35
Student Teacher Start AS Mathematics week 1-6 September 2014 City and Islington Sixth Form College Mathematics Department
Student Teacher
Start AS Mathematics week 1-6 September 2014
City and Islington Sixth Form College
Mathematics Department
1 | P a g e
Contents
Contents 1
Introduction 2
Summary Notes 3
WS Algebra 1 [Solving Linear and Quadratic Equations] 5
WS Algebra 2 [Factorising Quadratics] 6
WS Algebra 3 [Completing the Square] 8
WS Algebra 4 [Quadratic Formula, Discriminant] 10
WS Algebra 5 [Graph Sketching] 11
WS Algebra 6[Simultaneous Equations] 13
WS Numbers 1[Indices, Powers] 15
WS Numbers 2a [Directed Numbers] 17
WS Numbers 2b [Fractions] 18
WS Numbers 3 [Surds] 19
WS Sequences 1/2 [Arithmetic] 21
WS Sequences 3 [General Sequences and Summation Notation] 22
WS Geometry 1 [Areas of Triangles, Trapeziums, Sector] 23
WS Geometry 2 (Pythagoras, sin, cos, tan) 25
WS Geometry 3 [Sine Rule, Cosine Rule, Area] 27
WS Geometry 4 [Introduction to Coordinate Geometry] 29
WS Geometry 5(Equations of Straight Lines) 31
WS Geometry 6 [Circles] 32
Important Information 34
2 | P a g e
Introduction
Over the next 6 weeks you will be studying Number, Algebra, Sequences and Geometry.
You will have covered some of these topics at GCSE. At AS level you must ensure that you
achieve a high standard of written mathematics, which is clear, logical and fluent. You will
need to think deeply about the concepts and put in regular practice.
Week 2 ‘Skills Check’ test to help assess your starting level. [30 minutes]
Week 6 Test on Number and Algebra sections ONLY [1 hour]. HW6: Practice Test in your
C1 Homework Pack is an example of this test. You will need to work hard - the pass mark
for this test is 70%.
Welcome to AS Maths. We hope you enjoy the course and achieve your potential.
Number Algebra Sequences/Geometry
Week 1 Introduction to course Algebra 1(Equations:
linear and quadratic)
Sequences 1 (Arithmetic)
Week 2 Skills check (30 mins) +
Number 1 (Indices)
Algebra 2(Factorising,
graph sketching 1)
Sequences 2 (Proof of
sum)
Week 3 Number 2( Natural
numbers, integers and
fractions)
Algebra 3(complete the
square)
Sequences 3(Iteration)
Week 4 Number 3(Surds and
irrational numbers)
Algebra 4(Quadratic
formulae, discriminant)
Geometry 1
Week 5 Algebra5 (Graph
sketching 2)
Algebra 6(Simultaneous
equations)
Geometry 2
Week 6 TEST (1 hour) Coordinate Geometry 4
(mid points, gradients,
distance)
Geometry 3
Week 7 Coordinate Geometry 5
(Equations of lines)
Coordinate Geometry 6
(Circles)
Consolidation
Half term Half term Half term
3 | P a g e
Summary Notes
Number
Algebra
� � 4
�� � 8��� � 2� � 0
� � 2,�8
�� � 3� � 25
� � �3 � √25
� � 2,�8
Linear
2� � 3 � 11
2� � 8
Quadratic
� � 6� � 16 � 0 Factorise
�� � 3� � 9 � 16 � 0 Complete the Square
Formula� ����√������
�
Completed Square Form
� � � � 8� � 21 � � � � 8� � 21
� � �� � 4� � 16 � 21 � � �� � 4� � 16 � 21
� � �� � 4� � 5 � � �� � 4� � 5
�� � 3��2� � 5�� 2� � 11� � 15
�� � 3� � � � 6� � 9
�� � 3� � � � 6� � 9
�� � 3��� � 3� � � � 9
5� � 19� � 12
�5� � 4��� � 3�
Factorising
3� � 5� � 20
� � � � 5
Simultaneous Equations
� � 4� � 16 Elimination method
2� � � � 4 Substitution method
� Natural 1, 2, 3, .. [counting]
� Integers -2, -1, 0, 1, 2,… [counting ±]
� Rational
�, �
��
��, 86, 0 [fractions ±, all except Irrational]
� Real [All including irrationals numbers eg √2, �, �]
Irrational numbers cannot be written as fractions.
As decimals they are infinite and non-recurring.
�2 � 3 � �4 � 5
��
Fractions
Equivalent fractions �
��
��
!
" �
"#
" �
"!
" %&'
�
�(
�
�
�
�)
��
"
��
*
Directed Number Fractions
BIDMAS
( ) 2� � �5 � 1� � 24
� 5 ) 6 � 3 � 21
)( "#��
� 3
Indices (powers)
2� ) 2� � 2� 2�� �"
+
2 ( 2� � 2 2" � 2
�2�� � 2" 2# � 1
4 | P a g e
Geometry
Sequences
� � � � 9� � 22
� � �� � 11��� � 2�
Graph Sketching
When 22,0 −== yx
When 2,11,0 −== xy
-22 -11
2
(3, 1)
(x, y) m=2 Radius=4
�� � 3) + (� − 1) = 16
Line
� − 1 = 2(� − 3) � = 2� − 5 or 2� − � − 5 = 0
Circle
completing the square
(1, 3)
(5, 8)
Normal (perpendicular
line)
,-./0-12 = 31 + 52 , 3 + 82 4
.-52 = (5 − 1) + (8 − 3)
gradient," = 8 − 35 − 1 =
54
>[email protected]�12A0?,@B, = −45
CD = @ + (1 − 1). ED = 1
2 F2@ + (1 − 1).G ED = 1
2 F@ + BG
Arithmetic sequence,
first term a, common difference d, number of terms n
5 | P a g e
WS Algebra 1 [Solving Linear and Quadratic Equations]
Keywords: solve, factors, factorise, equation, linear, quadratic
Solve these linear equations. They are not hard but you should think very carefully about
the logic you use.
