Mathematics Intermediate Tier Paper 1 November 2001 (2 hours) CALCULATORS ARE NOT TO BE USED FOR THIS PAPER
1. Find the value of (a) 0. 2 x 0. 4 = 0. 08 (b) 8. 3 – 2. 47 8. 30 - 2. 47 5. 83
2. John saved £ 600. He spent ⅓ of this money on a bike and 2/5 of this money on clothes. What fraction of this money has he got left? 1 + 2 + x = 1 3 5 x = 1 - 1 - 2 3 5 x = 15 - 6 15 15 15 x = 4 15 Or 1 x 600 = 200 3 2 x 600 = 240 5 600 – 200 -240 = 160 = 16 600 60 = 4 15
3. (a) Write down the next two terms of the following sequence. 10 -40 110, 100, 80, 50, …… Difference increases by 10 3. (b) Simplify 6 a – 3 – 2 a + 8 = 4 a + 5
3. (c) Find the value of 5 x + 4 y when x = -3 and y = -2 = 5 x -3 + 4 x -2 = -15 - 8 = -23 (d) The diagram below represents a number machine. INPUT Add 9 Divide by 4 If the input is n, write down the output in terms of n. = n + 9 4 OUTPUT
4. Find the size of each of the angles marked x and y. 70º x y 2 x + 70 = 180 2 x = 180 - 70 2 x = 110 55+ y = 180 - 55 125 55 x = º y = º
5. Tim has a cube, which he has labelled P, a square-based pyramid labelled Q, a triangular prism labelled R and a tetrahedron labelled S. Complete the following table. One has been done for you. Properties of the shape Label on shape All its faces are square P It has two triangular faces and 3 rectangular faces All its faces are triangles It has exactly 5 vertices R S Q
6. A red bag contains five red balls numbered 1, 3, 4, 5 and 9 respectively. A black bag contains four balls numbered 2, 3, 6 and 8 respectively. In a game, a player takes one ball at random from each of the two bags. The score for the game is the sum of the numbers on the two balls. (a) Complete the following table to show all the possible scores. Black bag 8 9 11 6 7 3 2 9 12 10 13 11 17 15 4 6 7 8 12 3 5 6 7 11 1 3 4 5 9 Red bag
6. (b) (i) What is the probability that a player scores 7 = 3 20 (ii) What is the probability that a player does not score 7 Black bag 8 6 3 2 9 7 4 3 11 9 6 5 12 10 7 6 13 11 8 7 17 15 12 11 1 3 4 5 9 Red bag = 17 20 A player wins a prize by getting a score of 6 or less. (c) Brian plays the game once. What is the probability that he wins a prize? = 5 20 = 1 4
6. (d) (i) 600 people each play the game once. Approximately how many would you expect to win a prize? = 1 x 600 4 = 150 (ii) It costs 30 p to play the game once. The prize for getting a score of 6 or less is £ 1. If the 600 people each play the game once, approximately how much profit do you expect the game to make? Cost of playing = 150 x £ 1 = £ 150 Winnings = 600 x 30 p = 18000 p = £ 180 Profit = 180 - 150 = £ 30
7. Tony has some red blocks and some blue blocks. Every blue block weighs x grams. Every red block weighs 60 grams more than a blue block. (a) Write down, in terms of x, the weight of one red block. = x + 60 (b) Tony finds that 5 blue blocks weigh the same as 2 red blocks. Write down an equation that x satisfies. Solve the equation. Write down the weight of a blue block and the weight of a red block. 5 x = 2(x + 60) 5 x = 2 x + 120 5 x – 2 x = 120 3 x = 120 x = 40
8. Draw on the grid below, the enlargement of the given shape, using a scale factor of 3 and centre A A
9. When full, a jug holds 1 ⅓ litres. How many times can the jug be completely filled from a 15 litre container? = 15 ÷ 1 ⅓ = 15 ÷ 4 1 3 = 15 x 3 1 4 = 45 4 = 11 ¼ = 11 times
10. The points A and B have coordiates (-6, 7) and (4, 1) respectively and N is the foot of the perpendicular from A onto the –x axis. Y A (-6, 7) Diagram not drawn to scale. B (4, 1) N Write down the coordinates of (a) the mid-point of the line AB, (-1, 4) (b) The point N ( , ) -6 0 O X
11. Some of the ingredients needed to make enough Banoffi pie for 6 servings are listed below: 175 g of butter 30 g of plain chococlate 2 bananas 300 ml of double cream (a) How many bananas would be needed foe 18 servings? 6 servings need 2 bananas, 18 servings need 2 x 3 = 6 bananas (b) How much plain chocolate would be needed to make enough pie for 21 servings? 6 servings needs 30 g 1 serving needs 30 ÷ 6 = 5 g 21 servings needs 21 x 5 = 105 g Or 3 servings = 15 g 21 servings = 7 x 15 = 105 g
12. Solve the equation. 7 x + 15 =3(x+8). 7 x + 15 = 3 x + 24 7 x – 3 x = 24 - 15 4 x = 9 4 x = 2 ¼
13. The engine capacity, measured in cubic centimetres (c. c) and the time, in seconds, taken to accelerate to a certain speed, for each of 8 cars, are given in the table. Engine capacity (c. c. ) 1000 1100 1200 1300 1400 1600 1800 Acceleration time (s) 15. 4 14. 0 13. 4 11. 8 9. 1 6. 9 2000 6. 0 (a) On the graph paper, draw a scatter diagram to display these results. (b) What type of correlation does your scatter diagram show? Negative (c) The mean engine capacity is 1425 c. c. and the mean acceleration time is 11 seconds. Draw a line of best fit on your scatter diagram. (d) Use your line of best fit to estimate the acceleration time for a car with an engine capacity of 1750 c. c. = 7. 4 seconds
