Student Teacher AS STARTER PACK September 2016 City and Islington Sixth Form College Mathematics Department www.candimaths.uk
Student
Teacher
AS STARTER PACK September 2016
City and Islington Sixth Form College
Mathematics Department
www.candimaths.uk
2
CONTENTS
INTRODUCTION 3
SUMMARY NOTES 4
WS CALCULUS 1 ~ Indices, powers and differentiation 6
WS CALCULUS 2 ~ Applications of differentiation 8
WS CURVE SKETCHING 1 ~ Introduction 10
WS CURVE SKETCHING 2 ~ Sketching quadratics 13
WS CURVE SKETCHING 3 ~ Equation of a straight line 15
WS QUADRATICS 1 ~ Factorising, quadratic simultaneous equations 18
WS QUADRATICS 2 ~ Completing the square 20
WS QUADRATICS 3 ~ Quadratic formula and discriminant 22
WS 9 TANGENTS AND NORMALS 25
WS SEQUENCES 1 ~ General sequences and summation notation 28
WS SEQUENCES 2 ~ Arithmetic sequences 31
WS INTEGRATION 33
IMPORTANT INFORMATION 36
3
INTRODUCTION Over the next 6 weeks you will be studying new topics in mathematics. Each of these new topics
builds upon GCSE work. 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.
Homework: You will be given homework each week to support your learning. Some homework
will involve pre-learning that prepares you for the next lesson so it is very important that you
complete it. You will be expected to mark the homework yourself and your teachers will check the
working out and lay out of your work.
Week 2 Skills Test 1: This test is to check that you have not forgotten GCSE maths! If you do
not do well on this test then you will be given extra work to make sure you are ready for the A-
level mathematics.
Week 6 Skills Test 2: This test is to make sure you are ready for the harder parts of the course.
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%.
Lesson 1 Lesson 2 Lesson 3
Week 1 Induction 1 β Introduction
to course
Differentiation 1 Differentiation 2
Week 2 SKILLS TEST 1
Curves Sketching 1
Curve Sketching 2
(Quadratics)
Curve Sketching 3
(Equation of a straight line)
Week 3 Quadratics 1 Quadratics 2 Quadratics 3
Week 4 Tangents and Normals 1 Tangents and Normals 2 Induction 2 β ICT skills
Week 5 Sequences 1 Sequences 2 Induction 3 β Study skills
(folder check)
Week 6 SKILLS TEST 2 Integration Integration
Week 7 HALF TERM
Extra resources, links and digital copies of the booklets can be found at our website:
www.candimaths.uk
4
SUMMARY NOTES
Number
Algebra
2π₯ + 3 = 11
2π₯ = 8
π₯ = 4
π₯2 + 6π₯ β 16 = 0 π₯ + 8 π₯ β 2 = 0
π₯ = 2,β8
π₯2 + 6π₯ β 16 = 0 π₯ + 3 2 β 9 β 16 = 0
π₯ + 3 2 = 25
π₯ = β3 Β± 25
π₯ = 2,β8
Linear
Quadratic
Complete the Square
Formula π₯ =βπΒ± π2β4ππ
2π
Completed Square Form
π¦ = π₯2 + 8π₯ + 21 π¦ = π₯2 β 8π₯ + 21
π¦ = π₯ + 4 2 β 16 + 21 π¦ = π₯ β 4 2 β 16 + 21
π¦ = π₯ + 4 2 + 5 π¦ = π₯ β 4 2 + 5
2π₯2 + 11π₯ + 15 = π₯ + 3 2π₯ + 5
π₯2 + 6π₯ + 9 = π₯ + 3 2
π₯2 β 6π₯ + 9 = π₯ β 3 2
π₯2 β 9 = π₯ + 3 π₯ β 3
5π₯2 + 19π₯ + 12 = 5π₯ + 4 π₯ + 3
Factorising
3π₯ + 5π¦ = 20
π₯2 + π¦2 = 5
Simultaneous Equations
π₯ + 4π¦ = 16 Elimination method
2π₯ β π¦ = 4 Substitution method
β Natural 1, 2, 3, .. [counting]
β€ Integers -2, -1, 0, 1, 2,β¦ [counting Β±]
β Rational 2
3, β
43
67, 86, 0 [fractions Β±, all except Irrational]
β Real [All including irrationals numbers eg 2,π, π]
Irrational numbers cannot be written as fractions.
As decimals they are infinite and non-recurring.
