S3 Revision
In Preparation for:
S3 Exams
2016
 Pegasys 2013
EQUATIONS and INEQUATIONS
1.
Solve these equations
2x  12 = 3
(a)
2.
3(2x  4) = 6
6(a  1) = 4(a + 2)
(b)
7x > 42
3x  2 > 11
(b)
Solve these inequalities
9x + 2  6x + 11
(a)
5.
6y  9 = 2y + 5
(c)
Solve these inequalities
(a)
4.
5z + 9 = 4
Solve these equations by first multiplying out the brackets
(a)
3.
(b)
5(y  2) > 2(y + 4)
(b)
Solve these inequalities, giving your answer from the set {3, 2, 1, 0, 1, 2, 3, 4, 5, 6}
7x  3 > 2x  23
(a)
9(y + 2)  7(y + 4)
(b)
APPLYING the THEOREM of PYTHAGORAS
1.
Calculate the length of the side marked x in each triangle below
(a)
(b)
(c)
x
x
5
x
5
14
12
8
16
(d)
(e)
15
(f)
x
4
x
6
(g)
2
1 7
11
3 2
25
(h)
10
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x
13
x
(i)
25
x
07
x
2.
3.
B
Answer these questions about the framework opposite.
(a)
Calculate the length of BD.
(b)
Hence calculate the length of BC.
(c)
Calculate the area of triangle ABC.
26
A
24
D
12
A square snakes & ladders board has
100 squares and a diagonal of length
35 cm.
Find the length of side of one of the
small squares.
4.
The figure shows the cross section of a tunnel
Height of
O
.
the tunnel
2·5m
A
with a horizontal floor AB which is 2·4 metres
wide.
The radius OA of the cross section is 2·5
B
metres.
Find the height of the tunnel.
 Pegasys 2013
C
A
5.
B
Calum is making a picture frame, ABCD .
It is 25 cm high and 21·5 cm wide.
25cm
d
To check whether the frame is rectangular, he
measures the diagonal, d.
D
It is 31·5 cm long.
C
21·5cm
Is the frame rectangular?
6.
Calculate the perimeter of this field, which is made up
of a rectangle and a right angled triangle.
SURDS
147  5 3
1.
Simplify
(a)
2.
Express
2 5  20  45
3.
Express with a rational denominator
4.
Express as a fraction with a rational denominator
 Pegasys 2013
(b)
2( 3  2)  6
as a surd in its simplest form.
5
2 3
5
4 3
5.
In the rectangle ABCD, the diagonal AC is 8cm and the height BC is 4cm.
D
C
4 cm
8 cm
A
(a)
B
Calculate the length of the rectangle, giving your answer as a surd in its
simplest form.
(b)
6.
Calculate the area of triangle ABC.
Find the length of the diagonal, AB, of this rectangle leaving your answer as a surd in its
simplest form.
2 7
B
2 5
A
 Pegasys 2013
INDICES
m5
m3
1.
Simplify
2.
Simplify the expression below, giving your answer with a positive power.
m 5  m 8
p 3 ( p 2  p 3 )
3.
Express
in its simplest form.
4.
Simplify:
5.
y4  y
Express in its simplest form:
y 2
6.
Evaluate
7.
Simplify, expressing your answer with positive indices.
3a 2  2a
a2
3
16 4
( x 2 y 4 )  ( x 3 y 6 )
8.
Simplify
9.
Express
10.
Express
 Pegasys 2013
k 8  (k 2 ) 3
2
3
2
3

2
3
a (a  a )
1
2
1
a (a  )
a
in its simplest form.
in its simplest form.
SCIENTIFIC NOTATION / SIGNIFICANT FIGURES
1.
2.
3.
4.
Write the numbers in each of these sentences in standard form.
(a)
The mass of the moon is about 79 250 000 000 000 000 000 000 kg
(b)
The relative density of hydrogen is 0∙000 089 9
Write the numbers in each of these sentences in full.
