Credit Reasoning Practice
after ch1 area and volumes
1
The 5 chefs wanted to share a big rectangular chocolate.
However, it fell on the ground and when they opened it
they saw that it broke into 7 pieces. Molly ate the biggest
piece. Fatty and Sweety ate the same amount of chocolate
but Fatty ate three pieces while Sweety ate only one
piece. Belly ate a seventh of the whole chocolate, and
Spherie ate the rest.
Which chef ate which piece or pieces of the chocolate?
2
The table shows some interest rate for credit cards
Credit cards Interest Rates
Name of card
Monthly Rate
Annual percentage Rate (APR)
Flexicard
2.2%
29.8%
Shopcard
2.1%
Trustycard
23.9%
Read the following instructions carefully
The APR for FLEXICARD is obtained as follows
The amount outstanding each month is multiplied by 102.2%
MULTIPLYING FACTOR FOR 1 MONTH = 1.022, because 102.2% = 1.022
MULTIPLYING FACTOR FOR !” MONTHS =
1.02212 .
1.02212 = 1.298 CORRECT TO 3 DECIMAL PLACES.
The APR is therefore 29.8% correct to one decimal place.
a)
b)
Use the instructions shown above to calculate the APR for Shopcard
Calculate the monthly rate for TRUSTYCARD.
3
The diagram shows an open cylindrical tube which sits on top of a cuboid. A liquid is
poured into the top of the tube as shown. The liquid will fill the cuboid at the bottom before
filling up the cylindrical tube. It was the intention to put markers on the container to show
when the container would be
a) ¼ full
b) ½ full
c) ¾ full
of liquid but unfortunately these markings have been missed out. Trace the diagrams.
Perform calculations and show clearly exactly where these markings should go on the diagrams.
d=4cm
Water
in
20cm
5cm
4cm
Water
in
20cm
12cm
2cm
12cm
4cm
10cm
12cm
Page 1
4cm
2cm
10cm
water
Credit Reasoning Practice
after ch1 area and volumes
1
The 5 eat-all-we-can chefs use 3 major units of weight: chubby, heavy, and biggie. They
use the following rules of conversions: 5 heavies = 2 dekachubbies, and 7 chubbies = 11
decibiggies. (As we all know: 1 dekachubby = 10 chubbies, and 1 biggie = 10 decibiggies)
Put the 5 chefs in an order of their weights if Fatty is 2 kilochubbies, Sweety is 36
dekaheavies, Belly is 22 biggies, Spherie is 16 hectochubbies, Molly is 215 decibiggies.
2
My grandma harvested all her fruit in her huge garden. Half of all the fruit was plum, a
fifth was apricot, and the rest were apples. The apricot and the apples together were
500 kg. How much of each fruit did she harvest?
3
A security firm uses a guard dog to look after a compound. The compound
is a rectangular area measuring 90m by 40m as shown in the diagram.
On the first night, the dog is attached to a 40m chain which is tethered
to the wall at point A.
a)
Draw a diagram showing the area which the dog can patrol and
calculate this area.
40m
Compound
90m
A
40m
90m
B
40m
9
0
m
b) On the second night the dog is tethered at point B halfway down
one wall. Show and calculate the new area which the dog can patrol.
90m

c) After a breakout from the compound, it is decided to introduce a
second dog. This dog is also on a 40m chain. The dogs are tethered to the
walls as shown. Calculate the area now patrolled by the dogs.d) Are these
90m
the most effective positions to put the dogs in or can you think of a better
arrangement which would allow the dogs to patrol a greater area of the compound?
4
The diagrams show a number of pencils of various sizes.
Pencil length
5cm
Jane picks up three pencils at
8cm
random and forms a triangle.
12cm
a) How many different
13cm
triangles can she actually form?
15cm
b) How many of these triangles
17cm
will be right angled triangles?
24cm
40m
5
A man is riding his bicycle on a road that can be thought of as having four parts of equal
length. On the first part of his journey, which is level, he pedals at 10 kilometres per
hour. On the second part, an upward slope, he goes 5 kilometres per hour. On the third
part, a downward slope, he goes 30 kilometres per hour. On the last part, which is level
again-but with the wind pushing him-he goes at 15 kilometres per hour?
a)
What is the man’s average speed for the whole journey?
b)
A cyclist covers the first half of a journey at an average speed of ( v + 10) km/hr
and the second half of the journey at a speed of (v – 10) km/hr. Find his average speed
for the whole journey.
