MATHEMATICS PAPER 1 FORM FIVE. Question-Answer Book MATH PAPER 1. Index Number. St. Francis Xavier's College MOCK EXAMINATION ( )

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1 MATH PAPER 1 St. Francis Xavier's College MOCK EXAMINATION ( ) MATHEMATICS PAPER 1 FORM FIVE Index Number Class Class Number Question-Answer Book Time allowed: 2 hours Total Number of Pages: 24 INSTRUCTIONS 1. Write your Index Number in the space provided on Page This paper consists of THREE sections, A(1), A(2) and B. Each section carries 33 marks. 3. Attempt ALL questions in Sections A(1) and A(2), and any THREE questions in Section B. Write your answers in the spaces provided in this Question-Answer Book. 4. Write the question numbers of the questions you have attempted in Section B in the space provided on Page Unless otherwise specified, all working must be clearly shown. Section A Question No Section A Total Section B Question No.* Marker s Use Only Marks Marks 6. Unless otherwise specified, numerical answers should be either exact or correct to 3 significant figures. 7. The diagrams in this paper are not necessarily drawn to scale. Section B Total * To be filled by the candidate 2010-SFXC-MATH 1-1 1

2 FORMULAS FOR REFERENCE 2 SPHERE Surface area 4πr Volume 4 πr 3 3 CYLINDER Area of curved surface 2 πrh Volume πr 2 h CONE Area of curved surface πrl Volume 1 πr 2 3 PRISM Volume base area height h PYRAMID Volume 1 base area height 3 Answers written on this page will not be marked SFXC-MATH 1-2 2

3 SECTION A(1) (33 marks) Answer ALL questions in this section and write your answers in the spaces provided. 2 ( xy) 1. Simplify and express your answer with positive indices. 5 6 x y (3 marks) 2. 14x (a) Solve the inequality 2x x (b) Write down the greatest integer satisfying the inequality 2x 7. 5 (3 marks) 2010-SFXC-MATH 1-3 3

4 3. (a) Write down all positive integers of m such that m 3n 8, where n is a positive integer. (b) Write down all positive integers k such that 3x 2 8x k (3x m)( x n), where m, n are positive integers. (3 marks) 4. The following are the marks scored by a class of 35 students in a Mathematics test: a) Complete the stem-and-leaf diagram below. Stem (tens) Leaf (units) (2 marks) b) Explain briefly why the mode may not be a suitable measure of central tendency of the distribution of the marks in the mathematics test. (1 mark) 2010-SFXC-MATH 1-4 4

5 5. Factorize (a) (b) 4a b 2 2 ab 3, a ab 3b a b. (3 marks) 6. Nelson wants to buy the following items in a supermarket: Item Unit Price No. of Units A $ B $ C $ a) By correcting the unit price of each item to the nearest dollar, estimate the total amount he should pay. b) John claims that, in order to improve the accuracy, Nelson should correct the price of each item to the nearest dollar AFTER multiplying the Number of Units to the Unit Price. Briefly explain if John is correct or not. (4 marks) 2010-SFXC-MATH 1-5 5

6 7. The digital camera was marked at $3500. In a sale, the marked price was reduced by 20% and the shopkeeper still made a profit of 10%. (a) Find (i) the selling price, (ii) the cost price. (Give your answer correct to the nearest dollars.) (b) What is the profit percent the shopkeeper would have made if the digital camera had been sold at $3500. (Give your answer correct to the nearest 1 decimal place). (4 marks) 8. In the Figure 1, the coordinates of the point A are ( 2,5). A is rotated clockwise about the origin O through 90 to A '. A " is the reflection image of A with respect to the y-axis. (a) Write down the coordinates of A ' and A ". (b) Is OA" perpendicular to AA '? Explain your answer. (5 marks) Figure SFXC-MATH 1-6 6

7 9. In Figure 2, the pie chart shows the distribution of the numbers of traffic accidents occurred in a city in a year. In that year, the number of traffic accidents occurred in District A is 20% greater than that in District B. Figure 2 (a) Find the value of x. (b) Is the number of traffic accidents occurred in District A greater than that in District C? Explain your answer. (5 marks) 2010-SFXC-MATH 1-7 7

8 SECTION A(2) (33 marks) Answer ALL questions in this section and write your answers in the spaces provided Let f ( x) 5x 12x 9x 7 and g ( x) x 2x 3. (a) Find the quotient when f (x) is divided by g (x). (2 marks) (b) Let h( x) f ( x) ( mx n), where m and n are constants. It is given that h (x) is divisible by g (x). (i) Write down the value of m and n. (ii) Solve the equation h ( x) 0. (4 marks) 2010-SFXC-MATH 1-8 8

9 11. The cumulative frequency polygon in the Figure 3 shows the results of 200 students in a Mathematics test. Figure SFXC-MATH 1-9 9