Exercise A 1. 2954 =+x 2. 33557 +=+ xx
3. 21536 =− x 4. 72)7(8 =+x
5. 14)53(2 =−x 6. 5)29(5 =−− xx
7. 34)23(5 −=− x 8. 09)4(3)35(7 =++−− xx
9. 1235
=+x
10. 37
52=
−x
11. 432
95=
−
+
x
x 12.
35
14 xx=
−
13. 4
23
5
27 +=
+ xx 14. 7
4
2
3=+
xx
15. 14)43(21 =+x 16. 6)43(
54 =+x
Now solve the following quadratic equations by method of factorising.
For example: Solve 05112 2=+− xx
Exercise B
1. 02092=++ xx 2. 020122
=++ xx
3. 01892=+− xx 4. 018112
=+− xx
5. 01452=−+ xx 6. 01522
=−− xx
7. 0362=−x 8. 042
=− xx
Extension
9*. 0672 2=++ xx 10*. 015234 2
=+− xx
11* 3
457
−=−
xx 12* 3411)4(3 2
+=+ xx
13* 01092 23=+− xxx 14*
4
5
4
3
+=
− x
x
x
Answers ExA 6 14 3 2 4 2 4.9 3/4 45 13 7 3/7 2/13 8.4 8 7/6 ExB [-4, -5] [-2, -10] [3, 6] [2, 9] [2, -7] [-3, 5] [-6, 6] [0, 4] Ext [-3/2, -2] [3/4, 5] [-2, 12] [-7/3, -2]
[ 2,2
5,0 ] [ 4,
5
3 ]
5
05012
0)5)(12(
21 ==
=−=−
=−−
xorx
xorx
xx
6 | P a g e
WS Algebra 2 [Factorising Quadratics]
• How do you solve 021132 2=++ xx
• Write down the factors of 12, 30, 24, 13, 10.
• Below are a range of factorised and expanded quadratic expressions. You will need to
be able to factorise quadratics quickly. There are some simple patterns which will help.
Fold this worksheet along the line, first practice expanding [left to right] and then
factorising [right to left]. You will use factorising throughout your Maths A Level, so
return to this worksheet regularly until you can do them all without error.
Fold 1. Same signs ++
)5)(3( ++ xx 1582++ xx
)4)(2( ++ xx 862++ xx
)4)(3( ++ xx 1272++ xx
)8)(5( ++ xx 40132++ xx
2. Same signs --
)5)(3( −− xx 1582+− xx
)4)(2( −− xx 862+− xx
)4)(3( −− xx 1272+− xx
)8)(5( −− xx 40132+− xx
3. Signs +-
)5)(3( +− xx 1522−+ xx
)4)(2( −+ xx 822−− xx
)4)(3( +− xx 122−+ xx
)8)(5( −+ xx 4032−− xx
4. Same signs, same numbers
)3)(3( ++ xx 962++ xx
)3)(3( −− xx 962+− xx
)11)(11( ++ xx 121222++ xx
)11)(11( −− xx 121222+− xx
5. Difference of two squares
)4)(4( +− xx 162−x
)7)(7( +− xx 492−x
)85)(85( +− xx 6425 2−x
6. Don’t forget the easy ones
)35( +xx xx 35 2+
)47( −xx xx 47 2−
)35(2 +xx xx 610 2+
7 | P a g e
7. Mixed: Factorise then multiply to check
a) 1242−+ xx b) 20122
++ xx
c) 122−− xx d) 962
+− xx
e) xx 122− f) 642
−x
8. Hard: Factorise then multiply to check
a) 10113 2++ xx b) 21132 2
++ xx
c) 15196 2++ xx d) 649 2
−x
e) 5143 2−+ xx f) 18435 2
−− xx
9. Even Harder: Factorise then multiply to check
a) 15438 2++ xx b) 2121110 2
+− xx
c) 12239 2−+ xx d) 200833 2
++ xx
e) 25309 2++ xx f) 169400 2
−x
Answers
7 a) 1242−+ xx = (x + 6)(x – 2) 20122
++ xx = (x + 2)(x + 10)
c) (x +3)(x – 4) (x – 3)(x – 3) = (x – 3)2
e) x(x – 12) (x + 8)(x – 8)
8 a) 10113 2++ xx = (3x + 5 )(x + 2) 21132 2
++ xx = (2x + 7)(x + 3)
c) (3x + 5)(2x + 3) (3x + 8)(3x – 8)
e) (3x – 1)(x +5) (5x + 2)(x – 9)
9 a) (x + 5 )(8x + 3) (10x - 1)(x - 21)
9c) (9x – 4)(x + 3) (3x + 8)(x + 25)
e) (3x + 5)(3x + 5) = (3x + 5)2 (20x + 13)(20x – 13)
8 | P a g e
WS Algebra 3 [Completing the Square]
Keywords: express, solve, completed square form, solution, roots
Exercise A
Multiply the brackets (revision – try this as a mental arithmetic exercise)
1. )4)(4( ++ xx 2. )4)(4( −+ xx
3. 2)7( +x 4.