β2 + 3 β β4 = 5
+ β
Equivalent fractions
Fraction arithmetic
3
4+
5
6=
9
12+
10
12=
19
12 πΏπΆπ
3
4Γ·
6
5=
3
4Γ
5
6=
15
24=
5
8
Directed Numbers
BIDMAS
( ) 23 + 5 β 1 2 = 24
π₯2 5 Γ 6 β 32 = 21
ΓΓ· 10β4
2= 3
Indices (powers)
23 Γ 24 = 27 2β3 =1
23
25 Γ· 23 = 22 21 = 2
23 5 = 215 20 = 1
5
Geometry
Sequences
π¦ = π₯2 + 9π₯ β 22
π¦ = π₯ + 11 π₯ β 2
Graph Sketching
When
When
centre = 3, 1
π₯,π¦ π = 2 πππππ’π = 4
π¦ β 1 = 2 π₯ β 3
π₯ β 3 2 + π¦ β 1 2 = 16
Line
π¦ = 2π₯ β 5 or 2π₯ β π¦ β 5 = 0
Circle
1, 3
5, 8
Normal (perpendicular line)
πππ πππππ‘ = 1 + 5
2,3 + 8
2
πππ π‘2 = 5 β 1 2 + 8 β 3 2
gradient π1 =8 β 3
5 β 1=
5
4
ππππππππ‘ ππππππ π2 =β4
5
Arithmetic Sequence
First term: π,
Common difference: π
Number of terms: π
πth term: ππ = π + π β 1 π
Sum to π terms: ππ =π
2 2π + π β 1 π
ππ =π
2 π + π
6
WS CALCULUS 1 ~ Indices, powers and differentiation
Keywords BIDMAS, powers, indices, differentiation, evaluate
Exercise A
Simplify the following.
1. 3 Γ 3 Γ 4 2. 3 Γ 5 6 3. 20 7 Γ· 4 3 4. 7 3Γ2
3
5. 53
5 6. 2 3 7. 2 2 3 8. 2 2 3
9. 4 4 2 10. 3 2 4 2
Exercise B
1. Simplify the following, giving each term in the form , where and are constants.
(a) 2 (b) 3
2 (c)
1
3 3 (d)
1
3 3 β 2
(e) 2
3 (f)
3+
1
2 (g)
2 3
(h)
3 2β6
2
(i) 2 3 3
(j) 2 β + 2 (k) 3 2 2 + 2 (l) 3 β 2 (4 +
1
)
2. Use your answers to Q1 to differentiate each of the above expressions.
Exercise C β more challenging
1. Evaluate: (36
2 + 16
)
3
2. Solve to find : 3
2 = 2
Exam Questions
1. [C1 May 2014 Q2]
(a) Write down the value of 5
1
32 .
(1)
(b) Simplify fully 5
2
5 )32(
x .
(3)
7
2. [C1 Jan 2014 Q1]
Simplify fully
(a) 2
2 x
(1)
3. [C1 Jan 2014 Q2]
2 42 1y x
x
, x > 0
(a) Find d
d
y
x, giving each term in its simplest form.
(3)
Answers
EXA
1. 10 2. 15 7 3. 5 4 4. 14
3 5.
1
125
6. 6 7. 2 6 8. 8 6 9. 16 8 10. 12 2 8
EXB
1. (a) 2
2 (b) 3 β2 (c) 1
3 β3 (d)
1
3 4 β
2
3 3 (e) 2 β3
(f)
3 +1
2 β1 (g) 2 + 3 β1 (h)
3
2 β 3 β1 (i) 2
2 + 3
2 (j) 3 β 2 + 2
(k) 3 4 + 6 3 (l) 12 2 β 8 + 3 β 2 β1
2. (a) 1
(b) β
6
3 (c) β1
(d) 4
3 3 β 2 2 (e) β
6
(f) 1
3 β 23 β1
2 2 (g) β3
2 (h) 3
2+
3
2 (i) 5 3 +3
2 (j) 3 2 β 2 + 2
(k) 12 3 + 18 2 (l) 24 β 8 +2
2
EXC
1. 2 2. 9
Exam Questions
1. (a) 2 (b) 1
4 2 2. 4 3. 4 +2
β 3
8
WS CALCULUS 2 ~ Applications of differentiation
Exercise A - Complete the table
Equation of curve
Gradient Function
dy
dx
Gradient of the curve at these points
2x 1x 0x
Example = 4
= 4 3
4 2 3 = 32 4 β1 3 = β4 4 0 3 = 0
A = 5
B = 2 β 5
C = 3 β 9
D = 10 2
E = + 3 β 6
F = 2 + 3
G = 5 β
H = β 3 3
I = 19 + 4 2 β 3
J = 2 Γ 3
K = 1 β 3 + 4
L =1
2 2
M =1
4 + 2
N = 2 + 3 1 β
O =2
3 2
9
Exercise B - Complete the table
Equation of curve Gradient =
Solve for x
Example =1
3 3 β
5
2 2 + 8 2
= 2 β 5 + 8
β 2 β 5 + 8 = 2 β 2 β 5 + 6 = 0 β β 3 β 2 = 0 β = 2,3
A = 2 + 2 4
B = 3 3
4
C = 3 + 8 2 + 9 β 2 4
Answers
EXA EXB
A 5 4 80 5 0 A = 2
B 2 4 -2 0 B = Β±1
2
C 3 3 3 3
C = β1
3,β5
D 20 40 -20 0
E 2 β 3 1 -5 -3
F 1 + 2 5 -1 1
G 6 5 β 14 164 8 0
H β9 2 -36 -9 0
I 19 + 8 β 3 2 23 8 19
J 4 3 32 -4 0
K β3 + 4 3 29 -7 -3
L 2 -1 0
M 14β 1
4β 14β 1
4β
N β2 β 2 -6 0 -2
O 43β 8
3β β43β 0
10
WS CURVE SKETCHING 1 ~ Introduction
Keywords: curve, axes, intersection, maximum, minimum, linear, quadratic, cubic, reciprocal
Exercise A
Match the following graphs to their equations
A B C
D E F
G H I
Equation Graph Equation Graph
= = β 2
= 2 = β 1
= β =1
= 3 = + 1 β 2
= 2
11
Exercise B [There is more than one possible answer for some of these questions!]
