curve y = (x + k)(8 - x). y C (0,16) 0 A( 8,0) [1] x B(0,-6) Answer ( m =, c = [] ( k = [] ( [1] 7

5 4E/N Prelim 010 Paper A survey was carried out to find the distance of pupils homes from their school. The results are shown in the table below: Distance from school (km) 1 4 Number of pupils x ( If the mean is., find x. ( If the mode is, find the largest possible value of x. ( If the median is, find the largest possible value of x. Answer ( [] ( [1] ( [1] 11. Find the values of the constants a and b for the equation below. a 4b a b = Answer a =, b = [] 8

6 4E/N Prelim 010 Paper 1 1. The figure shows a trapezium such that AD is parallel to BC. BD and CA intersect at X. ( ( Prove that triangle BCX and triangle DAX are similar. Given that the area of triangle BXC is 8. cm, find the area of triangle ACD. Answer ( [] ( cm. [] 9

7 4E/N Prelim 010 Paper 1 1. In the diagram, BCD is a straight line, BC = 6 cm, AB = 8 cm, A B ˆC 90 and AC = CD. Find ( sin Bˆ AC, ( cos AC ˆ D, A 8 ( area of ACD. B 6 C D Answer ( [] ( [1] ( cm [] 14. ( Mr Spoon bought a vase for $70. When he moved overseas the following year, he sold it at a loss of 6.%. Find the selling price. ( Mrs Snow invested $0 000 in an account which pays 8% compound interest per annum. Find the total interest she will earn after investing for 6 years. Answer ( $ [] ( $ [] 10

8 4E/N Prelim 010 Paper 1 1. Given that ξ = {, 4, 6, 8, 10, 1, 14, 16, 18, A = x : x is a multiple of 4 B = x : 1 < x < 0 and C = {x : x is a multiple of } ( List the elements of (i) A B, (ii) A B. ( Find n(a B ). ( Find an element x such that x A, x B and x C. Answer ((i). [1] ((ii). [1] ( [1] ( x = [1] 16. The diagram shows a right-angled triangle OPQ. An arc of a circle with centre O and radius OP cuts OQ at R. Given that POQ radians and OP = 6cm, calculate ( ( the length of PQ. the area of the shaded region. Answer ( cm [1] ( cm [] 11

9 4E/N Prelim 010 Paper At the end of a company s training programme, participants have to pass a test to gain employment. The probability of passing the test at the first attempt is. Those who fail are allowed to take a re-test. The probability of passing the re-test is. No further attempts are allowed. The tree diagram below shows all the possible outcomes. 1 st Attempt ( ) Pass Fail nd Attempt Pass ( ) Fail ( (i) Complete the tree diagram above. [1] (ii) Find the probability that a participant gains employment. ( Two participants take the employment test. What is the probability that only one of them gains employment? Answer ( (ii) [] ( [] 1

10 4E/N Prelim 010 Paper 1 Speed (m/s) Time (s) The diagram shows the speed-time graph of an object which travels at a constant speed of m/s for seconds and then accelerates uniformly for 1 second before travelling at a constant speed of m/s for a further seconds. ( Find the speed of the object after 4 seconds. ( ( Calculate the total distance travelled in the seconds. On the axes below, sketch the distance-time graph of the object for the first seconds. Distance (m) Time (s) [] Answer ( m/s [] ( m [] 1

11 4E/N Prelim 010 Paper Study the following sequence of triangles drawn by a student. A A A Figure 1 Figure Figure The student continues the sequence by adding one extra line from vertex A to the base of the triangle. The table below shows the number of triangles formed and the number of lines added from vertex A for each of the above three figures. Figure Number of lines added from vertex A Number of triangles a b n ( State the value of a and of b. ( The number of triangles in Figure n can be expressed as 1 n(n + k). Find the value of k for this expression. ( Figure x in the sequence contains triangles. How many lines had been added from vertex A? Answer ( a = [1] b = [1] ( [1] ( [1] 14