Page 2
Credit Reasoning Practice
after ch1 area and volumes
1
Grandma walked from her house to the back of her garden to bring some water for the
workers. When she walked half the distance and another 50 metres, she stopped to rest
a little. When she walked half of the rest of the distance and another 50 metres, she
stopped to rest again. Then, finally, when she walked half of the rest of the distance and
another 50 metres, she got to the back of her garden. How far is the back of her garden
from her house?
2
The volume of an ice cube is numerically equal to its totals surface area. What is the
length of each side of the ice cube ?
3
A large floor is to be covered with black and
checked square tiles to make a pattern as
shown. The person laying the tiles must start
at the centre of the floor and work outwards.
a)
How many tiles are in the 4th pattern ?
b)
The formula of tiles T, needed to make the nth pattern is given by the
formula
T = 2N² + aN + b.
Find the values of a and b.
4
h metres
Three pipes are stored on horizontal ground as
shown in the diagram.
Each pipe has a circular cross-section with radius
1 metre.
Calculate the height, h metres, of stacked pipes.
( Ignore the thickness of the pipes.)
Give your answer in metres, correct to two decimal
places.
5
If x² - y² = 8 and x – y = 2, find the value of x + y.
Running
lanes
are all
1m wide
80m
Field
Track
b)
75.4m
6
The diagram shows an athletic track with
four running lanes. An athlete when running in a
lane will be assumed to be running in the centre of
her lane.
a)
Show that an athlete running in the first
lane will cover 400 metres in one lap of the track.
How far apart will the start positions in the other lanes be so that all athletes will run
400 metres in one lap of the track?
Page 3
Credit Reasoning Practice after ch1 area and volumes
1
90cm
This tree trunk is to be hollowed out to make a 4m long playground tunnel.
The tunnel will, externally, be a cylinder and will have a square based cuboid
hollowed out from within it. If the diameter of the cylindrical part of the
tunnel is 90cm and the volume to be removed is to be equal to the volume
left what is the length of a side of the entrance to the tunnel ?
2
A circular air vent has a diameter of 30cm. The diameter is to be enlarged by 4x cm.
Show that the increase in area is 4x( x -15 )cm².
3
A cable car is used to carry sightseers up a mountain.
For safety reasons, the cable car company must consider the total weight of sightseers
in a cable car.
They assume the average weight of an adult is 75kg and the average weight of a child is
35kg.
a)
Write down a formula for the total weight, W kilograms, of x adults and y
children.
b)
In the busy season, the company sets the following conditions.
i)
10 passengers must be carried at any one time.
ii)
Every child must be accompanied by at least one adult
iii)
The maximum total weight which can be carried is 700kg.
List all the combinations of adults and children which can now be carried in the
cable car to meet all the above conditions. Show all your working clearly.
4
5
2m
3.5m
The conservatory is to be 3 metres wide. The
height of the conservatory at the lower end is to be
2 metres and at the higher end 3.5 metres.
To obtain planning permission, the roof must slope
at an angle of ( 25 ± 2 ) degrees to the horizontal.
Should planning permission be granted ?
Justify your answer.
3m
6
A spacecraft has crash landed on the Moon 67 miles from its intended target, the Moon
Base at Armstrong Crater. The astronaut must leave his craft and
Moon Base
make his way unaided to the Moon Base. The spacecraft has on board,
Astronaut
12 cylinders full of oxygen. These cylinders could be attached to the
astronaut’s back and carried by him. Each cylinder can take the
67 miles
astronaut a distance of 20 miles and no more. Unfortunately the
astronaut can only carry two cylinders at a time. There is nothing to stop him leaving a cylinder
a distance from the craft and returning for others. Is it possible for the astronaut to reach
the Moon Base ?
Page 4
Credit Reasoning Practice
1
x is a whole number. Describe
a) an even number b) consecutive numbers
c) the sum of an even and odd number
d) the product of an even and odd number
2
A number pattern is given below :
1st term
2² - 0 ²
2nd term
3² - 1 ²
a)
Write down a similar expression for the 4th term
b)
Hence or otherwise find the nth term in simplest form
3
4
Using the sequence 1, 3, 5, 7, 9 …..
a)
Find S3, the sum of the first 3 numbers
b)
Find Sn, the sum of the first n numbers
c)
Hence or otherwise, find the ( n + 1 )th term of the sequence
a)
b)
c)
5
( 2005 )
( 2003 )
Solve the equation 2n = 32
A sequence of numbers can be grouped and added together as shown.