(a) On separate diagrams sketch the following the graphs. Make sure you label the axis correctly
and use a ruler where necessary.
(b) Try and write an equation for each graph (some are difficult!).
1. A linear graph that crosses the axes at 2, 0 and 0, 2 .
2. A quadratic graph that crosses the axes at 0, 3 only.
3. A cubic graph that crosses the axes at 0, β2 , β4, 0 , 1, 0 and 3, 0 .
4. A linear graph that has a gradient of 3 and crosses the -axis at 2.
5. A reciprocal graph that has asymptotes at = 3 and = 1.
6. A negative quadratic graph that passes through β1, 0 and the origin.
7. A cubic graph that crosses the origin and 2, 0 .
8. A reciprocal graph that passes through (0,1
2)
Exercise C
For each equation fill in the following table and use the results to sketch the curves.
0
0
Very big
Very small
1. = 3 + 1 2. = 2 + 1 3. =1
2 β 5
4. = + 1 β 4 5. = β2 + 2 6. = 3 β 1
Exam Question
1. [C1 Jan 2012 Q8]
The curve C1 has equation y = x2(x + 2).
(a) Find x
y
d
d.
(2)
(b) Sketch C1, showing the coordinates of the points where C1 meets the x-axis.
(3)
(c) Find the gradient of C1 at each point where C1 meets the x-axis.
(2)
The curve C2 has equation
y = (x β k)2(x β k + 2),
where k is a constant and k > 2.
(d) Sketch C2, showing the coordinates of the points where C2 meets the x and y axes.
(3)
12
Answers
EXA
EXB 1.
= β + 2 2.
= 2 + 3
3.
= β1
6 + 4 β 1 β 3
4.
= 3 β 6
5.
=1
β 3+ 1
6.
= + 3
7.
= β 2 2 8.
=1
+ 2
EXC 1.
2.
3.
4.
5.
6.
Exam question:
Exam question
(a) 3 2 + 4 (b) 0, 4 (c) crosses axes at: , 0 , β 2, 0 , 0, β 3 + 2
Equation Graph Equation Graph
= E = β 2 D
= 2 B = β 1 F
= β I =1
C
= 3 A = + 1 β 2 H
= 2 G
13
WS CURVE SKETCHING 2 ~ Sketching quadratics
Keywords: quadratic, negative, positive, factorise, intersection, axes, differentiation,
Exercise A
Factorise each quadratic and sketch each curve on a different set of axes, stating clearly the
coordinates of the points where the curve intersects the axes.
1. = 2 + 3 β 4 2. = 2 + β 6
3. = 2 β 2 + 1 4. = β 2 β 5 β 6
5. = 2 2 β 9 β 5 6. = 2 β 4
7. = β5 2 + 3 + 2 8. = 2 β 4
9. = 2 + 4 + 5
Exercise B
Using differentiation, find the minimum or maximum points of each curve in Exercise A and write
them on your diagrams.
Exercise C
Write down a possible equation for each of these curves.
1.
2.
3.
4.
5.
6.
7.
8.
9.
14
Answers
EXA/B 1.
= + 4 β 1
Crosses the axes at:
β4, 0 , 1, 0 and 0,β4
Minimum point:
(β3
2, β
25
4)
2.
= + 3 β 2
Crosses the axes at:
β3, 0 , 2, 0 and 0,β6
Minimum point:
(β1
2,
25
4)
3.
= β 1 2
Crosses the axes at:
1, 0 and 0, 1
Minimum point:
1, 0
4.