12 4E/N Prelim 010 Paper 1 0. The side AB of a triangle is drawn in the answer space below, where B is due east of A. ( A line l is parallel to AB and 6 cm due north of AB. Construct and label the line l. [1] ( C is a point on l, on a bearing of 0 from A. Mark the point C and complete the triangle ABC. Measure and write down the bearing of C from B. ( Using compasses and a ruler, construct the bisector of the angle CBA and the perpendicular bisector of the line AB. [] north A B Answer ( [] 1

13 Marking scheme 4E/S Prelim 010 Paper 1 Qn solution Marks Qn Solution Marks d) sig. fig p 9 4x k k 6 s(16 s s(4 s)(4 s) 11 y Ans: \ ) and y (1400) (190) S$ 1.40 S$ baht k F ( r) 16 or 1 9 \ \ 6. n cents 1 n or mn m( p ) mp to 0 cm from 0 to 1s 0 to 1.cm from 1 to 4s 1. to 0cm from 4 to 6 s 8. Let x be ext x x 180 9x 180 x 0 Or 180 ( t ) 860 t t 180t t = 18 m 4, c 6 At (0,16) 16 (0 k )(8 0) k 8 x x. 4 x 47 4x 4.x 1.7x 7 x 4

14 Qn Solution Marks Qn Solution Marks 10. Ans.: 10 Ans.: ai) aii) 6 8 4,1,16 8 x 11. 9, b a a b b a b a, 16. PQ=6.6979cm (6) 1 (6)(6.6979) 1 cm [M] 1. ) ( ). ( ) ( s alt DAX BCX s opp DXA vert BXC s alt ADX CBX by AAA, BCX and DAX are similar ) 6 ( 8. cm ACD area cm DXA area any 17. aii) 1, Both correct. 1 1 ) ( ) ( 18. s m v v / 4. 4 / 0m () (8) 1 () 0 to s & -s -s (curve) [B] 1. ˆ sin BAC cm AC 40 (10)8 1 cm area , b a $ ) (1 A 0000 $

15 0. Line drawn correctly and label mark C complete CBAbisector bisec torofab

16 4E/N Prelim 010 Paper 1. Caroline owns a gift shop. She bought 000 greeting cards for $600. She intends to sell them at 6 cents per card. ( Calculate the cost of each card. [1] ( Calculate the profit per card. [1] ( Find the profit she makes as a percentage of the cost price if she sells all her cards. [1] (d) What is the minimum number of cards she must sell if she wants to break even? [] (e) Caroline decides to sell 10% of the cards at cost price to a charitable organisation for fund raising. If she wants her net profit to remain the same, how much must she sell each of the remaining cards? []. ( Solve the equation 7p p = 0. [] ( Simplify 4 x x 0. [] x 6 ( Express the equation of the graph y = x 6x + 0 in the form y = (x + h) + k. Hence, write down the turning point for this curve. [] (d) C A O B In the figure, O is the centre of the semi-circle where AB is the diameter. A circle, centre C, is drawn to touch this semi-circle and also the semi-circles with diameters AO and OB. Given that AO = 6cm, calculate the radius of the circle, centre C. [] 4

17 4E/N Prelim 010 Paper. A petrol station sells different grades of petrol. Table 1 below shows the amount of petrol sold in litres over days. Table below shows the selling price per litre of petrol. Grade 9 Grade 97 X-Power Day Day Table 1 Selling Price/ litre Grade 9 1. Grade X-Power.1 Table It is given that P = and Q = ( Find PQ. [] ( Describe what is represented by each element of PQ. [1] All ABC credit card holders get 10% off the selling price per litre of petrol. ( Express the discounted selling price R as a x 1 matrix. [1] (d) Find P(Q R). [] (e) Describe what is represented by each element of P(Q R). [1]

