E Math (4016/01) Total marks : 80. x 1. Solve Answer x = [1]

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1 Requirement : Answer all questions Total marks : 80 Duration : hours x 1. Solve x 5 Answer x = [1]. Frank bought an antique vase for $345. One year later he sold it for a profit of 180% of the cost price. Calculate the selling price. Selling price Answer $ = [] 3. Alec has written down five numbers. The mean of these numbers is 7, the median is 5 and the mode is 4. The largest number is three times the smallest number. Find the five numbers. 4, 4, 5, x, 1 x Answer,,,, [] Prepared by Mr Ang, Oct 015 1

2 4. A field has an area of 1400 m. An average of 60 dandelion plants grow on each square metre. Each plant has an average of 0 flowers. Joel estimates that these plants will produce 8 10 seeds in total. Calculate an estimate of the average number of seeds produced by each flower. 8 average number of seeds produced by each flower Answer [] 5. Simplify 4x 3 5x x 44x 9 5x 4x 3 16x 18 45x 61x Answer [] 6. ={ integers x : 1 x 8 } A = { factors of 6 } B = { prime numbers } (a) List the elements in B A. ={ 1,, 3, 4, 5, 6, 7, 8 } A = { 1,, 3, 6 }, A = { 4, 5, 7, 8 } B = {, 3, 5, 7 }, B = { 1, 4, 6, 8 } B A= { 4, 8 } Answer [1] Prepared by Mr Ang, Oct 015

3 On the Venn diagram, shade the region which represent B A. [1] 7. Factorise fully 4ax 3ay 8bx 6by. 4x 3y b4x 3y 4x 3ya b 4ax 3ay 8bx 6by a Answer [] 3 8. A is the point (1, 4) and BA. 6 (a) Find the coordinates of point B. 1 OA 4 OB OA AB Calculate AB. Answer (a) B (, ) [1] AB BA Prepared by Mr Ang, Oct 015 3

4 Answer [1] 9. Simon weighed six apples. The mean mass of the apples was 137 grams. The standard deviation of the masses of the apples was 5.48 grams. The scales used by Simon were found to be inaccurate. The correct mass of each apple was 5 grams more than Simon recorded. Write down the correct values for the mean and standard deviation (SD). Since each apple is under recorded, the correct mean mass should be 16 grams Since there is no change in the difference between each apple, no change in SD. 10. Ravi invests $8 700 in an account. Answer Mean = SD = [] The balance, $A, of the account after t years is given by the formula A t. (a) Calculate A when t = 4. Give your answer correct to the nearest dollar. A Answer (a) $ = [1] Find the percentage increase in the balance over the 4 years. percentage increase % 13.9% ( 3 s.f.) Answer % [] Prepared by Mr Ang, Oct 015 4

5 11. Teresa is drawing a triangle. The second angle is 18 smaller than the first angle. The third angle is four times the size of the second angle. Form an equation and solve it to find the angles of the triangle. Let the first angle be x, x x 18 4x x x 45 x , 7, 108 Answer,, [3] 1. (a) Wong makes a fruit drink. He uses apple juice, peach juice and lemonade in the ratio 5 : : 8 respectively. He uses.4 litres of lemonade. (i) How much apple juice does he use? Answer (a)(i) litres [1] (ii) How much fruit drink does he make altogether? Answer (a)(ii) litres [1] Prepared by Mr Ang, Oct 015 5

6 Min makes a fruit drink using orange juice, lemonade and pineapple juice. The ratio orange juice : lemonade is : 5. The ratio lemonade : pineapple juice is 3 : 4. Find the ratio orange juice : lemonade : pineapple juice. orange juice : lemonade : pineapple juice : 5 3 : 4 6 : 15 : 0 Answer : : [1] 13. Gina can paint 7 fence panels in 5 hours. Lim can paint 6 fence panels in 4 hours. Gina and Lim work together to paint a total of 17 panels. If they continue to paint at the same rate, how long will it take them to paint the 17 panels? Give your answer in hours and minutes, to the nearest minute In 1 hour, Gina paints panels, Lim paints panels, together panels hours hours (nearest minute) 9 Answer hours minutes [3] Prepared by Mr Ang, Oct 015 6

7 14. (a) Express 450 as a product of its prime factors Answer (a) [1] Find two numbers, both smaller than 100, that have a lowest common multiple of 450 and a highest common factor Answer, [] 15. (a) Solve the inequalities 10 7 x x 6 17 x x 8 Answer (a) [] Write down all the integers that satisfy 10 7 x 1. 4, 5, 6, 7, 8 Answer [1] Prepared by Mr Ang, Oct 015 7

8 16. The diagram shows a hexagon. All of the sides have the same length. Three interior angles are each 96. The remaining three interior angles are equal x Find x. 96 Interior angle = [ ] x Answer x = [3] 17. Two bottles of water are geometrically similar. The larger bottle holds litres and the smaller bottle holds 1.5 litres. The height of the larger bottle is 33.5 cm. (a) Calculate the height of the smaller bottle. 1.5 height of the smaller bottle (3 s.f.) Answer (a) cm [] The ratio surface area of larger bottle : surface area of smaller bottle can be written in the form k : 1. Find the value of k. Give your answer to decimal places. k Answer k = [] Prepared by Mr Ang, Oct 015 8