= β + 3 + 2
Crosses the axes at:
β3, 0 , β2, 0 and 0,β6
Maximum point:
(β5
2,
1
4)
5.
= 2 + 1 β 5
Crosses the axes at:
β1
2, 0 , 5, 0
and 0,β5
Minimum point:
(β9
4 β
121
8)
6.
= β 4
Crosses the axes at:
0, 0 , 4, 0
Minimum point: 2,β4
7.
= β 5 + 2 β 1
Crosses the axes at:
β2 5, 0 , 1, 0 and 0, 2
Maximum point:
(3
10,
49
10)
8.
= + 2 β 2
Crosses the axes at:
β2, 0 , 2, 0 and 0,β4
Minimum point:
0,β4
9.
Canβt be factorised!
Crosses the axes at:
0, 5
Minimum point:
β2, 1
EXC
1. = + 1 β 3 2. = + 2 2 3. = β + 1 β 2 4. = β + 1 β 2 5. = + 3 6. = β β 2 7. = 2 β 1 + 1 8. = β 2 9. = β 1 + 1 β 3
15
WS CURVE SKETCHING 3 ~ Equation of a straight line
Keywords: gradient,
=
=
=
2 β 1
2 β 1
Exercise A
Work out the gradient of each of these lines
1.
2.
3.
4.
5.
6.
Exercise B
Find the gradient of the straight line between the following points:
1. 12, 3 14, 2. β1, 5 2, 8
3. 0, 1 5,β9 4. 3,β2 β1, 4
5. 2, 3 0, 6. β3,β1 β1, 11
7. 20, 1 15, 8. β1,β5 β2,β8
9. 4,β3 13, β8 10. β20,β4 0,β12
Exercise C
Use the equation β 1 = β 1 to find the equation of the following lines in the form
= +
1. Passing through 2, 3 with gradient 4 2. Passing through 1, 5 with gradient β2
3. Passing through β1, 0 with gradient 3 4. Passing through 2, β with gradient 1
2
5. Passing through 12, 3 and 14, 6. Passing through β1, 5 and 2, 8
7. Passing through β1, 5 and 1, 9 8. Passing through 8, 0 and β2, 5
9. Passing through 2, 2 and 5, β 10. Passing through 20, 10 and 35, 5
16
Exercise D
Find the equations of the following lines in the form + + = 0 where , , and are
integers:
Question Working Out Equation of line
1. Gradient is 4 and intercept is -2
2. Gradient is 1
2 and
crosses 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 β 3
10. Line passes through
1,4 and is
perpendicular to
= β + 2
17
Answers
EXA
1. 2 2. -1 3. 12β 4. -2 5. 3
4β 6. β52β
EXB
1. 2 2. 1 3. -2 4. β32β 5. -2
6. 6 7. β65β 8. 3 9. β5
9β 10. β25β
EXC
1. = 4 β 5 2. = β2 + 3. = 3 + 3 4. =1
2 β 8 5. = 2 β 21
6. = + 6 7. = 2 + 8. = β1
2 + 4 9. = β3 + 8 10. = β
1
3 +
50
3
EXD
1. 4 β β 2 = 0 2. β 2 + 10 = 0 3. 6 + β 2 = 0 4. 3 β + 5 = 0 5. + β 1 = 0
6. 2 β 3 β 6 = 0 7. 3 + β 20 = 0 8. 3 β + 11 = 0 9. + 2 β 3 = 0 10. β + 3 = 0
18
WS QUADRATICS 1 ~ Factorising, quadratic simultaneous equations
Key words Quadratic, Factorise, Simultaneous, Solve, Gradient function
Exercise A
Solve the following equation by factorisation.
