MA T H EMA T I C S
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2
© Pegasys 2008
Higher Still
Intermediate 2
Homework Pack
Contents
Intermediate 2 homework exercises
covering all 4 units.
Full written solutions for all the
exercises.
INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 1
PERCENTAGES
1.
Calculate the compound interest on £500 invested for 3 years at 5% per annum (each year).
2.
John has just put £700 into a savings account where the rate of interest is 4% per annum.
How much will his savings be worth after 3 years?
3.
Mary puts £1200 into an account where the annual rate of interest is 5∙5%.
How long will it be before she has at least £1400 in her account?
4.
My new car has just cost me £18,000. Its value will depreciate by 20% every year.
How much will it be worth when I trade it in 3 years from now?
5.
Due to inflation, house prices are expected to rise by 3∙6% each year.
What will the average house price be in 3 years if it is £142,000 today? (Answer to 3 sig figs)
6.
The pressure in my car tyre should be 30psi, but a nail in it is causing it to lose pressure at the rate
of 15% every mile that I drive.
How far can I drive before the pressure falls below 20psi?
7.
Hassan has been told his hourly pay is to increase from £5.70 to £6.00.
Calculate his percentage increase, giving your answer correct to 2 sig figs.
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INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 2
VOLUMES
Give answers correct to 3 significant figures unless otherwise stated.
Calculate the volume of a cuboid of length 5∙7cm, breadth 3∙2cm and height
3∙7cm, giving your answer correct to 2 sig figs.
1.
2.
(a)
Calculate the volume of the largest sphere which will fit inside a cube of side 15cm.
(b)
Calculate the volume of wasted space between the two. [Answer to nearest cm3]
3.
8cm
Calculate the volume of a cylinder with diameter 12cm and height 8cm.
12cm
4.
A cone has a base diameter of 16cm and a slant height of 17cm.
17cm
Calculate the volume of the cone, giving your answer correct to 3 sig figs.
16cm
5.
A cylindrical hosepipe has a diameter of 10cm and is 20m long.
How many litres of water does it take to fill it?
6.
A lead sinker is in the shape of a cone with a hemispherical base.
The total height of the sinker is 12cm and the diameter of the base is 10cm
12cm
Calculate the volume of lead required to make the sinker.
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10cm
INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 3
BRACKETS and FACTORISING
Simplify:
1.
(a)
3(x + 1) + 2(3x – 1)
(b)
3(2x – 1) – 2(x – 2)
2.
(a)
(x + 4)(x + 3)
(b)
(2x + 3)(x – 1)
(c)
(x + 4) 2
(d)
(2x – 3) 2
3.
(a)
(x + 1)(x 2 + 1)
(b)
(x + 1) 2 – (x – 1) 2
4.
(a)
(x + 2)(x 2 + x + 1)
(b)
(x 2 – 2)(x + 3)
Factorise fully:
5.
(a)
6x – 9x 2
(b)
10x 2 y + 25 xy 2
6.
(a)
4x 2 – 9y 2
(b)
8x 2 – 50y 2
7.
(a)
x 2 + 7x + 12
(b)
x2 – x – 20
(c)
2x 2 – 10x + 12
(d)
2x 3 + 5x 2 + 2x
8.
Find the value of x in this right angled triangle with sides
(x + 9) cm
x cm, (x + 7) cm and (x + 9) cm.
x cm
(x + 7) cm
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INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 4
1.
LINEAR RELATIONSHIPS
For each of the following lines state (i) the gradient and (ii) the coordinates of the point
where it crosses the y – axis.
2.
(a)
y = 2x + 3
(b)
y = −3x − 1
(c)
y=
(d)
y = 5 − 2x
(e)
y−x=3
(f)
2y = x + 4
y=x+1
(b)
y = 2x – 3
(c)
y = −x + 4
(d)
y = − 12 x + 2
y
y
y
y
2
2
2
2
0
-2
2
0
-2
x
-2
2
0
-2
x
2
-2
0
-2
x
-2
1
2
x
2
x
-2
2
3
4
Sketch the graphs of the lines with equations:
y = 3x − 2
(a)
4.
x+1
Match each of these equations with its graph.
(a)
3.
1
2
(b)
y=
1
2
x+1
(c)
y = −2x + 3
(d)
y=2−x
Write down the equations of these lines.
(a)
(b)
(c)
(d)
y
y
y
y
2
2
2
2
0
-2
-2
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2
x
-2
0
-2
2
x
-2
0
-2
2
x
-2
0
-2
INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 5
1.
CIRCLE 1 - SYMMETRY and CHORDS
Find the missing angles in each of these diagrams. Each circle has centre C.
