Worksheet 1 Laws of Integral Indices

Transcription

1 Worksheet 1 Laws of Integral Indices 1. Simplify a 4 b a 5 4 and express your answer with positive indices.. Simplify 6 x y x 3 and express your answer with positive indices. 3. Simplify x x 3 5 y 4 and express your answer with positive indices. 4. Simplify x y x 3 and express your answer with positive indices. 1

2 5. Simplify y x y x 3 and express your answer with positive indices. 6. Simplify y x y x and express your answer with positive indices. 7. Simplify b a b a and express your answer with positive indices. 8. Simplify ) ( n n m and express your answer with positive indices.

3 3 9. Simplify m n m and express your answer with positive indices. 10. Simplify y x x and express your answer with positive indices. 11. Simplify 3 5 ) ( y x y and express your answer with positive indices. 1. Simplify 4 3 ) ( y xy and express your answer with positive indices.

4 13. Simplify x 10 y ( x y) and express your answer with positive indices. 14. Simplify x ( x y 5 y 3 ) 4 and express your answer with positive indices. 15. Simplify ( xy 3 x y ) 5 and express your answer with positive indices. 4

5 5 16. Simplify ) ( y x y x and express your answer with positive indices. 17. Simplify ) ( ) ( y x y x and express your answer with positive indices. 18. Simplify ) ( ) ( y x y x and express your answer with positive indices. Come on! You can do it!

6 6 Useful Formulas (a) n m n m a a a + = (b) n m n m n m a a a a a = = (c) n m n m a (a ) = (d) n n n b ab = a ) ( (e) n n n b a b a =

7 Worksheet Formulas and Identities 5n 4m 1. Make m the subject of the formula =. 3. Make p the subject of the formula p + 3 = 4(p q). 3. Make a the subject of the formula 4a + b = b Make y the subject of the formula y = x 6y. 4 1

8 5. Make b the subject of the formula a + b = a(b + 5). 6. Make k the subject of the formula h(k + 6) = 3h k. 7. Make x the subject of the formula Px = (3Qx 1)R. 8. Make b the subject of the formula 3a + b 8 = 7. b

9 9. Make h the subject of the formula 9 5h 4k =. h 10. Make x the subject of the formula y = 4. 3 x 11. Make a the subject of the formula b + 1 = b. a s 1. Make t the subject of the formula = 5s. t 3

10 13. Make x the subject of the formula 4x + y = 3y. 1 x 14. Make q the subject of the formula 7 1 = 3. p q 15. Make k the subject of the formula 1 + = 1. h k 4

11 16. Make k the subject of the formula =. 8h 4k 17. Consider the formula y 3x =. 5 6 (a) Make x the subject of the above formula. (b) If the value of y is increased by 5, write down the change in the value of x. 5

12 18. Consider the formula a + 6b = (6 a). (a) Make a the subject of the above formula. (b) If the value of b is increased by 3, write down the change in the value of a. 19. Consider the formula 3a =. a + b (a) Make b the subject of the above formula. (b) If the value of a is decreased by 8, write down the change in the value of b. 6

13 0. Consider the formula 10(3c + 4d) = 5(c + d + 3). (a) Make d the subject of the above formula. (b) If the value of c is decreased by 18, write down the change in the value of d. I ve made it! Hooray! 7

14 Worksheet 3 Factorization of Polynomials 1. Factorize (a) x y + 3xy, (b) x y + 3xy + x + 6y.. Factorize (a) 6m n 4mn, (b) 6m n 4mn 9m + 6n. 3. Factorize (a) 4 a, (b) 4 a + b ab. 1

15 4. Factorize (a) 16m 9, (b) 4mn + 3n + 16m Factorize (a) 4 + 4x + x, (b) 4 + 4x + x y xy. 6. Factorize (a) p 10p + 5, (b) pq 5q + p 10p + 5.