16 14 Time (seconds) 12 10 8 6 4 2 0 1000 1100 1200 1300 1400 1500 Engine capacity (c. c. ) 1600 1700 1800 1900 2000
14. (a) Complete the table which gives the values of y = 2 x² + 4 x – 5 for values of x ranging from – 4 to 3. x - 4 - 3 Y = 2 x² + 4 x – 5 11 1 - 2 - 5 - 1 0 1 2 3 - 7 - 5 1 11 25 (b) On the graph paper draw the graph of y – 2 x² + 4 x – 5 for values of x ranging from -4 to 3. (c) Draw the line y = 8 on the same graph paper and write down the xvalues of the points where the two graphs intersect. -3. 7 a / and 1. 7 (d) Write down the equation in x whose solutions are the x-values you found in (c). 2 x² + 4 x – 5 = 8 2 x² + 4 x – 13 = 0
y 30 25 20 15 10 y = 8 5 -4 -3 -2 -1 0 -5 -10 1 2 3 x
15. Enid and George hide a box in their garden. They make a map of the garden, using a scale of 1 cm to represent 1 m. They give the map to some friends together with the following clues. The box is nearer the end A of the hedge than the end C. The box is less than 6 m away from the tree marked T. The box is nearer the garden wall AB than the hedge AC. On the map shown below, shade the region of the garden in which the box is hidden. A Hedge C T House wall Garden wall Scale: 1 cm = 1 m B
16. In a small pack of nine cards, the cards are numbered 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively. A fair cubical dice has faces numbered 1, 2, 3, 4, 5 and 6 respectively. Terry draws a card at random from the pack and rolls the dice. Calculate the probability that the number on the card is even and that the dice shows 5. = P (even) and P(5) = 4 x 1 9 6 = 4 54 = 2 27
17. Draw the image of the shape A after a translation of – 3 units in the x-direction and 5 in the y-direction. Label the image B. y 5 3 B 1 -5 -3 -1 -1 A -5 1 3 5 x
18. Sacks are filled with 50 kg of sand measured correct to the nearest kg. Write down the least and greatest amounts of sand there could be in the sack. 49. 5 kg Least …………………. 50. 5 kg Greatest …………………. (b) A person buys 20 sacks of sand. Write down the last and greatest amounts of sand he could receive. = 20 x 49. 5 Least kg = 990 = 20 x 50. 5 Greatest kg = 1010
19. Solve the simultaneous equations by an algebraic (not graphical) method. Show all your working. 4 x – 3 y = 20 6 x – 5 y = 22 1 2 Substitute y = 16 in equation 1 4 x – 3 y = 20 Multiply eqn 1 x 3 and eqn 2 x 2 1 x 3 12 x – 9 y = 60 3 4 x – 3 x 16 = 20 2 x 2 12 x – 10 y = 44 4 4 x – 48 = 20 3 - 4 -9 y - -10 y = 60 - 44 y = 16 4 x = 20 + 48 4 x = 68 4 x = 17
20. Each of the following quantities has a particular number of dimensions. Give the number of dimensions of each quantity. The first one has been done for you. Quantity The volume of a cone Number of dimensions 3 The perimeter of a polygon 1 The capacity of a bucket 3 How far a satellite travels in one orbit of the Earth The area of the cross-section of a prism 1 2
21. (a) Show, giving reasons, that the triangles ABC and XYZ below are not similar. You must show all your reasoning. Y B 16 cm 12 cm 8 cm A 6 cm C 12 cm Diagrams not drawn to scale. X 8 cm Z If similar then BA = AC = CB YZ ZX XY BA = 8 = 4 YZ 6 3 AC = 12 = 3 ZX 8 2 2 ≠ 3 therefore shapes not similar CB = 16 = 4 3 4 XY 12 3
(b) Every square is similar to every other square. Name another geometrical figure that has this property. Circle Equilateral triangle Regular pentagon Regular hexagon Regular polygon
22. (a) Simplify (2 a 4 c) x (5 a³c²). =10 a 7 c 3 (b) Expand the following expression, simplifying your answer as far as possible. (x – 2 ) ( x – 6 ) First Outside Inside Last = x² -2 x -6 x +12 = x² -8 x +12 (c) Make r the subject of the formula 3 t + 7 = 5 ( t – 2 r ) 3 t + 7 = 5 t – 10 r = 5 t -3 t -7 10 r = 2 t - 7 r = 2 t – 7 10
23. Glomo and Staybrite are two types of electric light bulbs. The lifetimes, in complete weeks, of eighty bulbs of each type were measured and recorded. The results for the Glomo bulbs are summarised in the following table. Lifetime in complete weeks 0 -9 10 -19 20 -29 30 -39 40 -49 50 -59 60 -69 70 -79 80 -89 Frequency 2 3 8 5 4 11 31 15 1 (a) Complete the following cumulative frequency table for the Glomo bulbs. Lifetime in complete weeks (less than) 10 20 30 40 50 60 Cumulative frequency 2 5 9 70 80 90 20 51 66 74 79 80
Cumulative frequency (b) The graph below shows the cumulative frequency diagram for the 80 Staybrite bulbs Using the same graph paper, draw a cumulative frequency diagram for the Glomo bulbs Glomo 80 60 40 20 (c) Use your cumulative frequency diagram to find the median and interquartile range for the Glomo bulbs. Upper Quartile Staybrite Median = 48 Lower Quartile Interquartile range = 56 – 40 16 0 0 20 40 60 80 Lifetime in complete weeks 100 (d) David wants a bulb that will last at least 75 weeks. If cost is not a factor, which type should he buy? Give a reason for your choice. Staybrite – only 54 bulbs have blown whereas 77 Glomo bulbs have blown in 75 weeks
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