2)6( −x
5. 2)32( +x 6.
2)53( −x
7. Bill thinks that 222)( pxpx +=+ , is he correct?
Exercise B
Express in completed square form qpxy ++=2)(
1. 162++= xxy 2. 3122
++= xxy
3. 2682+−= xxy 4. 32102
+−= xxy
5. 70162++= xxy 6. 852
++= xxy
7. 25112+−= xxy 8. 12
++= xxy
*9. 18282 2++= xxy *10. 1082
++−= xxy
Exercise C
Solve by completing the square and leave your answers as fractions or surds
1. 046142=++ xx 2. 01382
=−+ xx
3. 022102=+− xx 4. 018102
=++ xx
5. 030363 2=++ xx 6. 0652
=+− xx
*7. 01882 2=−+ xx *8. 06102
=++− xx
Extension tasks
1. Sketch the graphs of the functions below. Show the position of the vertex. [Hint:
express in completed square form first]
a) 1362
++= xxy
b) 23102+−= xxy
c) 1082++−= xxy
d) 11155 2+−= xxy
9 | P a g e
2. Check some of your answers for Ex C by substitution eg is 35 +=x
a solution for 022102=+− xx
3. Check that 54 +−=x is a solution for 01182=++ xx
4. Look up Completing the square on wikipedia!
Answers
ExA 1682++ xx 162
−x 49142++ xx 36122
+− xx
9124 2++ xx 25309 2
+− xx
ExB 8)3( 2−+= xy 33)6( 2
−+= xy 10)4( 2+−= xy 7)5( 2
+−= xy
6)8( 2++= xy 4
72
25)( ++= xy 4
212
211)( −−= xy 4
32
21 )( ++= xy
80)7(2 2−+= xy 26)4( 2
+−−= xy
ExC 37 ±−=x 294 ±−=x 35 ±=x 75 ±−=x
266 ±−=x 3,2=x 132 ±−=x 315 ±=x
Extension
3. Hint 022)35(10)35(2
=++−+
Expand and see if the LHS equals zero
x
y
)4,3(−
)13,0(
x
y
)2,5( −
)23,0(
x
y
)26,4(
)10,0(
x
y
)4
1,2
3( −
)11,0(
10 | P a g e
WS Algebra 4 [Quadratic Formula, Discriminant]
Keywords: real, distinct, roots, discriminant, quadratic formula
Exercise A Quadratic Formula
a
acbbx
2
42−±−
= Can also be written
a
bx
acb
2
42
∆±−=
−=∆
Example 01182=+− xx [6.24, 1.76]
Use the quadratic formulae to find the solution/ roots of the following equations (where possible).
1. (a) 01492=++ xx (b) 02452
=−− xx
(c) 063162=+− xx (d) 025102
=++ xx
(e) 01362=+− xx (f) 01072
=−+ xx
(g) 0362 2=+− xx (h) 0543 2
=−+ xx
What do you notice when there are no solutions?
2. Sketch the graphs for questions (b), (d), (e), (g)
Compare them with those of the person sitting next to you.
Exercise B discriminant [ acb 42−=∆ ]
Calculate the discriminant for each equation and state whether there are two real distinct roots,
one real root or no real roots.
1. (a) 0322=−+ xx (b) 0322
=++ xx
(c) 0962=+− xx (d) 0685 2
=++ xx
(e) 0342 2=−+ xx (f) 09124 2
=+− xx
Extension Questions
1. Write an equation that has no real roots then sketch the graph to show the vertex.
2. Write out the proof of the quadratic formulae.
Answers
ExA 1. [-2, -7] [8, -3] [9, 7] [-5, -5] [no solutions/ no real roots]
[1.22, -8.22] [2.37, 0.63] [0.79, -2.12]
2. We shall discuss these in the lesson
Ex B [two real distinct roots] [no real roots] [one real root]
[no real roots] [two real distinct roots] [one real root]
11 | P a g e
WS Algebra 5 [Graph Sketching]
Keywords: Vertex, Intercept, axes, coordinates, line of symmetry
1. Sketch the pairs of lines below. Calculate the intercept on the x and y axes. [Hint:
To find x-intercept, sub in y=0. To find y-intercept, sub in x=0.]
a) 123 += xy , 62 =+ yx b) 52 +−= xy , 105 =+ yx
c) 102 += xy , 122 −=+ yx d) 3=y , 4−=x
2. For each of the following factorise the quadratic and then sketch the curve. Show
the intercept with the x and y axes.