The sum of 2 numbers :
(1+2)
=4–1
The sum of 3 numbers :
(1+2+4)
=8–1
The sum of 4 numbers :
(1+2+4 +8)
= 16 – 1
Find a similar expression for the sum of 5 numbers.
Find a formula for the sum of the first n numbers of this sequence. ( 2002 )
1³ + 1 = ( 1 + 1 )( 1² - 1 + 1 )
2³ + 1 = ( 2 + 1 )( 2² - 2 + 1 )
3³ + 1 = ( 3 + 1 )( 3² - 3 + 1 )
Write down a similar expression for 7³ + 1
Hence write down an expression for n³ + 1
Hence find an expression for 8p³ + 1
( 2001 )
1, 3, 5, 7, …
a)
b)
c)
d)
7
4² - 2²
A number pattern is shown :
a)
b)
c)
6
3rd term
The first odd number can be expressed as
1 = 1² - 0
The second odd number can be expressed as
3 = 2² - 1²
The third odd number can be expressed as
5 = 3² - 2²
Express the fourth odd number in this form
Express the number 19 in this form
Write down a formula for the nth odd number and simplify this expression
Prove that the product of two consecutive odd numbers is always odd. ( 2000 )
A sequence of terms, starting with 1, is
1, 5, 9, 13, 17,………
Consecutive terms in this sequence are formed by adding 4 to the previous term.
The total of consecutive terms of this sequence can be found using the following pattern:
Total of the first 2 terms : 1 + 5
=23
Total of the first 3 terms : 1 + 5 + 9
=35
Total of the first 4 terms : 1 + 5 + 9 + 13
=47
Total of the first 5 terms : 1 + 5 + 9 + 13 + 17
=59
a)
Express the total of the forst 9 terms of this sequence in the same way.
b)
The first n terms of this sequence are added. Write down an expression, in n, for the
total.
( 1998 )
Page 5
Credit Reasoning Practice
A
after ch3 Similarity
1
A circus see-saw, AB is 7m long and is set as shown. When it
tips to the right the maximum height of the left seat A, is 1.8m
from the ground. The pivot point is 1.2m high. When it tips to
the left the right seat B, is x cm from the ground. What is the
maximum height the right seat can be from the ground ?
7m
1.8m
1.2m
B
3
Triangle ABC is right angled at C with
A
BC = 8cm. D and E are points on AB and AC
D
BC
B
4
P
8cm
E
xcm
Q
2cm
respectively such that DE is parallel to
2cm and AE = EC = 2cm
C
The diagram shows the cross-section of a petrol tank
A dipstick is used to check the level of the petrol in the
tanks.
The dipstick has marks to show empty (E), quarter full(¼), half full (½), three quarters full( ¾)
and totally full (F).
a)
Which dipstick 1, 2, 3 or 4 should be used with the tank ?
b)
Here is another petrol tank.
Sketch a graph to show how the depth of the
petrol varies with the volume of petrol in the tank.
5
A group of four people, when asked the time gave these different answers: - Peter said
it was 6.13pm.
Paul said it was 6.11pm. Mary said it was 6.18pm. John said it was 6.15pm.
They were all supposed to be telling the same time! They were all wrong. Their errors (in no
particular order) were 1, 2, 3 and 5 minutes.
a)
What was the correct time?
b)
Tom, Dick, Harry and Peter were asked the time. Tom said it was 8.16pm. Dick said it
was 8.24pm. Harry said it was 8.18pm and Peter said it was 8.23pm. They were all supposed to
be telling the same time! They were all wrong. Their errors (in no particular order) were 2, 3, 4
and 4 minutes. What was the correct time?
c)
Five people were asked if they knew the correct time. They gave the replies 9.37pm,
9.39pm, 9.40pm, 9.44pm and 9.46pm. They were all supposed to be telling the same time! They
were all wrong. Their errors (in no particular order) were 1, 2, 3, 4 and 5 minutes. What was the
correct time?
Page 6
Credit Reasoning Practice
after ch Ratio of Areas
R
1
T
P
18cm
2
6cm
S
Q
In the diagram, PST = PRQ, ST = 6cm, QR =
12cm and PQ = 18cm.