1. 2 + 6 + 5 = 0 2. 2 + 2 β 8 = 0
3. 2 2 + 11 + 5 = 0 4. 2 2 + + 5 = 0
5. 2 2 β 9 β 5 = 0 6. 2 2 + 9 β 5 = 0
7. 3 2 + 40 + 13 = 0 8. 3 2 β 8 β 11 = 0
9. β2 2 β 9 + 5 = 0 10. 2 + 2 β 11 = 0
Exercise B
1. Solve the simultaneous equations:
(a) = 4 + 2 (b) = 2 2 + 5 + 10 (c) = 3 2 + 30 β 10
= 2 + 9 + 16 = 5 β 2 = 1 β 2
2. 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
Exercise C
Find the minimum and maximum points of these curves by solving
= 0
1. = 3
3+
5 2
2+ 4 β
2. = 3
3β
7 2
2+ 12 β
3. =2 3
3β
23 2
2+ 11 + 3
4. =2 3
3β
13 2
2β + 350
5. 3 3
5β
2 2
5β
7
5β 10 = 0
19
Answers
EXA
1. = β1,β5 2. = 2,β4 3. = β1
2, β5 4. = β
5
2, β1
5. = β5
2, β1 6. =
1
2, β5 7. = β
1
3, β13 8. =
11
3, β3
9. =1
2, β5 10. = 1 Β± 3
EXB
1. (a) = β3,β4 (b) = β1
2, β5 (c) =
1
3, β11
2. (a) )1,1)(49,7( (b) )6,5)(1,2( (c) )6,1)(5,(169
43
(d) )3,1)(1,3( (e) )4,1)(6,3( (f) ),4)(2,1(54
53
(g) )4,2)(5,1( (h) )4,1)(1,4( (i) )0,5)(4,3(
EXC
1. = β1,β4 2. = 3,4 3. =1
2, 11 4. = β
1
2,
5. =7
3, β1
20
WS QUADRATICS 2 ~ Completing the square
Keywords express, solve, completed square form, solution, roots
Exercise A
Multiply the brackets (revision β 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
Exercise D
1. Sketch the graphs of the functions below. Show the position of the vertex. [Hint: express in
completed form first]
a) 1362 xxy b) 23102 xxy
c) 1082 xxy d) 11155 2 xxy
2. Check some of your answers for EXC by substitution e.g. is 35x a solution for
022102 xx ?
3. Check that 54x is a solution for 01182 xx
4. Look up βcompleting the squareβ on Wikipedia!
21
Answers
EXA
1. 1682 xx 2. 162 x 3. 49142 xx 4. 36122 xx
5. 9124 2 xx 6. 25309 2 xx 7. Not generally!
EXB
1. 8)3( 2 xy 2. 33)6( 2 xy 3. 10)4( 2 xy 4. 7)5( 2 xy
5. 6)8( 2 xy 6. 472
25)( xy 7. 4
212
211)( xy 8. 4
32
21)( xy
9. 80)7(2 2 xy 10. 26)4( 2 xy
EXC
1. 37x 2. 294x 3. 35x 4. 75x
5. 266x 6. 3,2x 7. 132 x 8. 315x
EXD
1. a) b)
c) d)
3. Hint: 022)35(10)35( 2 Expand and see if the LHS equals zero
x
y
)4
1,2
3(
)11,0(
x
y
)26,4(
)10,0(
x
y
)2,5(
)23,0(
x
y
)4,3(
)13,0(
22
WS QUADRATICS 3 ~ Quadratic formula and discriminant
Key words quadratic formula, discriminant, real, distinct, inequality, roots, solutions
Exercise A
For a quadratics equation in the form: = 2 + +
The quadratic formula: a
acbbx
2
42 can also be written
a
bx
acb
2
42
Example 01182 xx = 6 24, 1 6
Use the quadratic formulae to find the 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
2. Sketch the graphs for questions (b), (d), (e), (g)
Compare them with those of the person sitting next to you.
Exercise B
The 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. 0322 xx 2. 0322 xx
3. 0962 xx 4. 0685 2 xx
5. 0342 2 xx 6. 09124 2 xx
Exercise C
Solve the following by first sketching the graph (it will help to factorise these)
1. 2 + 6 + 5 0 2. 2 + 2 β 8 0
3. 2 2 + 11 + 5 0 4. 2 2 + 15 β 1 0
5. β 2 β + 2 0 6. β2 2 + 13 β 11 0
23
Exam questions
1. [C1 May 2006 Q2]
Find the set of values of x for which x2 β 7x β 18 > 0.
(4) 2. [C1 Jan 2005 Q3]
Given that the equation kx2 + 12x + k = 0, where k is a positive constant, has equal roots, find
the value of k.
(4)
3. [C1 Jan 2007 Q5]
The equation 2x2 β 3x β (k + 1) = 0, where k is a constant, has no real roots.
Find the set of possible values of k.
(4)
4. [C1 May 2007 Q7]
The equation x2 + kx + (k + 3) = 0, where k is a constant, has different real roots.
(a) Show that 01242 kk . (2)
(b) Find the set of possible values of k. (4)
5. [C1 Solomon B Q10] Figure 1
Figure 1 shows the curve = 2 β 3 + 5 and the straight line = 2 + 1.
The curve and the line intersect at the points P and Q.
(a) Using algebra, show that P has coordinates (1, 3) and find the coordinates of Q. (4)
(b) Find an equation for the tangent to the curve at P. (4)
(c) Show that the tangent to the curve at Q has the equation = 5 β 11 (2)
(d) Find the coordinates of the point where the tangent to the curve at P intersects
the tangent to the curve at Q. (3)
Exercise D
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.
24
Answers
EXA
1. (a) -2, -7 (b) 8, -3 (c) 9, 7 (d) -5, -5 (e) no real roots
(f) 1.22, -8.22 (g) 2.37, 0.63 (h) 0.79, -2.12
2.
