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 (b) 8.3 – 2.47 =

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? x = x = x = x = 4 15 Or 1 x = x = – = = = 4 15

3. (a) Write down the next two terms of the following sequence. 110, 100, 80, 50, …, …… 3. (b) Simplify 6a – 3 – 2a = 4a + 5 Difference increases by 10

3. (c) Find the value of 5x + 4y when x = -3 and y = -2 INPUTAdd 9Divide by 4OUTPUT If the input is n, write down the output in terms of n. (d) The diagram below represents a number machine. = 5 x x -2 = = -23 = n + 9 4

4.Find the size of each of the angles marked x and y. x = º y = º x y 70º 2x + 70 = 180 2x = x = y = 180 y =

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 shapeLabel on shape All its faces are squareP 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 Red bag

6. (b) (i) What is the probability that a player scores 7 (ii)What is the probability that a player does not score 7 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? = 3 20 = = 5 20 = Black bag Red bag

6. (d) (i) 600 people each play the game once. Approximately how many would you expect to win a prize? (ii) It costs 30p 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? = 1 x = 150 Cost of playing = 150 x £1 = £150 Profit = = £30 Winnings = 600 x 30p = 18000p = £180

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. (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. = x x = 2(x + 60) 5x = 2x x – 2x = 120 3x = 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 ÷ = 15 x = 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) N B (4,1) O X Diagram not drawn to scale. Write down the coordinates of (a) the mid-point of the line AB, (b) The point N (, ) (-1, 4) -6 0

11. Some of the ingredients needed to make enough Banoffi pie for 6 servings are listed below: 175g of butter 30g of plain chococlate 2 bananas 300ml of double cream (a) How many bananas would be needed foe 18 servings? (b) How much plain chocolate would be needed to make enough pie for 21 servings? 6 servings need 2 bananas, 18 servings need 2x3 = 6 bananas 6 servings needs 30g 1 serving needs 30 ÷ 6 = 5g 21 servings needs 21 x 5 = 105g Or 3 servings = 15g 21 servings = 7 x 15 = 105g

12. Solve the equation. 7x + 15 =3(x+8). 7x + 15 = 3x x – 3x = x = 9 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.) Acceleration time (s) (a) On the graph paper, draw a scatter diagram to display these results. (c) The mean engine capacity is 1425c.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 1750c.c. (b) What type of correlation does your scatter diagram show? = 7.4 seconds Negative

Engine capacity (c.c.) Time (seconds)

14. (a) Complete the table which gives the values of y = 2x² + 4x – 5 for values of x ranging from – 4 to 3. x Y = 2x² + 4x – (b) On the graph paper draw the graph of y – 2x² + 4x – 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 x- values of the points where the two graphs intersect. (d) Write down the equation in x whose solutions are the x-values you found in (c) x² + 4 x – 5 = 8 2x² + 4 x – 13 = a / and 1.7

x y y = 8

15. Enid and George hide a box in their garden. They make a map of the garden, using a scale of 1cm to represent 1m. 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 6m 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. B A Garden wall House wall Scale: 1cm = 1m C T Hedge

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 = 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. x y A B

18. Sacks are filled with 50kg of sand measured correct to the nearest kg. Write down the least and greatest amounts of sand there could be in the sack. Least …………………. Greatest …………………. (b) A person buys 20 sacks of sand. Write down the last and greatest amounts of sand he could receive. Least kg Greatest kg 49.5 kg50.5 kg = 20 x 49.5 = 990 = 20 x 50.5 = 1010

19. Solve the simultaneous equations by an algebraic (not graphical) method. Show all your working. 4x – 3y = 20 6x – 5y = Multiply eqn 1 x 3 and eqn 2 x x – 9y = 60 12x – 10y = 44 -9y - -10y = y = 16 Substitute y = 16 in equation 1 4x – 3y = 20 4x – 3 x 16 = 20 4x – 48 = 20 4x = x = 68 x = 68 4 x = 17 1 x 3 2 x 2

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. QuantityNumber of dimensions The volume of a cone3 The perimeter of a polygon The capacity of a bucket How far a satellite travels in one orbit of the Earth The area of the cross-section of a prism

21. (a) Show, giving reasons, that the triangles ABC and XYZ below are not similar. You must show all your reasoning. A B C X Y Z Diagrams not drawn to scale. 8cm 16cm 12cm 8cm 6cm If similar then BA = AC = CB YZ ZX XY BA = 8 YZ 6 AC = 12 ZX 8 CB = 16 XY 12 = 4 3 = 3 2 = ≠ 3 therefore shapes not similar 3 4

(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 (2a 4 c) x (5a³c²). (b) Expand the following expression, simplifying your answer as far as possible. (x – 2 ) ( x – 6 ) (c) Make r the subject of the formula 3 t + 7 = 5 ( t – 2 r ) = x² -2x -6x +12 = x² -8x +12 3t + 7 = 5t – 10r 10r = 5t -3t -7 10r = 2t - 7 r = 2t – 7 10 =10a 7 c 3 First Outside Inside Last

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 Frequency (a) Complete the following cumulative frequency table for the Glomo bulbs. Lifetime in complete weeks (less than) 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 Cumulative frequency Lifetime in complete weeks (c) Use your cumulative frequency diagram to find the median and interquartile range for the Glomo bulbs. (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 Median = 48 Interquartile range = 56 – Median Upper Quartile Lower Quartile Staybrite Glomo