16 7. Factorize (a) 6 + 5a + a, (b) 6 + 5a + a + 3b + ab. 8. Factorize (a) m m 3, (b) mn + n + m m Factorize (a) 9x 4y, (b) 9x 4y + 9x + 6y. 3

17 10. Factorize (a) 5r 16s, (b) 5r 16s 0r + 16s. 11. Factorize (a) a 8ab + 16b, (b) a 8ab + 16b + 5a 0b. 1. Factorize (a) 4x + 1xy + 9y, (b) 4x + 1xy + 9y 8x 1y. 4

18 13. Factorize (a) 4a 10b, (b) a 3ab 5b, (c) 4a 10b a + 3ab + 5b. 14. Factorize (a) 5p pq 4q, (b) 5p pq 4q p + q. 15. Factorize (a) 6m 13mn + 6n, (b) 6m 13mn + 6n 9m + 6n. 5

19 16. Factorize (a) r + 1r + 36, (b) r + 1r + 36 s. 17. Factorize (a) p + 14pq + 49q, (b) p + 14pq + 49q 4r. 18. Factorize (a) 5x 40xy + 16y, (b) 5x 40xy + 16y 9z. 6

20 19. Factorize (a) x 3 + x y + 5x, (b) x 3 + x y + 5x x y Factorize (a) a 3 + 3a b 4a, (b) a 3 + 3a b 4a a 3b Factorize (a) x 3 x y + x, (b) x 3 x y + x 9x + 9y 18. Come on! You can do it! 7

21 Useful Formulas (a) (i) (a + b) a + ab + b (ii) (a b) a ab + b (b) a b (a + b)(a b) 8

22 Worksheet 4 Percentages 1. There are 500 boys in a school and the number of girls is 0% more than that of boys. (a) Find the number of girls in the school. (b) What is the percentage of girls in the school?. A bag contains blue balls and red balls only. There are 4 red balls in the bag and the number of blue balls is 75% less than that of red balls. (a) Find the number of blue balls in the bag. (b) What is the percentage of blue balls in the bag? 3. There are 0 girls in a class and the number of girls is 0% less than that of boys. (a) Find the number of boys in the class. (b) What is the percentage of boys in the class? 1

23 4. There are 7 male workers in a company and the number of male workers is 60% more than that of female workers. (a) Find the number of female workers in the company. (b) What is the percentage of female workers in the company? 5. In a basket, there are 50 red beans and green beans. The number of red beans is 8% more than the number of green beans. Find the difference of the number of red beans and the number of green beans. 6. Ken is 0% heavier than Louis, and Louis is 0% lighter than Mary. Ken weighs 54 kg. (a) Find the weight of Louis. (b) Mary claims that she is lighter than Ken. Do you agree? Explain your answer.

24 7. Emily s monthly salary is 30% more than Fiona s, and Fiona s monthly salary is 30% less than Ginny s. Fiona s monthly salary is $ (a) Find the monthly salary of Emily. (b) Who has the greatest monthly salary? Explain your answer. 8. Henry s hourly wage is 5% lower than Ida s, and Ida s hourly wage is 5% higher than Jimmy s. Ida s hourly wage is $40. (a) Find the hourly wage of Jimmy. (b) Henry claims their total hourly wage is more than $60. Do you agree? Explain your answer. 9. The marked price of a pencil box is $54. It is given that the marked price of the pencil box is 0% higher than the cost. (a) Find the cost of the pencil box. (b) If the pencil box is sold at $4, find the percentage loss. 3

25 10. The marked price of a school bag is $150. It is sold at a discount of 30% on its marked price. (a) Find the selling price. (b) A profit of $5 is made by selling the school bag. Find the percentage profit. 11. The marked price of a calculator is $00. It is sold at a discount of 0% on its marked price. (a) Find the selling price. (b) If the percentage profit is 60%, find the cost. 1. The marked price of a music CD is $180. It is sold at a discount of 60% on its marked price. (a) Find the selling price. (b) If the percentage loss is 0%, find the cost. 4

26 13. The marked price of a toy is $40. It is sold at a discount of 40% on its marked price. (a) Find the selling price. (b) If the percentage profit is 0%, find the profit. 14. The cost of a desk is $400. If it is sold at a discount of 40% on its marked price, then the percentage profit is 0%. Find the marked price of the desk. 15. The cost of a dictionary is $50. If it is sold at a discount of 10% on its marked price, then the percentage profit is 44%. Find the marked price of the dictionary. 5