(a)
The number of seconds in a decade is about 3∙2  108
(b)
The size of a molecule of water is roughly 1  103
Calculate each of the following, giving your answers in standard form.
(a)
(42  1010)  (3  102)
(c)
(3  2  10 2 )  (4  5  10 3 )
3  10  6
(b)
4  2  10 5
8  10 1
The Earth is 93 million miles from the sun, which is one astronomical unit (AU).
The distance from the sun to Jupiter is 52 AU.
Calculate the distance in miles from the sun to Jupiter and give your answer
in standard form.
5.
A company’s profit for the year was £12  108.
Calculate the profit made per day, giving your answer to the nearest £.
 Pegasys 2013
6.
Use your calculator to find the following. Answer correct to 3 significant figures
(a)
(c)
84  (96  57)
58
(1  2  14)
(b)
20  (21 ÷ 59)
(d)
2500 × 1∙045³
ALGEBRAIC EXPRESSIONS with BRACKETS
Multiply out the brackets and simplify in each question.
16y  5(2y + 3)
(c)
7(s  2)  13
(b)
3m (8  m)
(c)
2y2(w  5y)
9(a + 5) + 7(2a + 7)
(b)
7(y  8) – 5(3y – 6)
(a)
(x + 4)(x + 7)
(b)
(y  9)(y  3)
(c)
(s + 12)(s  2)
(d)
(2a + 5)(a + 9)
(e)
(3w  8)(2w + 1)
(f)
(4x  3)2
(a)
(x + 1)(x 2 + 1)
(b)
(x – 2)(2x² – 3x – 2)
1.
(a)
3(x + 7) + 2x
2.
(a)
x (x3 + 2)
3.
(a)
4.
5.
 Pegasys 2013
(b)
FACTORISING an ALGEBRAIC EXPRESSION
Factorise each expression in the following:
1.
(a)
y2 + 5y
(b)
4x2  49
(c)
5s2  20
2.
(a)
x2 + 10x + 25
(b)
x2  10x  24
(c)
k2 + 5k  6
3.
(a)
12a2 + 7a  12
(b)
7w2  2w  9
(c)
4x2  11x + 6
4.
(a)
12x2 + 16x + 4
(b)
3m2  6m  9
(c)
3  3x  36x2
5.
(a)
x5  81x
(b)
a2 +3ab + 2b2
DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS
1.
The line CD passes through the points (0, 5) and (6, 0)
y
C
x
0
D
Calculate the gradient of CD.
 Pegasys 2013
2.
A line passes through the points A(2, 4) and B(8, 1).
Find the gradient of the line AB.
3.
Prove that the points A(0, 2), B(4, 4) and C(6, 11) all lie on the same straight line.
4.
The points S(k, 3), T(10, 2) and U(2, 5) are collinear. Find the value of k.
5.
Calculate the gradient of a line which is parallel to the line passing through F(3, 7)
and G(8, 2).
6.
The line which passes through (4, 1) and (7, 11) is parallel to the line through (2, y)
and (3, 3). Find the value of y.
7.
What is the gradient of the line perpendicular to the line with equation y = 3x – 5?
8.
The line which passes through (2, 2) and (6, 4) is perpendicular to the line through
(4, b) and (2, 2). Find the value of b.
 Pegasys 2013
WORKING with the LENGTH of ARC and AREA of a SECTOR of a CIRCLE
Give your answers correct to 3 significant figures where necessary.
1.
(a)
Find the length of the minor arc AB in this circle.
(b)
Calculate the area if the minor sector AOB.
O
8 cm
130o
A
2.
(a)Find the length of the major arc PQ
O
28o
P
 Pegasys 2013
5 cm
Q
in this circle.
(b) Calculate the area of the major sector POQ.
B
4. The area of sector OPQ is 100 cm2.
3.The length of arc CD is 88 cm.
Calculate the circumference.
Calculate x to the nearest degree
O
O
12 cm
125o
C
5.