(b)
(d)
(e)
(g)
Ex B
1. two real distinct roots 2. no real roots 3. one real root 4. no real roots
5. two real distinct roots 6. one real root
Ex C
1. β5 1 2. β4 2 3. β5 β1
2 4. β1
17
2
5. β2 1 6. 1 11
2
Exam questions
1. β2 or 9 2. = 6 3. β1
8 4. (b) β 2 or 6
5. (a) 4, 9 (b) = β + 4 (d) (15
6,3
2)
25
WS 9 TANGENTS AND NORMALS
Key words quadratic formula, discriminant, real, distinct, inequality, roots, solutions
Exercise A (Finding equations for the tangent and normal)
For each question follow these steps:
(a) Calculate the y coordinate (if necessary).
(b) Differentiate the function.
(c) Calculate the gradient at the point given.
(d) Write down the following , = =
(e) Find the equation of the tangent using β 1 = β 1 .
(f) Find the equation of the normal using β 1 = β1
β 1 as it has a perpendicular
gradient.
1. = 2 β + 12 5, 2
2. = β 2 + 4 + 5 = 3
3. = 3 β 2 β 6 = β1
4. =4
+ 3 β 5 2, 3
Exercise B (Sketching graphs)
For each question:
(a) Sketch the given curve.
(b) Calculate the tangent at the given point.
(c) Add this to your sketch (Remember sketches should include any points of axes intersection)
1. = 2 at 3, 9
2. = 2 2 β 11 + 5 at 2, β9
3. =1
when = 3
4. = 3 β 3 2 β 10 when = β3
5. = when = 4
Exercise C (Using completing the square)
Through completing the square, find the vertex of the following quadratics.
1. = 2 β 6 + 11 then calculate 3
2. = 2 + 4 β 1 then calculate β2 , what do you notice?
3. = 2 β 3 +13
4 Can you think of another way you could find this minimum point?
26
Exercise D (Working backwards to find coordinates)
Find the coordinates at which the following functions have their given gradient.
1. = 2 2 β 3 + 2
= β4
2. = β3 2 + β 5
= 13
3. = 3 β 2 2 + 5 β 1
= 9
4. = 2 3 +13
2 2 β + 2
= β2
5. =
=
1
3
Exercise E (Tangents, normals and simultaneous equations)
1. = 2 β 1
a) Sketch the quadratic function.
b) Find the equation of the normal to the quadratic when = 1
c) Find the coordinates where the normal to the quadratic intersects the curve again.
d) Add the normal line to your sketch, indicating the point of intersection.
2. = 3 + 3 2 β 4
a) Sketch the cubic function including all intersections with the coordinate axes.
b) Add on to your sketch the tangent to the curve at the origin.
c) Find
d) Find the tangent to the curve at the origin.
e) Find the other point that the tangent intersects the curve again.
3. = +3
a) Show that the point 1, 4 lies on the curve
b) Find
c) Show that the gradient of the tangent to the curve at is is β2
d) Find the equation of the normal to the curve at .
e) Find the point where the normal at intersects the curve again
Answers
EXA
tangent normal
1. = 3 β 13 = β1
3 +
11
3
2. = β2 + 14 β 2 + 13 = 0
3. = β + 3 = + 5
4. = 2 β 1 = β1
2 + 4
EXB
1. = 6 β 9 2. = β3 β 3 3. + 9 β 6 = 0
4. = 35 + 81 5. β 4 + 4 = 0
EXC
1. 3, 2 2. β2,β5 3. (β3
2, β1)
EXD
1. (β1
4,11
8) 2. β2,β19 3. (β
2
3, β
149
27) and 2,5 4. (
1
3,25
54) and (β
5
2,231
8)
5. (9
4,3
2)
EXE
1. Normal: + 2 β 1 = 0
Point of
intersection:(β3
2,5
4)
2. Tangent: = β4
Point of intersection: β3, 12 3. Normal: β 2 + = 0
Point of intersection:(6,13
2)
28
WS SEQUENCES 1 ~ General sequences and summation notation
Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing,
decreasing, recurrence relation
Exercise A
1. Write down the first five terms in each sequence for = 1, 2, 3,
a) = 2n β 1 b) = 3n + 1 c) = n2
d) = β3 e) = 20 β 5n f) = 2n + 2
g) = 5 + β1 h) = 1
2 i) =
1
β
1
1
2. Write down the first five terms in each sequence:
a) 1 = + 3, 1 = 2 b) 1 = 3 , 1 = 4
c) 1 = 5 β 2, 1 = 3 d) 1 = 2 β 1, 1 = 2
e) 1 =1
, 1 = 5 f) 1 = + 1 , 1 = 1
g) 1 = β , 1 = 5 h) 2 = 1 + , 1 = 1, 2 = 1
Exercise B β exam style questions
1. A sequence of positive numbers is defined by
1 = β 2 + 3 , 1 = 2
a) Find 2 and 3 in surd form.
b) Show that 5 = 4
2. A sequence 1, 2, 3 is defined by 1 = , 1 = 4 β where is a constant
a) Write down an expression for 2 in terms of .
b) Find 3 in terms of k, simplifying your answer.