12cm
Calculate the length PT and the value of the ratio
area of triangle PST : area of triangle PRQ.
If chord KL equals 24cm and the circle, centre O,
has a radius of 18cm, what will be the area of the
major arc KL ?
K
O
L
3
The Fjord Car Company have produced a new version of their top selling car and state
that it will travel an extra 3 miles per 5 litres of petrol. Grace is to test their claim.
She travelled 252 miles in the old model and 270 miles in the new model using the same
amount of petrol and found their claim to be true.
What is the mileage per 5 litres for the new model ?
2cm
4
(x+1)cm
xcm
A picture xcm broad and ( x + 1 )cm wide is to be
mounted on card. The area mounted on card. The
area of the border is to be 2cm² less than half of
the area of the original card used for the mount.
What are the dimensions of the picture ?
2cm
5
H
G
6
33°
44°
E
A signal is to be sent from Gerald(G), to Harry(H),
who is at the top of a mountain to Enid (E) who is
at the emergency centre to test clarity of
reception for the emergency services. The signal is
sent out at an angle of 33° and received at an
angle of 44°. If Gerald is 425km away from the
emergency centre what height is Harold above the
ground ?
A cylindrical can of soup is 14cm high and 6.8cm in diameter. A new design, which is to
hold the same volume, has a smaller height of 12.6cm. What is the diameter of the new
can ?
Give your answer correct to one decimal place ?
Page 7
Credit Reasoning Practice after ch8 quadratics
1
An architect advises that the extension should
have its length 2 metres more than its width.
a)
If the width of the extension is
w metres, write down an expression
for its length.
length
width
Planning regulations state that the area of the ground floor of the extension must not
exceed 40% 0f the area of the ground floor of the original house.
b)
The ground floor of the original house is 12 metres by 10 metres. Show that,
if the largest extension is to be built.
c)
Find the dimensions of the largest extension which can be built .(1994)
2
15m
3
.
The area of the sector is 200 square metres.
Find the length of the arc of the sector.
sensor
15m
The chefs are making cakes and they are cutting them into
4 cm x 4 cm pieces. They are leaving 1 cm spaces between
them so the pieces would not stick to each other. On the
smaller, square shaped plates 9 of these pieces fit so that
there is no more space around the edges. How many of
these pieces fit on the bigger plate which has 6 times as long
sides as the smaller plate?
4
On a table is a bag of nuts. Standing round the table are three boys Tom, Dick and Harry
and a gorilla called Hilda. The nuts are to be shared between them. Tom takes the bag of nuts.
He gives one to the Gorilla and takes half of what is left for himself, and then passes the bag
onto Dick. Dick opens the bag and takes half the nuts for himself before passing one to the
Gorilla. He then passes the bag to Harry. Harry eats one of the nuts when he gets the bag and
then shares the rest equally with the Gorilla. If the Gorilla got 17 nuts in total, how many nuts
were originally in the bag?
Page 8
Wall
Credit Reasoning Practice after ch8 quadratics
Sheep
1
Fencing
Farmer Jack has a field in which there is a long straight wall.
He also has 245 metres of fencing and wishes to use the fencing together with the wall,
to fence off a rectangular area in which he can keep his sheep.
a)
Calculate the largest area which he can fence off.
wall
b)
In another field there is a long wall. The farmer this
cows
goats pigs
time has 192 metres of fencing and he wishes to fence
fencing
of a rectangular area using the wall and the fencing. He
also wishes to have separate compartments for the different animals.
He would like to know the maximum area which he can fence off for his animals.
c)
Farmer Jill has 225m of fencing and wishes to have separate compartments for
goats, pigs, sheep cows and hens, advise her.
2
The Oklahoma Kite Company makes kites, using best quality rip-stop nylon to cover the
framework which is constructed of lightweight glass-fibre rods. In order to minimize costs the
company must use all the materials with as little waste as possible. One of the problems they
have in the company is how to fit the kite shape onto a long rectangular sheet
60cm
of nylon in the most economical way.
30cm

The dimensions of a kite are shown in the diagram and the nylon for making
the kites is available in rectangular strips which can be up to 4 m wide and

120cm
120m long.
a)
For each configuration shown, calculate the minimum width of nylon
necessary, the length required to produce 12 shapes and the percentage waste of material in
each case
    
    
A
C












B


Calculate the % waste in producing
a) 48 kites per strip of nylon
b) 120 kites per strip of nylon
c) 180 kites per strip of nylon
 


D


E




Which is the most economical method?