a) 1282+−= xxy b) 1272
++= xxy
c) 252−= xy d)* 62
+−−= xxy
3. For each of the following equations complete the square and then sketch the
graph. Show the coordinates of the vertex
a) 582++= xxy b) 1142
+−= xxy
c) 652+−= xxy d)* 3102
−+−= xxy
4 Write down possible equations for the following graphs. Then expand brackets and
then put them in the form cbxaxy ++=2
Extension
5. For each of the following sketch the curve showing the intercepts on x and y axes.
Then state the line of symmetry
a) 10173 2+−= xxy b) 2832
++−= xxy
6. Sketch the cubic graphs
a) )3)(1)(2( −−+= xxxy b) )5)(5( −+= xxxy
c) xxxy 65 23+−= d)
34 xxy −=
a)
(3, -2)
4 -2
c)
6 1
b)
-5 -12
d) (5, 100)
e)
-13
f)
12 | P a g e
Answers
4a) 822−−= xxy b) 60172
++= xxy
c) 672−+−= xxy d) 762
+−= xxy
e) 125102+−= xxy f) 169262
++= xxy
1a)
-4
12
b)
5
10
2a)
6 2
b)
-3 -4
d)
3
-4 6
3
2.5
2 b)
5
-10
-12
-6
5 -5
c)
2 -3
d)
(-4, -11)
3a) b)
(2, 7)
c)
(5/2, 1/4)
d)
(5, 22)
a)
5
b)
-4
12
12
-25
6
13 | P a g e
WS Algebra 6[Simultaneous Equations]
Key words: Coefficients, substitution, linear, quadratic
Exercise A (Two linear equations)
1. Solve the following pairs of simultaneous equations using the elimination method
a) 3225
193
=+
=+
yx
yx b)
93
835
=−
=+
yx
yx c)
1647
132
−=+
=+
yx
yx
2. Solve the following pairs of simultaneous equations using the substitution method.
a) 153
2
=+
=
yx
xy b)
1
01223
−=
=++
xy
yx c)
0545
8
=++
+=
yx
yx
d) 132
5
−=
+−=
xy
xy
Exercise B (One linear, one quadratic)
1. Solve each pair of simultaneous equations using substitution
a) 76
2
=+
=
yx
xy b)
104
1
=−
+=
xxy
xy c)
234
52
=+
+=
yx
xy
d) 10
2
22=+
−=
yx
yx e)
5
3
+=
−=
xx
y
xy
f) 082
23
2=+−
−=
yxy
yx
g) yx
yx
316
6
2−=
=+ h)
21
3
22=++
=−
yxyx
xy i)
043
52
2=+
=+
xyy
xy
Extensions questions
1. Solve each system of linear equations in 3 unknowns.
a)
4234
12
6
=+−
=−+
=++
zyx
zyx
zyx
b)
183
243
283
=++
=++
=++
zyx
zyx
zyx
2. Two positive numbers differ by 2 and have a product of 10. Find the two numbers exactly.
[surds]
14 | P a g e
3. What happens when you try to solve these equations? Explain.
a) 1832
232
=+
−=
yx
xy b)
32
042
−=
=+−
yx
yx
4. Try to sketch some of the graphs in Exercise B.
Answers
Ex A 1. )1,6( ),( 23
25 − )3,4(−
2. )6,3( )3,2( −− )5,3( − )1,6( −
Ex B 1. )1,1)(49,7(− )6,5)(1,2( −− )6,1)(5,( 169
43 −
)3,1)(1,3( −− )4,1)(6,3( −−−− ),4)(2,1( 54
53 −−
)4,2)(5,1( )4,1)(1,4( −− )0,5)(4,3(−
Extension 1. ( )3,2,1 ( )7,5,2 2. )111,111( +−
15 | P a g e
WS Numbers 1[Indices, Powers]
Keywords: BIDMAS powers, indices
Exercise A (Positive integer powers)
1. Evaluate the following
a) 27 b)
4)2(− c)
42− d) 35
e) 3)4(− f)
210− g)
3
2
1
h)
2
5
3
i) ( )3
431 j) ( )2
17
2. Simplify the following
a) 433
xxx ×× b) 653 yy × c)
37 420 yy ÷ d)
5
3
3
27
x
xx ×
e) 33
xx + f) 6
3
5
5 g)
32 )(y h)
32 )(2 y
i) 32 )2( y j)
24 )4( x k)
25
21 )( yx
l) 24 )2(3 xy
Exercise B (Integer powers)
1. Evaluate the following
a) 27−
b) 4)3( −
− c)
43−− d)
25−
e) 3)
2
1( f)
2
5
3−
g)
2
4
12
−
h) ( ) 2
17−
2. Simplify the following
a) 32
xx ×−
b) 726 −
× yy c)
214 −÷ yy
d) 1
2
10
18−
w
w
e) 3
17
4
32
x
xx−
× f)
2
24
3
33−
× g)
34 )( −x
h) 25 )3( −
y
i) 25 )(3 −
y j) 432 )(6 −
yx
Exercise C (Rational powers)
1. Evaluate the following
a) 100 b) 2
1
100 c) 9
4 d)
2
1
9
4
e) 3 1000 f) 3
1
125 g) 4
1
81 h) 4
1
10000
i) 21
36
25
j)
2
1
36
25−
16 | P a g e
2. Simplify the following
a) 2
13xxx ××
−
b) 1
4 2
1
−
×
x
xx c)
25
55 21
×
3. Evaluate the following
a) ( )3
36 b)
2
3
36 c) 32
125
Extension
a) 3
32
15
109 × b)
2
7
53
3
32−
cb
a
cb
cab c)
2
5
73
32
3
)1(
−
−−
ab
c
abc
ac
Answers
ExA 1. 49 16 16− 125 64− 100− 8
1
25
9
64
343 17
2. 10
x 715y
45y x3
14
32x 125
1
6y
62y
68y
816x 4
210yx
8212 yx
ExB 1. 49
1
81
1
81
1−
25
1
8
1
9
25
81
16
17
1
2. x 612 −y
16y
5
9 3w
3
2
3x
83 12−
x 109
1
y
10
3
y
1286 −−yx
ExC 1. 10 10 3
2
3
2 10 5 3 10
6
5
5
6
2. 2
3−
x 2
11
x 2
1
5−
3. 216 216 25
Ext 24 7
811
a
cb
7
118
b
ca−
17 | P a g e
WS Numbers 2a [Directed Numbers]
Keywords: N – Natural 1 2 3 … ,
Z – integers-2 -1 0 1 2 3 …,
Q – rational (fractions),
R – real numbers (includes irrational) eg 2
Surds – An expression involving irrational roots eg 543 +
Exercise A Answers
1) 72 − 2) 58 −− [-5,13]
3) 715+− 4) 5732 −++ [-8, 7]
5) 21846 +−−− 6) 135191220 −−−+−−− [23, 19]
7) 100054523200 +−− 8) 3.15.36.0 −+− [623, 1.6]
9*) 4
125
7
3512 −−++− 10*) 01.532.605.0 −−− [1, 46.86]
Exercise B
1) 74 −× 2) 87 −×− [-28, 56]
3) 1284 −÷ 4) 64512 −÷− [-7, 8]
5) 3)5(− 6)
4)3(− [-125, 81]
7) 12
48− 8)
7
217
−
− [-4, 31]
9*) 2103713 ÷+×− 10*) 18382742 −−×+÷ [-3, 1]
Exercise C Substitution
1) 23 −= xy 5,0,7 −=x [19, -2, -17]
2) 742+−= xxy 3,1,7 −−=x [28, 12, 28]
3*) 3
5
+
−=
x
xy 3,2,4 −−=x [
7
1− , -7, ?]