Given that 3 = 13
c) Find the value of k.
3. A sequence is defined by 1 = 3 β 5, 1 =
a) Find 2 and 3 in terms of
Given that β 3 1 = 92
b) Find .
29
4. A sequence is defined by 1 = + 2, 1 = 2
a) Find 2 and 3
Given that 3 = 6
b) Find the possible values of
5. A sequence is defined by 1 = β 3, 1 = 1, where is a constant
a) Find an expression for 2 in terms of a.
b) Show that 3 = 2 β 3 β 3
Given that 3 =
c) Find the possible values of .
6. A sequence 1, 2, 3 is defined by 1 = 1, 1 = + 5 where is a constant.
a) Write down an expression for 2 and 3 in terms of
b) Given that 3 = 41, find the possible values of
7. The sequence of positive numbers 1, 2, 3 is given by 1 = β 3 2, 1 = 1
a) Find 2, 3 and 4
b) Write down the value of 500.
8. A sequence is given by 1 = + , 1 = 1 where is a constant and 0
a) Show that 3 = 1 + 3 + 2 2
Given that 3 = 1,
b) Find the value of .
c) Write down the value of 2014.
Exercise C
1. UKMT Maths Challenge Question:
A sequence 1, 2, 3 is defined for positive integer values of by
3 = 2 + 1 β
Where 1 = 0, 2 = 2, and 3 = 1.
What is the sum of the first 100 terms of the sequence?
2. Calculate the first 10 terms of the following sequence 1 = 1, 2 = 1 and = β1 + β2. What is the name of this sequence and why is it so famous?
3. Research the Mandelbrot Set - this is all done using sequences.
30
http://www.youtube.com/watch?v=G_GBwuYuOOs
4. Calculate the first few terms of the following infinity series, where n is an integer.
a)
b)
c)
What do they value to they tend towards? Why?
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) , 10 4
4a) 2 = 4 β b) 3 = 16 β 35 c) = 3
5a) 2 = 3 β 5 3 = 9 β 20 b) = 9
6a) 2 = 2 + 2 3 = 2 2 + 2 + 2 b) = 1,β2
7a) 2 = β 3 c) = 5,β2
8a) 2 = + 5 3 = 2 + 5 + 5 b) = 4,β9
9a) 2 = 4, 3 = 1, 4 = 4 b) 500 = 4
10b) = β3
2 c) 2014 = β
1
2
31
WS SEQUENCES ~ Arithmetic sequences
Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing,
decreasing, recurrence relation
= + β 1 =1
2 + =
1
2 2 + β 1
Exercise A
Write an expression, in terms of and , for each of these statements:
1. 15th term is 104 2. 27th term is -5
3. 3rd term is 78 4. 109th term is 10
5. 11th term is 20 6. Sum of the first 6 terms is 34
7 Sum of the first 40 terms is 109 8. Sum of the first 17 terms is 80
Exercise B
Use either of the formulae at the top to calculate the missing quantity.
Note you can use = + β 1 directly 8 = +
1. = 3, = 5 Write out 1, 2, 3, 4 then calculate 4
2. = , = β2 Write out 1, 2, 3, 4, 5, 6 then calculate 6
Use and appropriate formulae to calculate the missing quantity.
3. = 3 4. = 5. = = 5 = 5 = 3 = 10 = 12 = 10 = 12 = 4 = 31
6. = 4 7. = 8. = 8 = = 3 = = 12 = 8 = 10 12 = 8 = 164 10 = β55
9. = 10. + 4 = 24 11. 4 = 15 = 3 + 9 = 38 11 = 1 = = = 140 =
32
Exercise C β Exam Questions
1. [C1 Jan 2012 Q9]
A company offers two salary schemes for a 10-year period, Year 1 to Year 10 inclusive.
Scheme 1: Salary in Year 1 is Β£P.
Salary increases by Β£(2T ) each year, forming an arithmetic sequence.
Scheme 2: Salary in Year 1 is Β£(P + 1800).
Salary increases by Β£T each year, forming an arithmetic sequence.
(a) Show that the total earned under Salary Scheme 1 for the 10-year period is
Β£(10P + 90T ).
(2) For the 10-year period, the total earned is the same for both salary schemes.
(b) Find the value of T.