27 16. The cost of a toy is $30. If the toy is sold at its marked price, then the percentage profit is 40%. (a) Find the marked price. (b) If the toy is sold at a discount of 40%, find the percentage profit or percentage loss. 17. The cost of a toolbox is $60. The toolbox is now sold and the percentage profit is 60%. (a) Find the selling price of the toolbox. (b) If the toolbox is sold at a discount of 0% on its marked price, find the marked price of the toolbox. 18. The cost of a watch is $300. The watch is now sold and the percentage profit is 45%. (a) Find the selling price of the watch. (b) If the watch is sold at a discount of 5% on its marked price, find the marked price of the watch. 6

28 19. A vase is sold, at a discount of 35% on the marked price, at $ (a) Find the marked price. (b) If the marked price is 0% above the cost, find the percentage profit or percentage loss. 0. A chair is sold, at a discount of 0% on the marked price, at $30. (a) Find the marked price. (b) If the marked price is 60% above the cost, find the percentage profit or percentage loss. I ve made it! Hooray! 7

29 Useful Formulas (a) A is m% more than B. A = B(1 + m%) C is n% less than D. C = D(1 n%) new value original value (b) Percentage increase = 100% original value original value new value Percentage decrease = 100% original value new value original value Percentage change = 100% original value (new value > original value) (original value > new value) (c) Profit = selling price cost selling price cost Percentage profit = 100% cost Selling price = cost (1 + percentage profit) (d) Loss = cost selling price cost selling price Percentage loss = 100% cost Selling price = cost (1 percentage loss) (e) Discount = marked price selling price marked price selling price Discount % = 100% marked price Selling price = marked price (1 discount %) 8

30 Worksheet 5 Linear Equations in Two Unknowns 1. If α + β = 5α + β = 1, find α and β.. The price of pears and 3 mangoes is $33 while the price of 4 pears and 7 mangoes is $73. Find the price of a mango. 3. The price of 3 oranges and 5 apples is $37 while the price of 7 oranges and 4 apples is $48. Find the price of an orange. 1

31 4. The prices of a pencil and a rubber are $3 and $4 respectively. A sum of $43 is spent buying some pencils and rubbers. If the total number of pencils and rubbers bought is 1, find the number of pencils bought. 5. The cost of 3 bottles of coffee is the same as the cost of bottles of peach tea. The total cost of 4 bottles of coffee and 6 bottles of peach tea is $18. Find the cost of a bottle of coffee. 6. The total number of books owned by Amy and Billy is 84. If Amy buys 1 books from a book store, the number of books owned by her will be 3 times that owned by Billy. Find the number of books owned by Billy.

32 7. In a classroom, the number of boys is times that of girls. If 10 boys and girls leave the classroom, then the number of boys and the number of girls are the same. Find the original number of girls in the classroom. 8. The number of stamps owned by Alan is 3 times that owned by Benny. If Alan gives 1 of his own stamps to Benny, they will have the same number of stamps. Find the total number of stamps owned by Alan and Benny. 9. In a party, the ratio of the number of men to the number of women is 8 : 7. If 11 men and 4 women leave the party, then the number of men and the number of women are the same. Find the original number of men in the party. 3

33 10. The ratio of the number of stickers owned by Anthony to the number of stickers owned by Heidi is 5 : 3. If Anthony gives 14 of his own stickers to Heidi, both of them will have the same number of stickers. Find the total number of stickers owned by Anthony and Heidi. 11. There are 84 helpers in an open day. The helpers are divided into 4 groups, and each group has the same number of helpers. In each group, there are 3 more male helpers than female helpers. Find the number of female helpers in the open day. 1. There are 140 guards in an international jewellery exhibition consisting of 5 zones. Each zone has the same number of guards. In each zone, there are 6 less female guards than male guards. Find the number of male guards in the exhibition. 4

34 13. In a basketball league, each team gains points for a win, 1 point for a draw and 0 point for a loss. The champion of the league plays 4 games and gains a total of 75 points. Given that the champion does not lose any games, find the number of games that the champion draws. 14. In a game rating system, Alice gains 500 points for a win and 00 points for a draw, and loses 300 points for a loss. She originally has performance rating 1300 points. If after playing 0 matches without any draws, Alice has performance rating 4900 points, find the number of matches that she wins. Come on! You can do it! 5

35 Worksheet 6 Compound Linear Inequalities 1. Find the range of values of x which satisfy x + 3 < 0 or 5(x 1) > 3x Find the range of values of x which satisfy x + 8 < 4 or 10 5x 1 x (a) Solve the inequality 7x + 1 < 3(3x + ). (b) Find all integers satisfying both the inequalities 7x + 1 < 3(3x + ) and x

36 15 x (a) Solve the inequality 6(x + 1) >. 15 x + 6 (b) Find all integers satisfying both the inequalities 6(x + 1) > and 3x 5 0. (6 x) 5. (a) Solve the inequality < x (b) Find all even numbers satisfying both the inequalities (6 x) < x + 4 and 9 4x x 1 3x 6. (a) Solve the inequality. 4 3 (b) Find all odd numbers satisfying both the inequalities 5x 1 3x 4 3 and x 9 < 1 x.