D
xo
P
Ornamental paving slabs are in the shape of part of a sector of a circle.
Calculate the area of the slab shown.
20 cm
35 cm
 Pegasys 2013
Q
3.2 RELATIONSHIP between the CENTRE, CHORD and PERPENDICULAR
BISECTOR
1.In each of the diagrams below AB is a diameter. Find the missing angles in each diagram.
A
jo
A
(a)
(b) A
ao
(c)
35
o
A
b
45
o
B
o
o
47 h
co
go
12o
(d)
27
ko lo o
o
eo
o
d
f
o72
o
io
B
B
B
2.Find the length of the diameter AB in each of the circles below, given the other 2 sides of
the triangle.
(a)
(b)
(c)
8cm
7cm
9cm
4cm
B
A
A
A
A
7cm
(d)
3cm
5cm
B
2cm
B
B
3.Use the symmetry properties of the circle to find the missing angles in the diagrams below.
In each diagram AB is a diameter.
A
(a)
(b)
(c)
mono
do
fo
57
A
o o
o
a b
lo
28
o
o
50
c
B
o
 Pegasys 2013
eo
B
jo ko
o
ho i
B
ooo
g
3.2 TANGENTS and ANGLES
1.
Calculate the sizes of the angles marked a, b, . . . . .r, in the diagrams below.
(a)
(b)
bo
120o
eo
do
o
70o c
o
a
fo
(c)
(d)
no
m
65o
o
45o
ho
go
o
k
2.
po
qo
ro
35o
In each of the diagrams below, PQ is a tangent which touches the circle at R.
Calculate the lengths of the lines marked x.
(a)
(b)
10 cm
P
8 cm
3.
O
O
x
5 cm
R
(c)
O
x
Q
P
12 cm
R
25 cm
7 cm
Q
P
R
Q
x
In each of the diagrams below, AB is a tangent which touches the circle at
Calculate x for each diagram.
(a)
C 27 cm
xo
A
(b)
(c)
B
B
18 cm
O
O
A
o
O 55
10 cm
x
x
C
 Pegasys 2013
C
o
18 cm
30
B
A
WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID
Give your answers correct to 3 significant figures where necessary.
1.
A cone has a base diameter of 16cm and a height of 17cm.
Calculate the volume of the cone, giving your answer correct to 3
17cm
sig figs.
16cm
2.
[Volume of cone =
1 2
r h ]
3
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
Calculate the volume of lead required to make the sinker.
12cm
[Volume of sphere =
3.
4.
4 3
r ]
3
10cm
(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]
A pyramid has a square base of side 6cm and a vertical height of 9cm.
Calculate the volume of the pyramid correct to 2 significant figures.
 Pegasys 2013
ANSWERS
EQUATIONS and INEQUATIONS
1.
(a)
x = 4·5
(b)
z = 1
2.
(a)
x=3
(b)
a=7
3.
(a)
x>6
(b)
x > 3
4.
(a)
x 3
(b)
y>6
5.
(a)
{3, 2, 1, 0, 1, 2, 3, 4, 5, 6}
(c)
y = 3·5
(b)
{3, 2, 1, 0, 1, 2, 3, 4, 5}
(d)
10∙2
CHANGING the SUBJECT of a FORMULA
1.
F=
9
(C  32) .
5
2.
m
50
H w
3.
 A5
x

 4 
4.
b
2
c
A 1
APPLYING the THEOREM of PYTHAGORAS
(a)
9∙43
(b)
21∙3
(c)
13
(g)
12∙4
(h)
26∙9
(i)
2∙4
2.
(a)
10
(b)
15∙6
(c)
180cm²
3.
2·5cm
1.
 Pegasys 2013
4.
4·7cm
(e)
1∙05
(f)
5∙07
5.
Frame is not rectangular
6.
270·6 metres
SURDS
1.
(a)
23
(b)
2.
5
3.
5 3
6
4.
5(4  3 )
13
5.