(a)
(b)
30
o
d
ao
jo
C lo
25o
fo
eo
ko
io
C
bo
mo
Use symmetry in the circle to find the missing angles in the circles (centre C) below.
(a)
(b)
(c)
e
61
co
go
C
do
ao bo
40o
ho
o
C
3.
ho
go
C
2.
(c)
co
o
C
io
o
25o
fo
Find x in each of the following:
(a)
(b)
(c)
C
C
30
o
C
25cm
x
x
17cm
37o
x
4.
20cm
14∙2cm
A cylindrical pipe is used to carry water underground.
The diameter of the pipe is 1m.
C
80cm
For safety reasons the maximum depth of water in the pipe is 80cm.
Calculate the width of the water surface when the depth is at its
maximum and state the maximum width of the water surface that
the pipe can accommodate.
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INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 5
1.
CIRCLE 2 - ARCS and SECTORS
Calculate the length of the minor arc AB and the area of the sector AOB in each of the following
where O is the centre of the circle.
(a)
(b)
(c)
120o
6cm
5cm
O
O
O
7cm
72o
2.
The length of arc XY is 8∙5cm.
3.
O
O
12cm
120o
X
xo
Y
Q
P
Calculate the circumference of the circle.
4.
Calculate the size of angle xo if the length of
arc PQ is 15∙1cm.
5.
O
18cm
O
X
150o
60o
Y
Q
P
The area of the sector is 23∙2cm2.
Calculate the area of the circle.
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Calculate the perimeter of sector XOY.
INTERMEDIATE 2 UNIT 1
HOMEWORK
OUTCOME 5
1.
CIRCLE 3 - TANGENTS and ANGLES
Calculate the sizes of the missing angles in each diagram.
(a)
(b)
ho
40o
20o
io
55o
co
go
bo
ao
2.
(a)
eo
do
fo
PR is a tangent to the circle, centre O, at T.
O
13cm
P
x
Calculate the length of the line marked x.
5cm
T
R
(b)
Calculate the diameter of the circle.
O
10cm
40o
P
T
R
C
3.
The length of OC is 8cm and the length
of CB is 15cm.
8cm
Find the length of AB.
O
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15cm
A
B
INTERMEDIATE 2 UNIT 1
HOMEWORK
MIXED EXAMPLES
1.
Mr Hamilton has just bought a classic car for £30,000. He hopes it will appreciate in value by 5%
each year.
How much more is he hoping it will be worth in 3 years time? (answer to 3 s.f.)
2.
A “Binnit” waste bin is in the shape of a cylinder with a semi-circle on top.
The diameter of the bin is 36cm and the total height is 70cm.
Calculate the volume of the bin giving your answer correct to
70cm
the nearest litre
36cm
3.
For each of the following lines, state its gradient and the coordinates of the point where it crosses
the y axis.
(a)
y = 2x − 4
(b) y = 5 − x
4.
Sketch each of the lines in Question 3.
5.
Express each of the following without brackets.
(a) (x + 3)(x − 2)
6.
(b) (x + 2)(x 2 − 2x + 3)
(c) x(x + 2) − 2(x − 1)
(b) ax 2 − 64a
(c) 15x 2 + 27x +12
Factorise each of the following.
(a) 2xy 2 + 4x 2 y
7.
(c) 2x + y = 5
Two identical circles with centres at P and Q
intersect at A and B.
If PQ = 24cm and AB = 10cm, calculate the
A
P
Q
24cm
radius of the circles.
B
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INTERMEDIATE 2 UNIT 2
HOMEWORK
OUTCOME 1
1.
TRIGONOMETRY
Write down the exact value of each of the following :
(a)
sin45°
(b)
cos60°
(c)
tan135°
(d)
sin120°
(e)
cos300°
(f)
tan225°
2.
Calculate the area of the triangle in the diagram.
7m
60°
9m
3.
Calculate the length of the shortest side in the triangle shown.
63°
35°
27cm
4.
A metal rod 82cm long is bent to form an angle of 125° at a point
37cm from one end.
37cm
125o
How far apart are the two ends of the rod now?
.
5.
The three sides of a triangle are 11∙2cm, 14∙3cm and 20∙4cm.
Calculate the size of the largest angle in the triangle.
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INTERMEDIATE 2 UNIT 2
HOMEWORK
OUTCOME 2
1.
The cost (C) of hiring a car from U-Drive is £30 plus £20 per day (D).
(a)
Copy and complete the table.
D
C
2.
SIMULTANEOUS EQUATIONS
0
30
1
50
2
3
4
(b)
Draw a graph of the relationship.
(c)
Write an equation of the form C =
(d)
Use your equation to find the cost of hiring a car for 7 days.
Use a graph to find the coordinates of the point where the line with equation y = 2x − 2 meets
the line with equation y = 8 − 12 x .
3.