If it is decided to use the maximum
available length of 120m, what are the %
waste figures now?
Page 9
 


 







     











     











Credit Reasoning Practice after ch9 Surds and Indices
1
Tom and Dick were going to cut the grass on their square lawn.
They agreed that each should cut one-half of the area. Tom went
first and cut a border 3 metres wide all the way round as shown in
the diagram.
After doing a few calculations, Dick agreed that exactly one-half
had been cut and happily cut the rest.
3m
a)
What was the area of the lawn?
b)
Suppose the border cut by Tom had actually been 5 metres wide all the way round,
what would the area of the lawn be now?
c)
If the border cut by Tom had been ‘a’ metres all the way round, show that the area
of the lawn is given by the expression A = 8a2(3 + 2√2).
2
The diagram shows two tower blocks 150m and 100m
L
O
high. The tower blocks are 200m apart. Power cables
150mC
run from the top of a tower to the foot of the opposite
K
1
tower as shown in the diagram. The cables cross a
h
200m
height ‘h’ above the ground.
a)
Determine the height h.
b)
Repeat the above calculations to determine ‘h’ or
towers of heights 125m and 95m which are 80m apart.
Show that if the towers have heights ‘a’ and ‘b’ and are a distance ‘c’ apart,
then the height ‘h’ is given by
.
ab
B
h
Power cables
gro
B
un
L
O 100m
d
C
K
2
ground
(a  b )
Use this result to show that if Block1 is twice as high as Block2 then the cables cross
two thirds of the way up Block2.
A
B
C
3
The diagram shows a circle of radius r. Two parallel chords (shown in
the diagram) divide the circle into three equal areas A,B and C
Show that the length L of the chords is given by L = 1.93r and that
the chords are approximately 0.26r from the centre of the circle.
4
The two main scales used for measuring temperature are the Fahrenheit and the Celsius
scales. Some commonly-used temperatures are shown as they would be shown on those scales.
Fahrenheit
Celsius
Boiling point of water is
212°F
100°C
Blood heat is
Freezing point of water
is
98.6°F
32°F
37°C
0°C
Letting C = aF + b, form two equations using some of the information in the table and find ‘a’ and
‘b’.
Using this formula or otherwise, can you determine:a)
At what temperature do both scales read the same?
b)
At what temperature is the Fahrenheit reading 100° more than the Centigrade
reading?
Page 10
c)
At what temperature is the Fahrenheit reading double the centigrade reading?
Page 11
Credit Reasoning Practice
1
This triangular array is known as Pascal’s Triangle
Row 1
1
Row 2
1
1
Row 3
1
2
1
Row 3
1
3
3
1
Row 4
1
4
6
4
1
a)
Write down the next two rows
Pascal’s triangle can be used to expand expressions like ( 1 + x )n ,where n is the whole
number.
( 1 + x )0
=1
1
(1+x)
=1+x
2
(1+x)
= 1 + 2x + x²
3
(1+x)
= 1 + 3x + 3x + x³
b)
Use Pascal’s triangle to expand ( 1 + x )4 .
2
Prove that the sum of three consecutive whole numbers is a multiple of 3
3
Prove that the sum of three consecutive even numbers is a multiple of 6
4
Prove that the difference between the squares of two consecutive whole numbers is odd.
5
There are five brothers in the Jackson family, each one 3 years older than the one
before him.
a)
Prove the sum of their ages is a multiple of 5
b)
Find their ages if these total 500
6
Consecutive cubic numbers can be added using the following pattern.
22  32
1 2 
4
3
32  42
1 2 3 
4
3
3
3
42  52
1 2 3  4 
4
3
3
3
3
3
a)
b)
Express 1  2  3  4  5  6  7 in the same way
Write down an expression for the sum of the first n consecutive cubic numbers.
c)
Write down an expression for
3
3
3
3
3
3
3
83  93  103  .........n 3
( 1997 )
7
Brackets can be multiplied out in the following way
( y + 1 )( y + 2 )( y + 3 ) = y³+ ( 1 + 2 + 3 )y² + ( 1  2 + 1  3 + 2  3)y + 1  2  3
( y + 2 )( y + 3 )( y + 4 ) = y³+ ( 2 + 3 + 4 )y² + ( 2  3 + 2  4 + 3  4)y + 2  3  4
( y + 3 )( y + 4 )( y + 5 ) = y³+ ( 3 + 4 + 5 )y² + ( 3  4 + 3  5 + 4  5)y + 3  4  5
a)
In the same way, multiply out ( y + 4 )( y + 5 )( y + 6 )
b)
In the same way, multiply out ( y + a )( y + b )( y + c )
( 1995 )
8
The sum Sn of the first n terms of a sequence, is given by the formula
a)
Find the sum of the first 2 terms
b)
When Sn = 80, calculate the value of n.