18 | P a g e
WS Numbers 2b [Fractions]
Keywords: Equivalent fractions, numerator, denominator, LCM – lowest common
multiple, mixed number
Exercise A -Simplify, where possible Answers
1) 12
8 2)
45
15 [
3
1,
3
2]
3) 72
45 4)
70
43 [
70
43,
8
5 ]
5) 8
32 6*)
156
42 [
26
7,4 ]
Write as mixed numbers and visa versa
7) 4
15 8)
7
20 [
7
62,
4
33 ]
9) 215 10)
734 [
7
31,
2
11]
Exercise B Evaluate: simplify first and then leave in lowest terms
1) 4
1
2
1+ 2)
12
1
8
1+ [
24
5,
4
3]
3) 5
2
7
3+ 4)
12
7
3
1
2
1−+ [
4
1,
35
29]
5) 4
1
20
1
12
15
4
3−−+ 6)
42
28
26
13
4
13 −−− [
12
19,
10
17]
7) 28
8
4
7× 8)
5
3
13
58
29
65×× [ 6,
2
1]
9) 15
8
5
6÷ 10)
−÷
+
4
35
18
152 [
3
2,
4
9]
Exercise C Substitution
1) 23 += xy 3
2,
4
1=x [ 4,
4
11]
2) 462−+= xxy
3
2,
2
1=x [
9
4,
4
3− ]
Extension Question (Essential Mathematics Web-book, Queen Mary College)
[21
1−]
19 | P a g e
WS Numbers 3 [Surds]
Keywords: N – Natural 1 2 3 … , Z – integers-2 -1 0 1 2 3 …, Q – rational
(fractions), R – real numbers (includes irrational) eg 2 Surds – An expression
involving irrational roots eg 543 + , Irrational numbers
1. Write down the square numbers from 21 to
215
2. Express as the product of prime factors eg 32212 ××=
(a) 40 (b) 100 (c) 98 (d) 360
3. Write the following surds in their simplest form
(a) 12 (b) 18 (c) 75 (d) 200
(e) 48 (f) 360 (g) 54 (h) 320
(i) 25
16 (j)
2
12
4. Simplify
(a) 34323 ++ (b) 818 + (c) 1275 −
(d) 5352 × (e) 2234 × (f) 21153 ×
5. Multiply and simplify
(a) )25)(23( ++ (b) )52)(543( ++ (c) )27)(27( −+
(d) )35)(35( +− (e) )37)(37( −− (f) )235)(24( +−
6. Rationalise the denominators and simplify.
(a) 5
1 (b)
7
7 (c)
48
12
(d) 54
3
+ (e)
73
5
− (f)
134
134
+
−
Extension Questions
1. (a) 22
8
3
6− (b)
2
21
21 )3( + (c)
35
2
+ (d)
32
325
−
−
2. Is 53 +=x a solution of 0462=+− xx ? Check by substitution.
3. Can you do the surds hexagon puzzle?
4. There is also a problem solving challenge for Indices and surds, ask your teacher
20 | P a g e
Answers
2. 523×
22 52 × 272 × 532 23
××
3. 32 23 35 210
34 106 63 58 54 3
4. 37 25 33 30 68 518
5. 2817 + 51126 + 47 22 31452 − 2714 +
6. 5
5 7
2
1
11
5312 −
2
7515 +
3
13829 −
Ext 1. )23(2 − 32
11+ 35 − 34 +
21 | P a g e
WS Sequences 1/2 [Arithmetic]
Keywords: un – nth term
a (= u1) – first term
d – common difference
Sn – sum of all terms from the 1st term up to and including the nth term
Use either of these formulae in the exercise below:
dnaU n )1( −+=
])1(2[2
dnan
Sn −+=
Note: You can use dnaU n )1( −+= directly, e.g. daU 78 +=
1. For the arithmetic sequence when 53 == da
a) Write out 432,1 ,, UUUU b) Calculate 4S
2. For the arithmetic sequence when 27 −== da
a) Write out 65432,1 ,,, UUUUUU b) Calculate 6S
Use the appropriate formulae to calculate the missing quantity.
3.
?