37 7. Consider the compound inequality x 5 > 3(x 5) or x + 0. (*) (a) Solve (*). (b) Write down the least positive integer satisfying (*). x 8. (a) Find the range of values of x which satisfy 4 7x 5(3 x) or 5 + < 0. (b) Write down the greatest integer satisfying the compound linear inequality in (a). 9. (a) Find the range of values of x which satisfy both (b) 4x 6(x ) and x Write down the greatest integer satisfying both the inequalities in (a). 3

38 10. (a) Find the range of values of x which satisfy both (b) x 1 x > 3 4 Write down the least integer satisfying both the inequalities in (a). and 3(x 1) (a) Find the range of values of x which satisfy both (b) 9 x +1 Write down the least integer satisfying both the inequalities in (a). x > 3(x 1) and > (a) Find the range of values of x which satisfy both 5(x 13) < 3 7x and 4(x 1) < 8. (b) How many positive integers satisfy both the inequalities in (a)? 4

39 13. (a) Find the range of values of x which satisfy both x 4x and x 1 < 7 3x. (b) How many positive integers satisfy both the inequalities in (a)? 14. (a) Find the range of values of x which satisfy 11 5x > 1 x or 3x (b) How many negative integers satisfy the compound linear inequality in (a)? 15. (a) Find the range of values of x which satisfy both 3 x + 3(x + 4) and (x + 10) > x. 5 (b) How many negative integers satisfy both the inequalities in (a)? I ve made it! Hooray! 5

40 Worksheet 7 More about Polynomials In this worksheet, students should try to score at least marks in each question. Section A(1) 1. Let f (x) = kx 3 7x + x + 3, where k is a constant. It is given that x 3 is a factor of f (x). (a) Find the value of k. (b) Find all the rational roots of the equation f (x) = 0. ( marks) ( marks). Let p(x) = x 3 + ax + bx + 1, where a and b are constants. When p(x) is divided by x + and when p(x) is divided by x, the two remainders are equal. It is given that p(x) = (x 1)(mx + nx 1), where m and n are constants. Find a and b. (4 marks) 1

41 3. Let f (x) = x 3 + x 5x + c, where c is a constant. When f (x) is divided by x + 4, the remainder is 18. (a) Is x a factor of f (x)? Explain your answer. (3 marks) (b) Someone claims that all the roots of the equation f (x) = 0 are integers. Do you agree? Explain your answer. ( marks) 4. Let f (x) = 3x 3 + 0x + kx 1, where k is a constant. When f (x) is divided by x + 1, the remainder is 4. (a) Is x + 3 a factor of f (x)? Explain your answer. (3 marks) (b) Someone claims that all the roots of the equation f (x) = 0 are rational numbers. Do you agree? Explain your answer. ( marks)

42 Section A() 5. Let f(x) = x 3 + kx 3x 3, where k is a constant. It is given that f (x) = (x 1)(ax + bx + c), where a, b and c are constants. (a) Find a, b and c. (4 marks) (b) Someone claims that all the roots of the equation f (x) = 0 are real numbers. Do you agree? Explain your answer. (3 marks) 6. Let f(x) = 1x 3 11x + kx +, where k is a constant. It is given that f (x) = (3x + 1)(px + qx + r), where p, q and r are constants. (a) Find p, q and r. (4 marks) (b) Someone claims that all the roots of the equation f (x) = 0 are real numbers. Do you agree? Explain your answer. (3 marks) 3