Page 12
Sn = 3n – 1
Credit Reasoning Practice
after ch12 Triangle Trig
1
The diagram shows the goalposts on a
rugby field.
To take a kick at goal, a player moves
from T to position P.
40°
B
TP is perpendicular to TB.
Angle TPA = 40° and angle APB = 10°
The distance AB between the goal
posts is 5.6metres.
10°
Find the distance from T to P. (1993)
y
P
2
5.6m
m
A
T
y = -x
B
A
0
3
What is the area of triangle OAB ?
y=x
x
y=
2
a) The drawing shows a country’s flag.
It consists of a white cross on a red
background. Both arms of the white cross are the same width. The flag is a square of
side 64cm. If the area of the red is equal to the area of the white, what would be the
width of the white arms?
100cm
b) A new flag design. The diagram shows a new design for a flag.
The flag measures 100cm by 90cm. The flag consists of a red cross
90cm
with a white background. The thickness of the vertical strip is twice
that of the horizontal one. The area of the cross is half the area of
the flag. By making an equation or otherwise determine the thickness of the vertical
strip.
4
The diagram shows a barn in a large field of grass. The barn has a length of 12m and a
breadth of 7m. A goat is tied to a stake at the corner of the barn, marked A. The rope used to
tether the goat is 17m long.
7m
a)
Sketch, showing the grazing area available to the goat.
Grass
b)
Would a larger area of grass be available to the goat if it had
B
12m A
Rope=17m
been tethered to the barn at point B ( midway along the length ) or C
a
( midway along the breadth) or at any other point on the barn.
r
Find the area available to a goat for grazing, if
the barn has length 15m, breadth 9m and rope 23m.
Barn

B
C
Page 13
n
Credit Reasoning Practice
after ch Ratio of Areas
1
The diagram shows two storage jars which are
mathematically similar. The volume of the large jar is 1.2
litres. Find the volume of the smaller jar.
Give your answer in litres correct to 2 sig. fig. (1993KU)
24cm
30cm
2
The volume of water, V millions of gallons, stored in a reservoir during any month is to be
predicted by using the formula
V= 1 + 0.5cos(30t)°
Where t is the number of the month. ( For January t = 1, February t = 2…)
a)
Find the volume of water in the reservoir in October.
b)
The local council would need to consider water rationing during any month in which
the volume of water stored is likely to be less than 0.55 million gallons.
Will the local council need to consider water rationing ?
Justify your answer
(1993)
3
PQ is parallel to BC and EQ = x cm.
a)
Show that PQ = 2( x + 2 )cm
b)
Find an expression in terms of x for the area of triangle APQ and show that the
area of quadrilateral BPQC = 12 – 4x - x²
4
A cuboid has dimensions 3x cm by 2x cm by x cm and a cube has edge 2x cm. Find the
ratio of the surface area of the cube to the surface area of the cuboid.
5
A ship is steaming at 16km/hr on a course 075°.
L shows the position of a lighthouse. At
0600 the ships is at A, which is on a bearing
of 316° from the lighthouse and one hour
later it is at B, which is due North of the
lighthouse.
N
75°
N
A
At what time is the ship nearest to the lighthouse ?
5
A rectangle is x metres broad and its perimeter is 44 metres. If the area of the
rectangle is 120m² what are the dimensions of the rectangle ?
Page 14
B
L
316°
Credit Reasoning Practice
after ch9 Surds and Indices
C
1
Calculate the radius if the circle which passes
through A, B and C
6cm
4.5cm
A
7.5cm
2
3
B
Given ( 2.3  10p )  (2.3  10q ) = 6.9  104. List all the possible values of p and q.
xcm
In the diagram all the angles are right angles. If the area of
this shape is 126cm² what is the perimeter ?