10
5
3
10 =
=
=
=
U
n
d
a
4.
47
12
5
?
12 =
=
=
=
U
n
d
a
5.
31
?
3
7
=
=
=
=
nU
n
d
a
6.
?
12
7
4
12 =
=
=
=
S
n
d
a
7.
164
8
3
?
8 =
=
=
=
S
n
d
a
8.
55
10
?
8
10 −=
=
=
=
S
n
d
a
9*.
140
?
3
7
=
=
=
=
nS
n
d
a
10.
?
?
389
234
=
=
=+
=+
d
a
da
da
11.
?
?
110
153
=
=
=+
=+
d
a
da
da
Answers 1) 3, 8, 13, 18 42 2) 7, 5, 3, 1, -1, -3 12 3) 48 4) -8 5) 9 6) 510 7) 10 8) -3
9) 8 10) 11, 3 11) 21, -2
22 | P a g e
WS Sequences 3 [General Sequences and Summation Notation]
Key words Sequence, Arithmetic, Geometric, Converge, Diverge, Oscillating,
Periodic, Increasing, Decreasing, Recurrence relation
Read the chapter on Sequences in your text book p93 - 104
1. Write down the first five terms in each sequence for 1 = 1, 2, 3, … a) uJ = 2n − 1 b) uJ = 3n + 1 c) uJ = n d) uJ = (−3)J e) uJ = 20 − 5n f) uJ = 2n + 2J g) uJ = 5 + (−1)J h) uJ = J(JK")
i) uJ = "J −
"JK"
2. Write down the first five terms in each sequence
a) LDK" = LD + 3,L" = 2 b) LDK" = 3LD,L" = 4 c) LDK" = 5LD − 2,L" = 3 d) LDK" = LD − 1,L" = 2 e) LDK" = "
MN ,L" = 5 f) LDK" = LD(LD + 1),L" = 1 g) LDK" = −LD,L" = 5 h) LDK = LDK" + LD,L" = 1, L = 1
Sigma notation
3. Write out the following and calculate the sums
a) ∑=
5
1
3r
r b) ∑=
5
1
2
r
r c) ∑=
−4
1
35r
r d) ∑=
+3
1
)1(r
rr
e) ∑=
10
6
2
r
r f) ∑=
6
3
1
r r g) ∑
=
5
1
3r
r h) ∑=
−10
1
)1(r
rr
Extension
1. Try page 103 Ex 3B in your text book.
2. Find out about the Mandelbrot Set - this is all done using sequences.
http://www.youtube.com/watch?v=G_GBwuYuOOs
3. Find out about the Fibonacci Sequence.
Answers 1.a) -1, 1, 3, 5, 7 b) 4,7,10,13,16 c) 1, 4, 9, 16, 25 d) -3, 9, -27, 81, -243 e) 15, 10, 5, 0, -5
f) 3, 8, 14, 24, 42 g) 4, 6, 4, 6, 4 h) 1, 3, 6, 10, 15 i)1/2, 1/6,
1/12, 1/20,
1/30
2a) 2, 5, 8, 11, 14 b) 4, 12, 36, 108, 324 c) 3, 13, 63, 313, 1563 d) 2, 3, 8, 63, 3968
e) 5, 1/5, 5, 1/5, 5 f) 1, 2, 6, 42, 1806 g) 5, -5, 5, -5, 5 h) 1, 1, 2, 3, 5
3a) S5= 45 b) S5= 55 c) S4=38 d) S3= 20 e) 330 f) 19/20 g) 3S5=45
h) S10= 5
23 | P a g e
WS Geometry 1 [Areas of Triangles, Trapeziums, Sector]
Keywords: Area, Sector, Arc length, Trapezium, Circle, Polygon
Area of a circle = 2
rπ Area trapezium = 2
)( hba + Area triangle = bh
2
1
Take 142.3=π
Find the area A or the length x in the shapes below. In some cases you will need to
rearrange the formulae.
You need your calculator for this exercise.
8 cm 2 cm
13 cm
8 cm
14 cm
8 cm
3 cm
4 cm
6 cm
9 cm
Area 63
cm2
8 cm
x cm
20 cm
8 cm
Area A
cm2
Area A
cm2
Area 180
cm2
35°
8 cm
Area A
cm2
x cm
4.
1. 2.
3.
7.
5.
6.
8.
65°
Area 72 cm2
x cm Area 28 cm2
x cm
Area A
cm2
Area A
cm2
24 | P a g e
Extension -
Calculate the surface area or the solids below. Can you find a general formulae for the
areas? What about a square based pyramid?
Answers
1. 252cm 2. cm7 3.
244 cm 4. cm12
5. cm15 6. 221 cmπ 7.
25.19 cm 8. cm3.11
9. 27.240 cm 10.
22.60 cm 11. 24.59 cm 12.
24.105 cm
Extension 1. 2256 cm 2.
2192 cmπ 3. 2160 cmπ
4.6 cm
1.5 cm
255°
10.4 cm
9.
11.
1.
15 cm
8 cm
75°
Area A
cm2
12.
2. 3.
5.2 cm
12 cm 4 cm
5 cm
6 cm
10 cm 12 cm
8 cm
255°
Area A
cm2
25 | P a g e
WS Geometry 2 (Pythagoras, sin, cos, tan)
Key words Pythagoras, hypotenuse, ratio, similar triangles, sin, cos, tan
Pythagoras 222
bac +=
Trigonometry ratios
b
a
adj
oppx
c
b
hyp
adjx
c
a
hyp
oppx
==
==
==
tan
cos
sin
Exercise A
Find the value of x.