9 18. The diagram shows the positions of three towns P, Q and R. PQ is 46.5 km, PR is 3.6 km and QR is 56.0 km. The bearing of Q from P is 041. Q Calculate the bearing of R from P. N QPR cos P 3.6 R QPR bearing of R from P Answer [4] 19. ABCD is a major segment of a circle, centre O and radius 17 cm. BD = 5 cm. Angle ABD = 90. Calculate the area of the segment. D O. 5 A B C OB AB Area of triangle AOC AOC sin (3 d.p.) Since , angle AOC is obtuse. Prepared by Mr Ang, Oct 015 9

10 AOC Reflex AOC (3 d.p.) Area of major sector AOC (3 d.p.) (Alternatively), Area of major segment (nearest whole) OB Answer cm [5] AB AOB sin AOC Reflex AOC (3 d.p.) Area of major sector AOC (3 d.p.) 8 30 Area of triangle AOC 10 Area of major segment (nearest whole) 0. P is the point (0, 3), Q is the point (4, 11) and R is the point (a, 3). (a) The product (gradient of PQ) (gradient of QR) = 1. Use this information to show that a = 0. Answer (a) a a 4 Prepared by Mr Ang, Oct

11 a a 0 [] The line PQ is perpendicular to the line QR. Use vectors to find the coordinates of the point S, so that PQRS is a rectangle OR ; OP ; OQ OS OR RS OR QP OR OP OQ Answer (, ) [1] (c) Calculate the area of the rectangle PQRS OR ; OP ; OQ PQ OQ OP ; PQ OR OQ QR ; QR area of the rectangle PQRS (Alternatively), Consider triangle PQR, PR 0, altitude from Q = 11 3 = 8. Answer (c) unit [1] Prepared by Mr Ang, Oct

12 area of the rectangle PQRS (a) The surface area of a solid is given by A p p q Make q the subject of the formula.. A p p q A q p p Answer (a) [] Another solid is made from a cylinder and two hemispheres. The cylinder has radius r and length r and the hemispheres have radius r. The total surface area of the solid is twice the total surface area of the cone with radius r and slant height l. Find l in terms of r. r r Surface area of the solid 4r r r 8 r r l Surface area of the cone r l r r l 8r 4 r ( r r 3 r r l l 3r r l) r l Answer l = [3] Prepared by Mr Ang, Oct 015 1

13 . In supermarket A, water costs $1.50 per litre, milk costs $.40 per litre and the cola costs $1.40 per litre. In supermarket B, water costs $0.0 more per litre, milk costs $0.40 less per litre and cola costs $0.10 less per litre. This information can be represented by the matrix A B W Q M C (a) Natalie and Hadi go shopping. Natalie buys 4 litres of water, litres of milk and 3 litres of cola. Hadi buys 3 litres of water and 4 litres of cola. Represent their purchases in a 3 matrix P. 4 P Answer (a) P [1] Evaluate the matrix R = PQ R R Answer R [] (c) How much money would Hadi save by shopping in supermarket A? $0.0 Answer (c) $ [1] Prepared by Mr Ang, Oct

(d) Natalie shops in supermarket B. She has a shopping voucher giving a discount of 10%. How much does she pay altogether for her items? 15 0.3 0.9 13. 3 Answer (d) $ [] 3.

14 (d) Natalie shops in supermarket B. She has a shopping voucher giving a discount of 10%. How much does she pay altogether for her items? Answer (d) $ [] 3. This design is drawn using a large circle and semicircles. The diameters, in centimetres, of two of the semicircles are shown. kr r (a) Show that the total area, A, of the large circle is given by the formula k A r 1. Answer (a) Diameter of the large circle kr r rk 1 r k 1 Area of the large circle A r k 1 Find, in terms of π and r, the difference in area between the shaded section and the unshaded section when k =. difference in area = shaded circle unshaded circle r r 3 r Answer cm [4] Prepared by Mr Ang, Oct

15 4. A bag contains 10 marbles, n of which are red and the rest are yellow. A marble is chosen at random and not replaced. (a) Write down, in terms of n, the probability that the marble is yellow. 10 n P(yellow marble) 10 Answer (a) [1] A second marble is chosen at random. Find, in terms of n, the probability that both marbles are yellow. 10 n 9 n P(both yellow marbles) n9 n 90 Answer [1] (c) (i) 1 The probability that both marbles are yellow is. 15 Show that n 19n Answer (c)(i) 10 n9 1 n n n 19n n 84 0 (ii) Solve the equation n 19n 84 0 to find the number of yellow marbles in the bag. n 1n 7 0 n 7 or n 1 Since n 10, n 7 Yellow = 10 7 = 3 [] Answer (c)(ii) [3] Prepared by Mr Ang, Oct

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