Without drawing a graph, solve the systems of equations:
(a)
2x + y = 5
(b)
3x − y = 10
4.
4x + 3y = 1
3x + 4y = 6
250 minutes of calls and 50 texts cost Colin £43.00 on his mobile phone.
180 minutes of calls and 40 texts cost him £31.40.
How much does each minute of calls and each text cost?
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INTERMEDIATE 2 UNIT 2
HOMEWORK
OUTCOME 3
1.
Calculate
(a) the range
STATISTICS
(b) the mean
(c) the median
and (d) the mode
of the following data.
40
2.
41
34
50
66
46
37
40
Copy this table, add a cumulative frequency column and find the median of the data.
Score
Frequency
2
2
3
11
4
15
5
8
6
1
7
0
8
1
Totals
3.
Find the median, the upper and lower quartiles and the semi-interquartile range for:
(a)
1
2
4
7
7
10
13
(b)
12
13
15
20
23
23
25
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26
27
4.
Calculate the mean and standard deviation for the marks of five pupils in a Maths test.
19
19
In the previous test their marks were
24
35
24
48
36
39
40
41
Make an appropriate statement on the distribution of marks in the two tests.
5.
A bag contains the following beads:
Black
White
Red
Blue
5
7
9
4
(a) What is the probability that a bead drawn at random from the bag will be:
(1)
white
(2)
black or white (3)
not blue ?
(b) A red bead is drawn from the bag and not replaced.
What is the probability that the next bead drawn will be red?
(c) The bead drawn in (b) was red and again it was not replaced.
What is the probability that the next bead to be drawn will also be red?
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INTERMEDIATE 2 UNIT 2
HOMEWORK
OUTCOME 4 GRAPHS CHARTS and TABLES
1.
The back-to-back stem and leaf diagram shows the age and gender of diners in a restaurant.
AGE
MALE
7 7 5
3
5 4 4
7 5
2.
3.
5
2
4
5
FEMALE
9
4
2
1
3
1
2
3
4
5
8
3
0
1
0
3
3
4
2
(a)
How many males were aged 40 or more?
(b)
What was the modal age of the males?
(c)
What was the median age of the females?
(d)
What was the range of the diners’ ages?
4
5
2
4
5 8 9
7 7
8
7 7 8
The data below shows the marks gained by seven pupils in two class tests.
Maths
10
35
60
42
24
17
56
Physics
23
57
88
62
40
33
85
(a)
Show the data on a scattergraph.
(b)
Draw a line of best fit
(c)
State the equation of the line of best fit.
(d)
Use your line to estimate the Physics mark of a pupil whose Maths mark was 50.
For the following data:
6
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6
7
7
(a)
make a five figure summary (b)
(c)
calculate the semi-interquartile range.
8
8
12
12
12
13
15
20
20
draw a box plot and
4.
Copy and complete the following table adding a cumulative frequency column. State the median,
and upper and lower quartiles.
x
f
0
1
1
5
2
8
3
12
4
6
5
6
6
2
c.f.
Totals
5.
The following table gives information about a pie-chart.
(a)
Copy and complete the table.
(b)
Draw the pie-chart.
Programme
Number of Votes
Comedy
10
Film
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160
Sport
36
Documentary
4
Totals
Angle at centre of
pie chart
INTERMEDIATE 2 UNIT 2
HOMEWORK
MIXED EXAMPLES
80o
1.
6∙1
Calculate the area of this triangle.
70o
6∙9
2.
Two ships leave port at the same time.
One sails on a course of 028º at 10 mph.
The other sets a course of 130º at 8mph.
Make a neat sketch to illustrate this and calculate how far apart the ships are after 3 hours.
3.
2y + 3x = −5
Solve the system of equations :
5y + 2x = 4
4.
Construct a box plot to illustrate the following data.
5
5.
7
2
8
7
3
6
7
9
5
In the gym, 8 people recorded how many press-ups they managed in a given time. Their results
were
17
20
16
23
24
19
20
21.
Calculate the mean and standard deviation of their results.
For sit-ups they recorded the same mean and the standard deviation was 3∙2.
Comment on their performance on the two activities.
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INTERMEDIATE 2 UNIT 3
HOMEWORK
OUTCOME 1 ALGEBRAIC OPERATIONS
1.
Simplify:
(a)
2.
3x
6
(d) 12 x 2
24 x
( x  1)( x  2) (b)
( x  2)
(c) x 2  5 x  6
(2 x  1)
( x  3)
(2 x  1)(2 x  1)
(d)
x2  x  6
x2  4x  4
m
3
 n4
(b)
m3 m2