Exercise B
Find the value of p, giving its units.
b
x
a
c
8
6
x
8
17
8
14
x
x
(a) (b)
(c)
3.7 cm
3.2 cm
p
5.3 cm
8.6 cm
p
6.7 cm
4 cm
p
53.5°
3.1 cm
p
(a)
(b)
(d)
(c)
26 | P a g e
Find the value of p, giving its units.
Extension
1. (a) If 5
3sin =θ , what is θtan ? (Hint – draw a triangle)
Answers
ExA 10 15 332 =11.49
ExB 38.04° 4.19 cm 40.86° 53.34° 3.82cm 9.54cm 12.31cm
7.70cm
Ext 1. 0.75 60cm2 21.22cm2 cmcm 90.927 =
3.7 cm 8.6 cm
14.6 cm
62.3°
5 cm
8.7 cm
3 cm
8 cm
B
A
Calculate the length of the diagonal AB
13 cm
10 cm
Calculate the area of the triangle
x
25.6°
44.1°
x
An Equilateral triangle has sides length 7 cm Calculate the area
32.5°
x
x
(f)
(h)
(e)
(g)
(d)
(b) (c)
Research Kepler’s model of the Planets [1571-1630] What is the radius of an octahedran with side 5cm ?
2.
27 | P a g e
WS Geometry 3 [Sine Rule, Cosine Rule, Area]
Keywords: Pythagoras, hypotenuse, ratio, similar triangles, sin, cos, tan theta θ
Exercise A: Sine rule
Calculate x or θ
Exercise B: Cosine rule
Calculate x or θ
A B
C
b a
c
AbcArea
Abccba
B
b
A
a
sin area
cos2rule cosine
sinsinsinrule
21
222
=
−+=
=
8 cm
68°
9 cm 12 cm
34°
7 cm
12 cm
118°
15 cm
86°
65°
8 cm
x
x
θ
θ
43°
θ
65°
θ
x
x
5 cm
7 cm 7 cm
98°
5 cm
8 cm
11 cm 8 cm
7 cm 4 cm
4.
1.
2.
3.
4.
1.
2.
3.
28 | P a g e
Exercise C: Area
Calculate the areas
Extension
Answers
ExA 7.60cm 7.27cm 46.15° 33.07°
ExB 7.43cm 10.57cm 113.58° 29.99°
ExC 6.83cm2 733.66cm2 6.95cm2
Ext 17.15cm [56.44°, 123.56°] obtuse
3.5 cm
58°
4.6 cm
4 cm
105°
3.6 cm
θ
10 cm
30°
6 cm
The triangle below can be drawn in two
different ways. This is called the
ambiguous case. Calculate both
possible values of theta and draw the
triangles.
Try to make up a question that is
ambiguous. Can you devise a rule for
this?
x20 cm
57°
77°
1.
1. 3.
2.
3.
θ
18 cm
8 cm
15 cm
Is theta acute or obtuse? Change the numbers
and make up two different questions, one with
theta acute the other obtuse.
Calculate x
35 cm
48°
24.2 cm
72°
2.
29 | P a g e
WS Geometry 4 [Introduction to Coordinate Geometry]
Two points Gradient of line
joining them
Gradient of
perpendicular line
Midpoint of line Distance between the
points
Example: ( )2,6− and ( )10,2−
24
8
62
210==
−−−
−
2
1−
( )6,42
102,
2
26−=
+−−
( ) ( ) 22228421062 +=−+−−−
5480 ==
( )2,5 and ( )8,13
( )6,1 and ( )12,3
( )7,3 − and ( )4,1 −−
( )2,3 − and ( )6,1−
( )1,0 and ( )____, ( )4, 4
( )____, and ( )9,12 9
,62
( )11,3− and ( )0__, 5
30 | P a g e
Answers
Two points Gradient of line
joining them
Gradient of
perpendicular line
Midpoint of line Distance between the
points
Example: ( )2,6− and ( )10,2−
24
8
62
210==
−−−
−
2
1−
( )6,42
102,
2
26−=
+−−
( ) ( ) 22228421062 +=−+−−−
5480 ==
( )2,5 and ( )8,13
4
3
3
4− ( )5,9 10
( )6,1 and ( )12,3
3 3
1−
( )9,2 102
( )7,3 − and ( )4,1 −−
3
4−
4
3
−
2
11,1
5
( )2,3 − and ( )6,1−
2− 2
1
( )2,1 54
( )1,0 and ( )7,8
4
3
3
4− ( )4, 4 10
( )0,0 and ( )9,12
4
3
3
4− 9
,62
15
( )11,3− and ( )15,0−
3
4
4
3− 3
13,2
−
5
31 | P a g e
WS Geometry 5(Equations of Straight Lines)
Find the equations of the following lines in the form 0=++ cbyax :
Question Working Out Equation of line
1. Gradient is 4 and y
intercept is -2
2. Gradient is 1
2 and
crosses y axis at 5
3. Gradient is -6 and
goes through ( )0, 2
4. Gradient is 3 and
passes through ( )1,2−
5. Gradient is -1 and
passes through ( )4, 3−
6. Gradient is 2
3 and
passes through ( )6, 2
7. Line passes through
( )4,8 and ( )3,11
8. Line passes through
( )2,5− and ( )1,14
9. Line passes through
( )3,0 and is
perpendicular to 2 3y x= −
10. Line passes through
( )1,4 and is
perpendicular to 2y x= − +
Answers
1 4� − � − 2 = 0 6 2� − 3� − 6 = 0 2 � − 2� + 10 = 0 7 3� + � − 20 = 0 3 6� + � − 2 = 0 8 3� − � + 11 = 0 4 3� − � + 5 = 0 9 � + 2� − 3 = 0 5 � + � − 1 = 0 10 � − � + 3 = 0
32 | P a g e
WS Geometry 6 [Circles]