18 4E/N Prelim 010 Paper 4. m of water is needed to fill a bath. The hot water tap supplies water at a rate of x m per minute and the cold water tap supplies water at a rate of y m per minute. 1 Given that the bath is filled in minutes using both taps, x ( show that y. [] Given also that filling the bath using only the hot water tap takes 4 minutes longer than filling the bath using the cold water tap, ( form an equation in x and show that it reduces to 10x 16 x + = 0. [] ( (d) Find the values of x which satisfy this equation, giving your answer correct to two decimal places. [] Calculate the time, to the nearest minute, taken to fill the bath using only the hot water tap. []. B A C O E 18 F CE is a diameter of the circle ABCDE, centre O, as shown in the diagram. F is the point on CE produced, where EF = EB and cuts the circle at A. D BFC 18. The straight line BF ( Find ADE. [] ( Find AEC. [] ( Find CBA. [] (d) Show that BE bisects AEC. [] (e) Given also that DBE 7, calculate AED. [] 6

19 4E/N Prelim 010 Paper 6. A, B, C and D are points on level ground, with A due North of B, o BAD 8, o ADB 6, BD 6 km and CD = 10 km. North A 8 10 km D 6 6 km B C Given that ADC is a straight line, calculate ( BC, [] ( DBC, [] ( the bearing of B from C, [1] (d) the shortest distance from D to BC. [] A vertical tower stands at D. The angle of elevation of the top of the tower, T, from B is o.. Find (e) the height of the tower in metres, [] (f) the maximum angle of elevation of the top of the tower, T, when observed along BC. [] 7

20 4E/N Prelim 010 Paper 7. Diagram I shows a tent comprising a cylindrical side with a conical roof. The tent is secured to the ground by ropes. Diagram II is a vertical cross-section of the tent. The diagrams are not drawn to scale. P, E, D, C and Q lie on horizontal ground. AD is a vertical pole that supports the tent, the ropes at B and F are taut and secured to the ground at Q and P respectively. ABF forms an isosceles triangle and BCEF forms a rectangle. A F B P E D C Q Diagram I Diagram II Given that FE. 6 m, AD 4. 9 m, EC 8m and FP BQ 8. m, ( Calculate (i) the length of AF, [] (ii) the total curved surface area of the roof and the walls, [] (iii) the internal volume of the tent. [] ( BF forms the diameter of a circle. Ropes, such as FP and BQ are attached to holes on its circumference. For two particular holes, the distance apart, along the circumference, is m. Calculate (i) the angle subtended at the centre of the circle by these two holes, [] (ii) the shortest distance between these two holes. [] 8

21 4E/N Prelim 010 Paper 8. In the diagram, PQRS is a parallelogram. The diagonals PR and QS intersect at T. U is a point on QR such that QR = QU. V is the midpoint of QT. QU a and RT b. P S T V b Q a U R ( Express as simply as possible in terms of a and/or b (i) QP, [1] (ii) QV, [1] (iii) PV. [1] ( Show that PV produced passes through U. [] ( Calculate the value of (i) Area of Area of PQS PVS, [1] (ii) Area of PVS Area of PQRS. [1] 9

22 4E/N Prelim 010 Paper 9. ( The following box-and-whisker diagrams show the distribution of marks of Chinese, English and Mathematics tests of a class of students. (i) Find the median mark for Chinese test. [1] (ii) Which test has the highest interquartile range of marks? [1] (iii) Compare the performance in the English and Mathematics tests in two ways. [] (iv) Shuxin obtained 70 marks in all three tests. Explain clearly in which test she performed the best. [1] ( Machine A is used to fill up packets of flour weighing kg. In order to check the accuracy of the machine, a technician inspects the weights of 00 packets of flour with the results shown in the following table. Weight of packs (kg) Frequency 4.7 x x x x x. 4 (i) Two packets of flour are chosen at random. Calculate the probability that A) both weighed less than 4.8 kg [] B) one weighed less than 4.8 kg and the other weighed more than.0kg. [] 10

