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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
3
+ 2
5
+ x
x= 1 - 1 - 2
3
5
x = 15 - 5 - 6
15 15 15
x=
4
15
= 1
Or
1 x 600 = 200
3
2 x 600 = 240
5
600 – 200 -240 = 160
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
6a – 3 – 2a + 8
= 4a + 5
3. (c) Find the value of 5x + 4y 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
2x + 70 = 180
55+ y = 180
2x = 180 - 70
2x = 110
y = 180 - 55
x = 55 º
y = 125 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
R
All its faces are triangles
S
It has exactly 5 vertices
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
9
11
12
13
17
6
3
2
7
4
3
1
9
6
5
3
10
7
6
4
11
8
7
5
15
12
11
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 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?
Cost of playing = 150 x £1 = £150
Winnings = 600 x 30p = 18000p = £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.
5x = 2(x + 60)
5x = 2x + 120
5x – 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 ÷ 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)
X
N
Write down the coordinates of
(a) the mid-point of the line AB,
(-1, 4)
(b) The point N
( -6 , 0 )
O
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?
6 servings need 2 bananas, 18 servings need 2x3 = 6 bananas
(b) How much plain chocolate would be needed to make enough pie
for 21 servings?
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 + 24
7x – 3x = 24 - 15
4x = 9
x= 9
4
x=2¼
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
11.4
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 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.
= 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 = 2x² + 4x – 5
for values of x ranging from – 4 to 3.
x
-4
-3
Y = 2x² + 4x – 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 – 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 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).
2x² + 4 x – 5 = 8
2x² + 4 x – 13 = 0
y
30
25
20
15
10
y=8
5
-4
-3
-2
-1
0
-5
-10
1
x
2
3
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.
A
Hedge
C
T
House wall
Garden
wall
Scale:
B
1cm = 1m
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)
= 4x1
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
-1
A
-5
3
5
x
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.
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
= 990
kg
= 20 x 50.5
Greatest
= 1010 kg
19. Solve the simultaneous equations by an algebraic (not graphical)
method. Show all your working.
4x – 3y = 20
6x – 5y = 22
1
2
Substitute y = 16 in equation 1
4x – 3y = 20
Multiply eqn 1 x 3 and eqn 2 x 2
1x3
12x – 9y = 60
3
4x – 3 x 16 = 20
2x2
12x – 10y = 44
4
4x – 48 = 20
3 - 4
-9y - -10y = 60 - 44
y = 16
4x = 20 + 48
4x = 68
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
16cm
12cm
8cm
6cm
C
A
12cm
Diagrams not drawn to scale.
X
8cm
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
(2a4c) x (5a³c²).
=10a7c3
(b) Expand the following expression, simplifying your answer as far as
possible.
(x – 2 ) ( x – 6 )
First Outside Inside Last
= x² -2x -6x +12
= x² -8x +12
(c) Make r the subject of the formula
3t+7=5(t–2r)
3t + 7 = 5t – 10r
10r = 5t -3t -7
10r = 2t - 7
r = 2t – 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
(c) Use your cumulative frequency
diagram to find the median and
interquartile range for the Glomo
bulbs.
Upper Quartile
Staybrite
40
Median
Median = 48
20
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