9cm
6cm
xcm
4
If y varies directly as the square of x and inversely as z, what will be the effect on y if x
is doubled ?
P
5
1cm
30°
2cm
Q
6
S
PS is an altitude of right angled triangle
QPR. Find the exact length of PS. Surds
R
This is an architects model for a new office and
shopping complex. Marked on each block is the time
it will take to build. Each block is to be built by a
different team of builders, so two more blocks can
be worked on at the same time.
a)
From figure 2 it can be seen that block D cannot
be started until blocks B and C are both finished.
How long should it take to complete blocks B, C
and D ?
b)
The whole complex must be finished by 1st May 2008. Find the latest possible
starting date. Justify your answer. (1992)
Page 15
Credit Reasoning Practice
1
A driver uses the track shown in figure 1 to test a new car.
In all tests, the driver has to drive the car at
100mph on all the straight bends.
The graph in figure 2 shows how the speed of
the car changes as it travels around the track
on its first lap.
a)
Did the driver start his first lap at A, B,
C, or D ( see figure 1)
Explain your answer clearly.
b)
The car is tested on a different track
under the same test conditions.
The track is shown below in figure 3.
S is the starting point.
Sketch a graph to show how the speed of the car changes on the second lap of this track.
(1992)
2
In a competition, each team plays every other team twice – once at home, once away.
The total number of games played in the competition is given by the expression t² - t
Where t represents the number of teams entered.
a)
If the total number of games played in a competition was 380, how many
teams entered the competition
b)
Is it possible to run a competition like this in which the total number of
games is exactly 200 ?
Explain your answer clearly. (1992)
3
A cylindrical tank, 5 metres long, is used to store a
hazardous liquid as shown in figure 1.
A dipstick is used to measure the depth of the
liquid.
The dipstick passes through the centre O,
of a cross-section of the tank as shown in
figure 2. The diameter of the cross-section of the tank is 2 metres.
To satisfy safe storage conditions, the horizontal surface area of the liquid
in the tank must not be less than 2 square metres.
a)
Explain why the width of the horizontal surface of the liquid must not
be less than 0.4 metres.
figure 2
b)
The depth of the liquid in the tank is found to be 1.8 metres
Can this volume of liquid be stored safely in the tank ?
Justify your answer.(1992)
Page 16
Credit Reasoning Practice after ch9 Surds and Indices
1
It is required to design a cardboard box in the shape of a cuboid which satisfies the
following requirements : The box is to have a volume of 1200cm3, and the length of
the box is to be 3 times the breadth. The area of cardboard used to make
the box is to be a minimum.
V=1200cm3
a)
b)
h 
400
x2
xcm
3xcm
Show that the height h is given by
Show that the total area of cardboard required to make the box is A  3200  6x 2
x
Use this equation and the Table function on your graphic calculator, or otherwise, to determine
the dimensions of the box which will have the minimum surface area.
c)
Find the dimensions of the box satisfying the following requirements:i)
It is to hold 4 litres.
ii)
The length is to be twice the breadth.
iii)
The area of material used to make the box is to be a minimum.
A
2
The rectangular sheet of paper below has an interesting
property. When it is cut in half along AB, it forms two
rectangular pieces of paper each of which is similar in
shape to the original sheet.
A
A
A
B B
B
a)
b)
B
If this is true for a rectangular sheet of paper with its longer side measuring
40cm, what is the length of the shorter side ?
If the long and short sides of the original sheet measures p centimetres and q
centimetres respectively, prove that p² = 2q².
30cm
c)
15√2cm
30.3cm
Sheet1
sheet2
21.7cm
Do the two rectangular sheets of paper
shown have this property ?
Justify your answer. (1991)
3
Evaluate the expression
B
4cm
2 tan A
1  tan2 A
3cm
where A is the angle shown in this diagram.
5cm
A
Page 17
C
Credit Reasoning Practice
1
Sophie aims to send a ball at A towards a ball sitting over a pocket at B. This pocket is
135cm away from A and the angle between AB and the side of the table is 65°. She
makes an error and the ball goes off in a direction which makes an angle of 5° with AB. It
strikes the opposite side of the table at C, as shown in figure 1.
a)
Calculate the distance from C to the pocket B.
b)
From C the ball moves of in a different direction such that the
angles marked in figure 2 are equal.
c)
There is a ball at D, 55cm from B, such that the
ball has been hit hard enough from A, will it hit
the ball at D ? (1992)
2
Due to tidal variations, the depth of water in a harbour is given by the formula
D = 6 + 4cos(32t + 108 )°
Where D is the depth of water in metres and t is the time in hours after midnight on
Monday night. A boat needs at least 4m of water to leave the harbour. Can it leave the
harbour at 3pm on Tuesday ? Justify your answer. (1992)
3
An empty rectangular swimming pool is filled using a
hosepipe which delivers water at a constant rate.