23 4E/N Prelim 010 Paper (ii) (iii) Find the mean and standard deviation of the weights of the 00 packets of flour inspected. [] Another machine B is also used for the same purpose. The mean and standard deviation of the weights of the packets of flour filled up by machine B is kg and 0.11 kg respectively. Which machine is more reliable? Explain your answer. [] 10. Answer the whole of this question on a sheet of graph paper. The table below shows some values of x and the corresponding values of y for the function y x(6 x)( x ). x y 80 7 p q ( (i) Find the value of p and the value of q. [] (ii) Using a scale of cm to 1 unit, draw a horizontal x-axis for 4 x. Using a scale of cm to 10 units, draw a vertical y-axis for 10 y 80. On your axes, plot the points given in the table and join them with a smooth curve. [] ( By using your graph, find two values of x for which y = 10. [] ( By drawing a tangent, find the gradient of the curve at the point where x =. [] (d) By drawing a suitable straight line, use your graph to solve x(6 - x)(x + ) x 0 = 0. [] 11

24 Marking Scheme E-maths Prelim 010 Paper Qn Solution Marks Qn Solution Marks 1.. p (7 p) 0 d) e). d) e) $0.0 4cents 0 4 % x x or y y $ (1.) 70(1.8) 800(.1) 100(1.) 80(1.8) 00(.1) represent the total sales of all types of petrol in Day represents the total sales ($) of all types of petrol in Day they represent the total discount given out on day 1 & day resp. d) 4. d) p 0 or x ( x )( x ) 4( x ) 0 ( x )( x ) 4x 8 ( x )( x ) ( x ) 4 x 11 turning pt is (,11) ( r ) r r (6 r) 6r 9 61r r x y x, 10x 10y 6 4 x y 0x, 10x 6-10x y 10 x 10 4 x x ( x) 10x 4 x( x) 6 0x 1x 0x ( 6) x or min s 9 (shown) -16x 0 (shown) 16 4(10)()

25 Qn Solution Marks Qn Solution Marks. ABE 18( isos) ADE 18( s) [M1[ 6. BC 10 6 (10)(6)cos14 Ans: 14.km d) CAE 90( s in same segment) 7 CBE 90( s in semi - circle) CBA BEF 180 (18) ( isos ) 144 BEC ( s in str line) 6 AEB since BEC AEB, BE bisects AEC d) e) sin D BC ˆ sin DBC. 6 Ans: No penalty for missing 0. d or sin d sin.7 6 shortest distance.49km(sf) TD tan. 6.67m(sf) e) 7. ai) aii) iii) bi) AED ( s in cyclic quad) 10 *deduct 1m if no reason given for a,b,c and e. AF. AF m 4 (4).6 (4) m (sig fig) 1 (4).6 (4). 4Q Q 0.rad (sig fig) or 8.6 [M] [M] f) 8) ai) ii) iii) tanq ~ QP a~ b 1 QV ( a ~ b ~ ) PV ( a ~ b ~ ) 1 VU ( a ~ b ~ ) PV ( vu) PV vu ii) x (4) (4 ) cos0.rad shortest dist 1.98m ci) 4 ii) 8

26 Qns Solution Marks Qns Solution Marks 9. ai) Median = 4 marks (Chinese) 10. ai) p (6 )( ) 0 q (6 )( ) ii) iii) iv) highest IQR, mathematics test Median for English test is lower at 8 marks compared to median marks for mathematics at 6. (pupils do better for math) Based on IQR, the spread of data is more consistent for English test compared to math test. There is greater variation in marks for math test. (wider spread) She performed best at English as she is in the top % in her class. OR Chinese as her score is much greater than median as compared to other subjects. OR Chinese as her score is closer to max mark obtain. Any 1. ii) d) Draw axes/plot all given points. P (allow P1 if not more than errors.) Draw smooth curve C1. x 0. or 0. 6 gradient *accepts 1 to 17. Draw tangent and attempt at n t draw y x 0 or B if (d) drawn x.7,1.6, [B1,B1] bi) ii) 0(4.7) (4.8) 8(4.9) 76(.0) 4(.1) kg or [B] SD = 0.110kg iii) Machine A performs better. There is no significant difference in the mean for both machines but the lower SD for machine A tells us that there is less variation in its performance.

Mathematical Formulae. r 100. Total amount = Curved surface area of a cone = rl. Surface area of a sphere = Volume of a cone = Volume of a sphere =

Mathematical Formulae. r 100. Total amount = Curved surface area of a cone = rl. Surface area of a sphere = Volume of a cone = Volume of a sphere = 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