43 7. Let f (x) = x 3 + kx bx 0, where k and b are constants. It is given that f (x) = (x + 4)(ax + bx + c), where a and c are constants. (a) Find a, b and c. (4 marks) (b) Someone claims that all the roots of the equation f (x) = 0 are rational numbers. Do you agree? Explain your answer. ( marks) 8. Let f (x) = (x 3) (x + h) + k, where h and k are constants. When f (x) is divided by x 3, the remainder is 7. It is given that f (x) is divisible by x 4. (a) Find h and k. (3 marks) (b) Someone claims that all the roots of the equation f (x) = 0 are integers. Do you agree? Explain your answer. (3 marks) 4

44 9. Let f (x) = (x 1) (x + h) + k, where h and k are constants. When f (x) is divided by x 1, the remainder is 4. It is given that f (x) is divisible by x. (a) Find h and k. (b) Factorize f (x). (3 marks) (3 marks) 10. Let f (x) be a polynomial. When f (x) is divided by x 1, the quotient is x 4x. It is given that f (1) = 6. (a) Find f (). (3 marks) (b) Factorize f (x). (3 marks) 5

45 11. Let f (x) be a polynomial. When f (x) is divided by x +, the quotient is 3x 5x. It is given that f ( ) = 8. (a) Find f ( 1). (b) Factorize f (x). (3 marks) (3 marks) 1. Let g (x) be a polynomial. When g (x) is divided by x 3, the quotient is 3x + 11x + k. It is given that g ( 3) = 40 and g (3) = (a) Find g. 3 (3 marks) (b) Solve the equation g (x) = 0. (3 marks) 6

46 13. (a) Find the quotient when x 3 11x + 13x + 9 is divided by x 6x + 8. ( marks) (b) Let g (x) = (x 3 11x + 13x + 9) (ax + b), where a and b are constants. It is given that g (x) is divisible by x 6x + 8. (i) Write down the values of a and b. (ii) Solve the equation g (x) = 0. (4 marks) 14. (a) Find the quotient when x 3 x 5x + 5 is divided by x 4x + 1. ( marks) (b) Let g (x) = x 3 x (5 + a)x + (5 b), where a and b are constants. It is given that g (x) is divisible by x 4x + 1. (i) Write down the values of a and b. (ii) Solve the equation g (x) = 0. (4 marks) 7

47 15. Let f (x) = 3x 3 + ax + bx + 1, where a and b are constants. It is given that f (x) is divisible by 3x 4, and the quotient is px + x + q, where p and q are constants. (a) Find p and q. ( marks) (b) Find all rational roots of the equation 3x 3 + ax + (b + 3q)x + (1 4q) = 0. (3 marks) 8

48 16. (a) Find the value of k such that x 3 is a factor of x 3 kx + 45x 54. ( marks) (b) The figure shows the graph of y = x 1x B is a variable point on the graph in the first quadrant. A and C are the feet of the perpendiculars from B to the x-axis and the y-axis respectively. y y = x 1x + 45 C O B A x (i) Let (b, 0) be the coordinates of A. Express the area of the rectangle OABC in terms of b. (ii) Are there three different positions of B such that the area of the rectangle OABC is 54? Explain your answer. (5 marks) 9

49 17. (a) Find the value of k such that x 4 is a factor of x 3 + kx 18 = 0. ( marks) (b) In the figure, V is the vertex of the graph of y = x + 8. P is a variable point on the graph in the first quadrant, and S is the reflection image of P with respect to the axis of symmetry of the graph. Q and R are the feet of the perpendiculars from P and S to the x-axis respectively. y y = x + 8 S P R O V Q x (i) Let (p, 0) be the coordinates of Q. Express the area of the pentagon VPQRS in terms of p. (ii) If the area of the pentagon VPQRS is 18, find the coordinates of P and S. (6 marks) Come on! You can do it! 10

50 Worksheet 8 Variations In this worksheet, students should try to score at least marks in each question. Section A() 1. Let $C be the cost of manufacturing a cubical flower pot of side s cm. It is given that C is partly constant and partly varies as the square of s. When s = 1, C = 100; when s = 0, C = 8. (a) Find the cost of manufacturing a cubical flower pot of side 15 cm. (4 marks) (b) If the cost of manufacturing a cubical flower pot is $60, find the length of a side of the flower pot. ( marks). The cost of a carpet of perimeter p metres is $C. It is given that C is the sum of two parts, one part varies directly as p and the other part varies directly as p. When p = 5, C = 1600; when p = 8, C = (a) Find the cost of a carpet of perimeter 6.5 metres. (4 marks) (b) If the cost of a carpet is $1390.5, find the perimeter of the carpet. ( marks) 1