Keywords: Euclid, circle theorems, Centre, diameter, radius, completing the
square
Skills required
• Completing the square and general algebra
• Mid points, distance between points
• Circle theorems and basic geometry
Exercise A
1. Write down the equations of the circles.
(a) Centre (1,5) radius 3 (b) Centre (-4,6) radius 7
(c) Centre (-3,-20) radius 8 (d) Centre (2,7) radius 35
2. Find the centre and radius from the equation of the circle.
(a) 022146 22=++++ yyxx (b) 364 22
=−+− yyxx
(c) 24102 22=−++ yyxx (d) 091022
=+−+ xyx
3. Find the equations of the circles in the form 22
1
2
1 )()( ryyxx =−+−
(a) Diameter A (2,7) B (10,1) (b) Diameter A (-3,1) B (5,7)
(c) Diameter A (-6,7) B (18,-3) (d) Diameter A (2,-3) B (8,1)
Extension
1. The line y = 3x – 4 is a tangent to the
circle C, touching C at the point
P(2, 2), as shown in Figure 1.
The point Q is the centre of C.
(a) Find an equation of the straight
line through P and Q.
Given that Q lies on the line y = 1,
(b) show that the x-coordinate of Q
is 5,
(c) find an equation for C.
y = 3x – 4
P(2, 2)
O x
y
C
Q
33 | P a g e
2. The points A and B lie on a
circle with centre P, as shown in
Figure 3.
The point A has coordinates
(1, –2) and the mid-point M of
AB has coordinates (3, 1).
The line l passes through the
points M and P.
(a) Find an equation for l.
(b) Given that the x-coordinate
of P is 6, use your answer to
part (a) to show that the y-
coordinate of P is –1,
(c) find an equation for the circle.
3. (a) Using the fact that a tangent of a circle is perpendicular to the radius at the point of
contact, find the equation of the tangent to the circle 25)4( 22=++ yx at the point )1,4( −−
.
(b) Show that the circle C with centre (7, 3) and passing though the point (12, -5) has equation
0111201222=+−−+ yxyx . The line 7−= xy intersects the circle at the points A and B.
Find the coordinates of A and B.
4. a) Investigate the following – you could use a computer to look at the graphs
2522=+ yx 253 22
=+ yx 254 22=+ yx 255 22
=− yx
b) What about 36222=++ zyx ?
c) Internet research: Euclidean Geometry, Conic Sections
Answers
1. 9)5()1( 22=−+− yx 49)6()4( 22
=−++ yx
64)20()3( 22=+++ yx 75)7()2( 22
=−+− yx
2. Centre (-3,-7) radius 6 Centre (2,3) radius 4
Centre (-1,5) radius 25 Centre )0,5( radius 4
3. Centre (6, 4) radius 5 Centre (1, 4) radius 5
Centre (6,2) radius 13 Centre (5,-1) radius 13
Extension
1. a) 083 =−+ yx c) 10)1()5( 22=−+− yx http://youtu.be/JSbKOCgZ6c8
2. a) 932 =+ yx , c) 26)1()6( 22=++− yx http://youtu.be/bT3J5Nine8U
3. a) 01334 =+− yx , b) A(2, -5) and B(15, 8)
M
x
A
B
O
y
P
(1, –2)
(3, 1)
l
34 | P a g e
Important Information Maths and Computer Science Teachers:
room email
Ali Ali 218 [email protected]
Bill Alexander 232 [email protected]
Forough Amirzahdeh 214 [email protected]
Najm Anwar 218 [email protected]
Elliot Henchy 232 [email protected]
Ceinwen Hilton 232 [email protected]
Greg Jefferys 214 [email protected]
Dan Nelson 214 [email protected]
Mike Thiele 214 [email protected]
Sally Ville 201 [email protected]
Homework
Work outside lessons should take 4-5 hours. You will be set homework on all the main topics.
Complete the set work thoughtfully, it is for your benefit. Remember to check and mark your
answers, write any comments or questions to the teacher on your work and submit it on time.
You should also review notes, revise for future tests and plan ahead as part of your homework. Support – to help you succeed
The department runs several support workshops at lunchtimes and after college where you
can get extra help. This is also an opportunity for you to get to know other teachers and
students.
Expectations
Students take increasing responsibility for their learning at the Sixth Form. Do join in the
classes, volunteer answers and ask questions. Spend time at home organising your equipment,
notes and learning. Learning demands, courage, determination and resourcefulness. Use other
text books, YouTube, websites, work with other students and talk with teachers.
Moodle pages
Remember to visit our Mathematics Moodle page. This is where you can find copies of all
these worksheets, packs and department documents.
Links
www.examsolutions.net Most popular site with past exam papers and video
solutions. Also clear explanations of topics
www.physicsandmathstutor.com Exam revision site
www.numberphile.com Short video clips of popular maths
www.nrichmaths.org Problem solving challenges
www.geogebra.org Geometry, graphs and animations
www.mathscareers.org.uk/ Careers linked to mathematics
www.supermathsworld.com Multiple choice practice with cartoons