10 (a) Draw the box-and-whisker diagramn of the give set of data. (3 marks) (b) Describe the distribution of the set of data. (2 marks) 2010-SFXC-MATH

11 (c) Complete the table below. Marks ( x ) Class Mark Frequency 30 x x x x x x x Hence, calculate the mean and standard deviation of the given set of data. (4 marks) 2010-SFXC-MATH

12 12. In the Figure, ABCD is a parallelogram. The diagonals AC and BD intersect at E. Figure 4 (a) Write down the coordinates of point E and D. (b) By considering the area of BCD, find the area of parallelogram ABCD. (c) If AB is the base of the parallelogram ABCD, find its height. (2 marks) (5 marks) (2 marks) 2010-SFXC-MATH

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14 13. Figure 5 shows a lampshade, which can be spread out as in the figure when it is cut along a slant height. The figure is formed by removing a small sector from a large sector. It is given that the upper and lower radii of the lampshade are 18cm and 27cm respectively, and the height is 40cm. Figure 5 (a) Find the external surface area of the lampshade, and express the answer in terms of. (5 marks) 2010-SFXC-MATH

15 (b) If two identical figures that are exactly the same as the one in the Figure 5 are combined as shown in Figure 6, and folded to make a new lampshade. Find the height of the new lampshade. Figure 6 (4 marks) 2010-SFXC-MATH

16 SECTION B (33 marks) Answer any THREE questions in this section and write your answers in the spaces provided. Each question carries 11 marks. 14. John takes a course with two independent tests. In each test, the score is an integer from 0 to 5. John estimates that the probability of getting scores in the two tests are as follows: Test Score Test Test 2 x x (a) Write down the value of x. (1 mark) (b) John wants to apply for a scholarship. John estimates that the probability of successfully getting the scholarship depends on his total scores in the two tests as follows: Total Score or below Probability Find the probability that John successfully gets the scholarship. (4 marks) (c) Suppose that the passing score of each test is 3. John can pass the course if he passes both tests. Find the probability of the following events (i) John passes at least one test. (ii) John passes the course, given that he passes at least one test. (6 marks) 2010-SFXC-MATH

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18 15. In the Figure 7, C is the centre of circle C 1, AB is a diameter of the circle and Q is a point on chord PR such that AQ is perpendicular to PR. (a) Prove that (i) ΔAPB~ΔAQR, (ii) AP AR AB AQ. Figure 7 (4 marks) Figure 8 (b) A rectangular coordinate system, with O as the origin, is introduced such that the x-axis is the tangent to circle C 1 at A. The coordinates of A are (8,0), AP=15, AQ=10 and AR =12. Another circle C 2 touches C 1 and the x-axis at D and E(20,0) respectively. (i) Find the coordinates of the centre and the radius of C 1. (ii) Find the equation of C 2. (iii) Find the coordinates of D. (7 marks) 2010-SFXC-MATH

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20 16. (In this question, the answers are given correct to 2 decimal places, if necessary.) Katharine spent $2000 through her credit card this month. Here are the conditions for the use of the credit card: Interest charge on the amount owed: 24% p.a., compounded monthly Monthly minimum payment: The GREATER between $50 or 5% of the sum of remaining loan and interest. (Hints: You may consider using the credit card to borrow money from the bank, and the interest will be charged if you do not FULLY paid the bill on time.) a) Write down the interest, and the minimum payment in the first bill. (2 marks) b) Suppose Katharine decides to pay the minimum payment in the first three bills. Fill in the following table: n th bill Remaining Loan Interest Minimum payment 1 $2000 Answers in a) Answers in a) 2 3 (1 mark) c) Suppose Katharine pays the minimum payment for each month and from the m th bill onward, the minimum payment will be $50. (i) Write down the remainingloan in the ( n 1) th bill in terms of n, where 1 n m. (ii) Find the value of m. (iii) Let $ M be the opening balance in the Show that the opening balance in the th ( m 1). ( m k th 1 ) bill is k k $ M (1.02) 50 for k 1, 2, 3, Hence, find the number of years that the Loan is fully repaid. (8 marks) 2010-SFXC-MATH

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22 17. In the Figure 9, ABDC is a sign board of area 78 m 2 standing vertically on the horizontal ground along the east-west direction. AC = 8m and BD = 5m. When the sun shines from N 40 W with an angle of elevation 30, the shadow of the sign board on the horizontal ground is CDEF. Figure 9 (a) Find the length of CD, CF and DE. (5 marks) (b) Find the area of CDEF. (Hint: CDEF is NOT a right-angle trapzium) (2 marks) (c) In each of the following, determine whether the area of the shadow of the sign board would be greater than, smaller than or equal to the area obtained in (b). (i) The sun shines from N W, where unchanged , but its angle of elevation remains (ii) The sun still shines from N 40 W, but its angle of elevation is, where (4 marks) 2010-SFXC-MATH 1-22

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