51 3. The production cost of a greeting card of perimeter s centimetres is $C. It is given that C is the sum of two parts, one part varies as s and the other part varies as the square of s. When s = 40, C = 13; when s = 80, C = 4. (a) Find the production cost of a greeting card of perimeter 60 centimetres. (4 marks) (b) If the production cost of a greeting card is $6.5, find the perimeter of the greeting card. ( marks) 4. When Helen sells n pieces of jewellery in a month, her income in that month is $S. It is given that S is a sum of two parts, one part is a constant and the other part varies as n. When n = 5, S = 10 00; when n = 8, S = (a) When Helen sells pieces of jewellery in a month, find her income in that month. (4 marks) (b) Is it possible that when Helen sells a certain number of pieces of jewellery in a month, her income in that month is $30 000? Explain your answer. ( marks)

52 5. When David sells n watches in a month, his income in that month is $S. It is given that S is a sum of two parts, one part is a constant and the other part varies as n. When n = 3, S = 9040; when n = 30, S = 000. (a) When David sells 5 watches in a month, find his income in that month. (4 marks) (b) Is it possible that when David sells a certain number of watches in a month, his income in that month is $8 000? Explain your answer. ( marks) 6. Let $C be the production cost of a banner of surface area A m. It is given that C is partly constant and partly varies as A. When A = 10, C = 185; when A = 13, C = 1. (a) Find the production cost of a banner of surface area 8 m. (4 marks) (b) There is a larger banner which is similar to the banner described in (a). If the perimeter of the larger banner is times that of the banner described in (a), find the production cost of the larger banner. ( marks) 3

53 7. Let $C be the cost of painting a vessel of surface area A m. It is given that C is partly constant and partly varies as A. When A =, C = 19; when A = 5, C = 103. (a) Find the cost of painting a vessel of surface area 4 m. (4 marks) (b) There is a smaller vessel which is similar to the vessel described in (a). If the volume of the smaller vessel is 8 1 times that of the vessel described in (a), find the cost of painting the smaller vessel. ( marks) 8. Let $C be the cost of making a product of volume V m 3. It is given that C is the sum of two parts, one part varies as V and the other part varies as V. When V = 1.96, C = 98; when V =.56, C = 10. (a) Find the cost of making a product of volume 3.4 m 3. (4 marks) (b) There is a smaller product which is similar to the product described in (a), and the surface area of the smaller product is 16 1 times that of the product described in (a). Heidi claims that the cost of marking the smaller product is greater than $10. Do you agree? Explain your answer. (3 marks) 4

54 9. It is given that f (x) is the sum of two parts, one part varies as x and the other part varies as x. Suppose that f (5) = 60 and f (8) = 10. (a) Find f (x). (3 marks) (b) Solve the equation f (x) = 30. ( marks) 10. It is given that f (x) is the sum of two parts, one part varies as x and the other part is a constant. Suppose that f ( 3) = 9 and f (6) = 90. (a) Find f (4). (4 marks) (b) P (4, p) and Q ( 4, q) are points lying on the graph of y = f (x). Find the area of PQR, where R is a point lying on the x-axis. (4 marks) 5

55 11. It is given that f (x) is the sum of two parts, one part varies as (x + ) and the other part is a constant. Suppose that f ( 4) = and f (3) = 1. (a) Find f ( 9). (4 marks) (b) A ( 9, a) and B (5, b) are points lying on the graph of y = f (x). Find the area of VAB, where V is the vertex of the graph of y = f (x). (4 marks) 1. It is given that f (x) is the sum of two parts, one part varies as (x 4) and the other part is a constant. Suppose that f (1) = 0 and f (11) = 80. (a) Find f (9). (4 marks) (b) The graph of y = f (x) passes through P (9, p) and Q ( 1, q), and cuts the x-axis at the points R and S, where x-coordinate of R < x-coordinate of S. Find the area of the quadrilateral PQRS. (4 marks) 6

56 Worksheet 9 Measures of Dispersion In this worksheet, students should try to score at least marks in each question. Section A(1) 1. The table below shows the distribution of the numbers of toy cars owned by some children. Number of toy cars Number of children Find the median, the mode and the standard deviation of the above distribution. (3 marks). The following table shows the distribution of the numbers of hours spent on using computer by a group of students on a certain day. Number of hours Number of students Find the median, the mode and the standard deviation of the above distribution. (3 marks) 1