3
2
4 x 1
 2
x
x
(d)
(c)
1 2

a2 a
(d)
(c)
n
(c)
x 1 x  2

x  2 x 1
Express each of the following in its simplest form.
(a)
5.
(c)
Simplify:
(a)
4.
w
w2
Simplify:
(a)
3.
(b)
17
51
7 9k

3k 21
(b)
3x
2
 2
5 9x
2x 4x2

y
3y
Change the subject of each formula to x:
(a)
y
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5
x
(b)
y
5 x
2
(d)
1
(2 x  5)
3
w
3
2 x
INTERMEDIATE 2 UNIT 3
HOMEWORK
OUTCOME 1 SURDS and INDICES
1.
Simplify:
(a)
2.
12 + 27
2
3
(c)
50 
(c)
5
15
(d)
3 27
32 − 18
(b)
32
(d)
13
52
(b)
3
5
(d)
3
4 3
4 2  43
(b)
(c)
x7  x4
3 
4 5
(d)
3a 2
Simplify the following expressions giving your answers with positive indices.
(a)
6.
4 12
Write the following in its simplest index form.
(a)
5.
(c)
Express with a rational denominator:
(a)
4.
75
Simplify:
(a)
3.
(b)
20
3
2
3
4
(b)
x 
2 2
(c)
m 4
m 3
(d)
Find the value of:
1
(a) 49 2
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(b)
2
3
83
(c) 9 2
(d) 16
3
4
2a 2
4a 1
INTERMEDIATE 2 UNIT 3
HOMEWORK
OUTCOME 2
1.
For each of the following, factorise the equation and then
(a)
2.
(1)
Sketch the graph
(2)
State where it crosses the x axis (the roots)
(3)
State the coordinates of the turning point
(4)
State the equation of the axis of symmetry.
y  x2  4x
y  x2  6x  5
(b)
(c)
y  8  2x  x2
For each of the following write down:
(a)
3.
QUADRATICS
(1)
the turning point and its nature
(2)
the equation of the axis of symmetry.
y  ( x  3) 2  2
(b)
y  ( x  1) 2  3
y  ( x  1) 2  4
(c)
Write down the equations of the following parabolas.
(a)
(b)
y
y
5
O
5
x
O
4.
Solve, giving your answer correct to 2 decimal places
3x 2  8 x  2  0
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5 x
INTERMEDIATE 2 UNIT 3
HOMEWORK
OUTCOME 3
1.
FURTHER TRIGONOMETRY
Write down the equations of the following graphs.
(a)
(b)
y
y
4
3
0
180o
360o
x
-4
2.
180
o
-3
Make a neat sketch of each of the following for 0 ≤ x ≤ 360, showing all important points.
(a) y  3 sin( x º 45)
3.
360o
0
(b)
y = 2 cos xº + 1
A graph of the form y  a sin( x  b)º passes through the points (0, − 2) , (30, 0) and (120, 4).
Draw a neat sketch of the graph for 0 ≤ x ≤ 360 showing all important points, and state
the values of a and b.
4.
Solve for 0 ≤ x ≤ 360, giving your answer correct to 3 significant figures.
(a) sin xº = 0∙839
5.
(b) 4cos xº + 7 = 6
(c) tan 2 xº = 25
Prove the following identities:
(a) (sin xº + cos xº)2 = 1 + 2 sin xºcos xº
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(b)
tanxº  sinxº =
1 − cos xo
cos x
x
INTERMEDIATE 2 UNIT 3
HOMEWORK
MIXED EXAMPLES
x3
2
x  5x  6
1.
Simplify:
(a)
2.
Express as a single fraction in its simplest terms:
7 x2

x x 1
(a)
3.
(b)
(b)
(a)
Simplify: 3 5  2 20  180
(b)
Evaluate:
2 x 2  5x  2
8x 2  2
x 1 x 1

x2 x2
2
(1)
83
(2)
(2 2 ) 3
(3)
27
1
3
4.
Sketch the graph with equation y  5  4 x  x 2 . State the coordinates of the turning point and
the equation of the axis of symmetry of your graph.
5.
A parabola with equation of the form y  ( x  a) 2  b has its minimum turning point at (5, 2).
With the help of a diagram if necessary:
(a)
state the values of a and b
(b)
state the equation of the axis of symmetry
(c)
find the point A where the graph crosses the y-axis
(d)
find the coordinates of point B, the image of A under reflection in the axis of
symmetry.
6.
Solve, giving your answer correct to 3 significant figures: 2 x 2  5 x  6  0
7.
Solve:
3sin 2 xº = 2, 0 ≤ x ≤ 3
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