(4) For this value of T, the salary in Year 10 under Salary Scheme 2 is Β£29 850.
(c) Find the value of P.
(3)
2. [C1 Jun 2013 Q7]
A company, which is making 200 mobile phones each week, plans to increase its production.
The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to
220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week N.
(a) Find the value of N.
(2) The company then plans to continue to make 600 mobile phones each week.
(b) Find the total number of mobile phones that will be made in the first 52 weeks starting from
and including week 1.
(5)
Answers
Exercise A 1. a + 14d = 104 2. a + 26d = -5 3. a + 2d = 78 4.a + 108d = 10 5.a + 10d = 20 6. (1/2)(6) (2a + 5d) = 34 7. (1/2)(40) (2a + 39d) = 109
8. (1/2)(17) (2a + 16d) = 80
Exercise B 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
Exercise C 1. (b) T = 400 (c) P = Β£24 450 2. (a) N = 21 (b) 27 000
33
WS INTEGRATION
Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing,
decreasing, recurrence relation
=
β =
1
+ 1 1 + , β1
β
β
34
Exercise A
Write the following in a form that they can be integrated.
1. = 2 =1
3 3. =
5
2
4. =1
5. =
4
6. =
1
2
7. = 3 + 2 2 8. = 2 β 1 3 9. = + 1
10. = β 2 β 3 11. = 2 1 +2
*12. =
23 2 3
Exercise B
Now integrate the expressions in Exercise A (donβt forget the constant of integration!).
Exercise C (Finding the constant of integration)
1. A function passes through 1, 8 and has gradient function
= 5 , find the equation of the curve.
2. A function passes through 1, 9 and has gradient function
= 6 2 + 5 , find the equation of the
curve.
3. A function passes through 2, 1 and has gradient function
=
4
2 , find the equation of the curve.
4. Solve the differential
= ( +
1
)2
given the point 1, 3 .
Exercise D - Exam Questions
1. [C1 May 2012 Q1]
Find
x
xx d5
26
2
2, giving each term in its simplest form.
(4) 2. [C1 Jan 2012 Q7]
A curve with equation y = f(x) passes through the point (2, 10). Given that
f β²(x) = 3x2 β 3x + 5,
find the value of f(1).
(5)
3. [C1 May 2011 Q6]
Given that x
xx
2
5
36 can be written in the form 6x p + 3xq,
(a) write down the value of p and the value of q.
(2)
Given that x
y
d
d =
x
xx
2
5
36 and that y = 90 when x = 4,
(b) find y in terms of x, simplifying the coefficient of each term.
(5)
35
Answers
EXA
1. =
2 2. =1
3 β1 3. = 5 β2 4. = β
2 5. = 4 β1 6. =1
2 β
2 7. = 9 2 +
12 + 4
8. = 8 3 β 12 2 + 6 β 1 9. = 3
2 +
2 10. = β 3 + 3 2 11. = 2 + 4
2 12.
= 2
2 + 3
2
EXB
1. 2
3
3
2 + 2. 3.β5 β1 + 4. 2
2 + 5. 6.
2
7. 3 3 + 6 2 + 4 + 8. 2 4 β 4 3 + 3 2 β + 9. 2
5
2 +2
3
3
2 + 10. β
4+ 3 +
11. 2 +8
3
3
2 +
12. 24
29
2
2 +36
17
2 +
EXC
1. = 5 + 3 2. = 2 3 + 5 + 2 3. = 3 β4
4. =
1
3 3 + 2 β
1
+
5
3
EXD
1. 2 3 β 2 β1 + 5 + 2. 1 =5
2 3. (a) =
1
2, = 2 (b) = 4
3
2 + 3 β 6
36
CORE 3 FORMULA SHEET
Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and
C2.
Logarithms and exponentials
xax alne
Trigonometric identities
BABABA sincoscossin)(sin
BABABA sinsincoscos)(cos
))(( tantan1
tantan)(tan
21
kBA
BA
BABA
2cos
2sin2sinsin
BABABA
2sin
2cos2sinsin
BABABA
2cos
2cos2coscos
BABABA
2sin
2sin2coscos
BABABA
Differentiation
f(x) f (x)
tan kx k sec2 kx
sec x sec x tan x
cot x βcosec2 x
cosec x βcosec x cot x
IMPORTANT INFORMATION
Maths and Computer Science Teachers: room email
Ceinwen Hilton 232 [email protected]
Elliot Henchy 232 [email protected]
Flo Oakley 232 [email protected]
Greg Jefferys 218 [email protected]
Dan Nelson 214 [email protected]
Najm Anwar 214 [email protected]
Nadya de Villiers 214 [email protected]
Vijay Goswami 214 [email protected]
Website
Please take some time to visit our website: www.candimaths.uk
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 β 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.
Other 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