57 3. The following table shows the distribution of the numbers of hours spent on reading books by a group of teenagers during weekends. Number of hours Number of teenagers Find the median, the mode and the standard deviation of the above distribution. (3 marks) 4. The bar chart below shows the distribution of the numbers of family members of the students in class 6A. Distribution of the numbers of family members of the students in class 6A 10 Number of students Number of family members (a) Find the mean, the inter-quartile range and the standard deviation of the above distribution. (4 marks) (b) A student leaves class 6A. The number of family members of this student is 6. Find the change in the standard deviation of the numbers of family members of the students in class 6A due to the leaving of this student. (1 mark)

58 5. The bar chart below shows the distribution of the numbers of good friends of the students in class 6B. Distribution of the numbers of good friends of the students in class 6B 10 Number of students Number of good friends (a) Find the mean, the inter-quartile range and the standard deviation of the above distribution. (4 marks) (b) A student leaves class 6B. The number of good friends of this student is 1. Find the change in the standard deviation of the numbers of good friends of the students in class 6B due to the leaving of this student. (1 mark) 6. The box-and-whisker diagram below shows the distribution of the times taken by a large group of students of an athletic club to finish a 00 m race: m n 38. Time (s) The inter-quartile range and the range of the distribution are 7. s and 14.8 s respectively. (a) Find m and n. ( marks) (b) The students join a training program. It is found that the longest time taken by the students to finish a 00 m race after the training is 4.4 s less than that before the training. The trainer claims that at least 5% of the students show improvement in the time taken to finish a 00 m race after the training. Do you agree? Explain your answer. ( marks) 3

59 7. The box-and-whisker diagram below shows the distribution of the heights of a large group of students: 18 a b Height (cm) The inter-quartile range and the range of the distribution are 13 cm and 33 cm respectively. (a) Find a and b. ( marks) (b) Four years later, the heights of the students are measured again. It is found that the shortest height of the students is 1 cm taller than that in the first measurement. Someone claims that at least 5% of the students grow taller 4 years after the first measurement. Do you agree? Explain your answer. ( marks) 4

60 Section A() 8. The box-and-whisker diagram below shows the distribution of the scores (in marks) of 30 students in a mathematics test. It is given that the mean of this distribution is 60 marks Score (marks) (a) Find the range and the inter-quartile range of the above distribution. (3 marks) (b) Since four students did not attend the above test, they have to take a make-up test. Their scores in the make-up test are 4 marks, 56 marks, 67 marks and 75 marks. The mathematics teacher includes these scores in the distribution. Find the new mean and the new median of the scores in the test. (3 marks) 5

61 9. There are 34 antique vases in an antique shop. The box-and-whisker diagram below shows the distribution of the prices (in thousand dollars) of the antique vases in the antique shop. It is given that the mean of this distribution is 56 thousand dollars Price (thousand dollars) (a) Find the range and the inter-quartile range of the above distribution. (3 marks) (b) Six antique vases of respective prices (in thousand dollars) 37, 4, 46, 60, 68 and 83 are now donated to a museum. Find the mean and the median of the prices of the remaining antique vases in the antique shop. (3 marks) 6

62 10. The following stem-and-leaf diagram shows the distribution of the ages of the members in a club. Stem (tens) Leaf (units) (a) Find the mean, the median and the range of the distribution. (3 marks) (b) Two more members now join the club. It is known that the mean and the range of the distribution are both increased by 1. Find the ages of these two members. (4 marks) 7

63 11. The following stem-and-leaf diagram shows the distribution of the heights (in cm) of the members in a basketball club. Stem (tens) Leaf (units) (a) Find the mean, the median and the range of the distribution. (3 marks) (b) Two more members now join the club. It is known that the mean and the range of the distribution are both increased by cm. Find the heights of these two members. (4 marks) Come on! You can do it! 8

64 Useful Formula For a set of data x 1, x, x 3,, x n, where x is its mean, (i) variance = ( x x) + ( x x) ( xn x n 1 ) (ii) standard deviation σ = ( x x) + ( x x) ( xn x n 1 ) 9