Which of the graphs best describes this
information ?
Explain your choice carefully.
40cm
36cm
(1992)
3
The diagram shows a cake which consists of three tiers. The diameter of the
bottom tier is 40cm. The diameters of the other two tiers are 30cm and 20cm
respectively. On every cake, the height of a tier is the same as the radius of that
tier. The cake has now to be iced. All the cake which is visible will be iced.
a)
What area of the cake will be covered by icing?
b)
What area of the cake will be covered by icing for the cake shown on the left
which has tiers of diameter 36cm, 26cm and 18cm respectively.
c)
What area of the cake will be covered by icing for a
which has tiers of diameter d1, d2 and d3 respectively.
Page 18
Errors
http://www.math.vanderbilt.edu/~schectex/commerrs/
http://teacher.scholastic.com/maven/
http://www.algebra.com/testing/scripts/st.mpl
http://score.kings.k12.ca.us/standards/sixth.html#reasoning
http://www.nzmaths.co.nz/brightsparks/doubletrouble.asp?applet
http://www.gcschool.org/pages/program/Abacus.html
http://www.actionmath.com/wordproblemsamples/newwordproblems.htm
http://www.mathplayground.com/
http://www.satmathpro.com/Practice.html
http://whyslopes.com/
Websites
http://www.learningwave.com/abmath/
http://www.tki.org.nz/r/wick_ed/quizit/feature_quizits.php
http://www.blis.canberra.edu.au/mathstat/site/maths_main_puzzles.htm
http://www.mathsnet.net/puzzles.html
http://trunks.secondfoundation.org/files/psychic.swf
Word problems
http://www.hawaii.edu/suremath/intro_algebra.html
http://www.themathlab.com/Algebra/expressions%20equation%20solving%20and%20grahing/w
ackwordprbs.htm
http://www.stfx.ca/special/mathproblems/welcome.html
lots diff levels here
http://tlfe.org.uk/solveit/
http://teachingtreasures.com.au/maths/yr7-puzzle1.htmProblem solving websites
http://classroom.jc-schools.net/mather/ppt.html
http://www.rhlschool.com/math.htm
http://www.studyworksonline.com/cda/content/article/0,,NAV2-95_SAR1867,00.shtml
http://www.studyworksonline.com/cda/content/explorations/0,,NAV2-95_SEP1237,00.shtml
http://www.studyworksonline.com/cda/explorations/main/0,,NAV2-95,00.html
http://www.stfx.ca/special/mathproblems/welcome.html
http://www.cut-the-knot.org/cgi-bin/dcforum/ctk.cgi?conf=DCConfID1
http://www.louisiana.edu/Academic/Sciences/MATH/problems/dec04pblms.html
http://www.neatherd.org/mathsfun/
http://homeschooling.about.com/od/basicmath/
http://www.figurethis.org/challenges/math_index.htm
Word problems
http://www.algebra.com/algebra/homework/word/
http://www.internet4classrooms.com/word_problems_quest.htm
http://www.stfx.ca/special/mathproblems/welcome.html
Page 19
Answers
 Strange Containers:- Solution
Container 1
¼ mark is 0.94 cm from the ground
½ mark is 1.87 cm from the ground
¾ mark is 11.3 cm from the ground
Container 2
¼ mark is 1.75 cm from the ground
½ mark is 3.50 cm from the ground
¾ mark is 9.01 cm from the ground
Container 3
¼ mark is 1.52 cm from the ground
½ mark is 7.09 cm from the ground
¾ mark is 14.5 cm from the ground
5
The diagram is part of the intersection of
two motorways. The motorway running
North-South crosses the one running East-West
by a bridge. Drivers coming from the West
intending to go North must go via A, P and D.
Drivers coming from the North intending to go
West must go via D, C ,Q ,B and A.
All arcs are circular with a radius of x metres.
Show that difference between the lengths of the two routes is x(4 + )
N
P
A
C
Page 20
D
B
Q