(a)
6.
4√3
2
4√3
(b)
8√3
INDICES
1.
m2
2.
1
m3
3.
p5 – 1
4.
6a
5.
y7
6.
8
7.
x5
y2
8.
k²
9.
a 3 1
4
3
 12
10.
a2  a
(c)
4∙8 × 105
SCIENTIFIC NOTATION / SIGNIFICANT FIGURES
1.
(a)
7∙925 × 1022
(b)
8∙99 × 105
2.
(a)
320 000 000
(b)
0∙001
3.
(a)
1∙26 × 109
(b)
5∙25 × 105
4.
4∙836 × 108
6.
(a)
2∙15
 Pegasys 2013
5.
£328 767
(b)
7∙12
(c)
3∙45
(d)
2 850
ALGEBRAIC EXPRESSIONS with BRACKETS
1.
(a)
5x + 21
(b)
6y  15
(c)
7s  27
2.
(a)
x4 + 2x
(b)
24m  3m²
(c)
2y2w  10y³
3.
(a)
23a + 94
(b)
 8y – 26
4.
(a)
x² + 11x + 28
(b)
y²  12y + 27
(c)
s² + 10s  24
(d)
2a² + 23a + 45
(e)
6w²  13w – 8
(f)
16x²  24x + 9
(a)
x³ + x 2 + x + 1
(b)
2x³ – 7x² + 4x + 4
5.
FACTORISING an ALGEBRAIC EXPRESSION
1.
(a)
y(y + 5)
(b)
(2x  7)(2x + 7)
(c)
5(s  2)(s + 2)
2
(a)
(x + 5)(x + 5)
(b)
(x  12)(x + 2)
(c)
(k+ 6)(k  1)
3.
(a)
(4a  3)(3a + 4)
(b)
(7w  9)(w + 1)
(c)
(4x  3)(x  2)
4.
(a)
4(3x + 1)(x + 1)
(b)
3(m  3)(m + 1)
(c)
3(1  4x)(1+ 3x)
5.
(a)
x(x² + 9)(x – 3)(x + 3)
(b)
(a + b)(a + 2b)
DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS
1.

5
6
2.
1
2
4.
k=6
5.

 Pegasys 2013
9
11
3
2
3.
Proof [gradients  ]
6.
y = 17
7.

1
3
8.
6
WORKING with the LENGTH of ARC and AREA of a SECTOR of a CIRCLE
1.
(a)
18∙1cm
(b)
72∙6cm²
2.
(a)
29∙0cm
(b)
72∙4cm²
4. 80o
3.25∙3cm
5. 785cm²
RELATIONSHIP between the CENTRE, CHORD and PERPENDICULAR
BISECTOR
(a)
90o
(b)
45o
(c)
90o
(d)
55o
(e)
90o
(f)
43o
(g)
90o
(h)
18o
(i)
90o
(j)
63o
(k)
90o
(l)
78o
2.
(a)
9∙9 cm
(b)
8∙5 cm
(c)
6∙4 cm
(d)
9∙2 cm
3.
(a)
40o
(b)
40o
(c)
50o
(d)
33o
(e)
33o
(f)
57o
(g)
28o
(h)
62o
(i)
62o
(j)
118o
(k)
118o
(l)
31o
(m)
31o
(n)
31o
(o)
31o
1.
TANGENTS and ANGLES
(a)
90o
(b)
20o
(c)
110o
(d)
90o
(e)
60o
(f)
30o
(g)
35o
(h)
35o
(k)
90o
(m)
65o
(n)
90o
(p)
55o
(q)
90o
(r)
45o
2.
(a)
6 cm
(b)
13 cm
(c)
24 cm
3.
(a)
33∙7o
(b)
10∙4 cm
(c)
14∙3 cm
1.
WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID
1.
1140cm³
3.
(a)
2.
1770cm³
 Pegasys 2013
445cm³
(b)
1610cm³
4.
110cm³
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