0

Student

Teacher

A2 STARTER PACK September 2016

City and Islington Sixth Form College

Mathematics Department

www.candimaths.uk

1

Contents

INTRODUCTION 2

SUMMARY NOTES 3

WS TRIGONOMETRY 1 ~ Reciprocal Trig 5

WS TRIGONOMETRY 2 ~ Reciprocal Trig 6

WS TRIGONOMETRY 3 ~ Compound Angle Formula 8

WS TRIGONOMETRY 4 ~ Double Angle Formula 10

WS EXPONENTIAL FUNCTIONS 12

WS DIFFERENTIATION 1~ Trigonometric and Exponential Functions 16

Review Exercise for 1-1’s Questions based on Jun 2014 (R) C3 paper 18

WS DIFFERENTIATION 2 ~ Chain Rule 20

WS DIFFERENTIATION 3 ~ Product Rule 22

WS DIFFERENTIATION 4 ~ Quotient Rule 24

WS NUMERICAL METHODS 26

WS MULTIPLE TRANSFORMATIONS 28

WS MODULUS FUNCTION ~ Sketching Graphs 30

WS TRIGONOMETRY ~ Applications of Compound Angle Formulae 34

CORE 3 FORMULA SHEET 36

IMPORTANT INFORMATION 37

2

INTRODUCTION Over the next 7 weeks you will be studying new topics in mathematics. Each of these new

topics builds upon AS work. At A2 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 AS – A2 test: This test is to check that you have not forgotten AS maths and C3

functions and algebra! 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 8 C3 mock exam: This test is to make sure you are ready for the harder parts of

the course. HW7: Practice Test in your C3 Homework Pack is an example of this test. You

will need to work hard - the pass mark for this test is 50%.

Lesson 1 Lesson 2 Lesson 3

Week 1 Reciprocal Trigonometry

Trigonometric Identities

and proof

Week 2

AS-A2 TEST Trigonometry –

Compound angle

formulae

Trig double angle

formulae

Careers lesson

Week 3

Exponential functions 1 to 1’s – Test feedback

Exponentials-

applications

Differentiating

trigonometry , ex &

Week 4

Differentiation:

Chain Rule

Differentiation:

Product Rule

Differentiation:

Quotient rule

Week 5

Numerical Methods

Numerical Methods

Multiple graph

transformations

Week 6

Modulus functions:

Sketching

Solving modulus

equations

Trig - Compound

angle

Week 7 HALF TERM – There will be a C3 mock in the first week back after half term

Extra resources, links and digital copies of the booklets can be found at our website:

www.candimaths.uk

3

SUMMARY NOTES

Number

Algebra

2𝑥 3 = 11

2𝑥 = 8

𝑥 = 4

𝑥2 6𝑥 − 16 = 0 𝑥 8 𝑥 − 2 = 0

𝑥 = 2,−8

𝑥2 6𝑥 − 16 = 0 𝑥 3 2 − 9 − 16 = 0

𝑥 3 2 = 25

𝑥 = −3 ± 25

𝑥 = 2,−8

Linear

Quadratic

Complete the Square

Formula 𝑥 =−𝑏± 𝑏2−4𝑎𝑐

2𝑎

Completed Square Form

𝑦 = 𝑥2 8𝑥 21 𝑦 = 𝑥2 − 8𝑥 21

𝑦 = 𝑥 4 2 − 16 21 𝑦 = 𝑥 − 4 2 − 16 21

𝑦 = 𝑥 4 2 5 𝑦 = 𝑥 − 4 2 5

2𝑥2 11𝑥 15 = 𝑥 3 2𝑥 5

𝑥2 6𝑥 9 = 𝑥 3 2

𝑥2 − 6𝑥 9 = 𝑥 − 3 2

𝑥2 − 9 = 𝑥 3 𝑥 − 3

5𝑥2 19𝑥 12 = 5𝑥 4 𝑥 3

Factorising

3𝑥 5𝑦 = 20

𝑥2 𝑦2 = 5

Simultaneous Equations

𝑥 4𝑦 = 16 Elimination method

2𝑥 − 𝑦 = 4 Substitution method

ℕ Natural 1, 2, 3, .. [counting]

ℤ Integers -2, -1, 0, 1, 2,… [counting ±]

ℚ Rational 2

3, −

43

67, 86, 0 [fractions ±, all except Irrational]

ℝ Real [All including irrationals numbers eg 2,𝜋, 𝑒]

Irrational numbers cannot be written as fractions.

As decimals they are infinite and non-recurring.

−2 3 − −4 = 5

−

Equivalent fractions

Fraction arithmetic

3

4

5

6=

9

12

10

12=

19

12 𝐿𝐶𝑀

3

4÷

6

5=

3

4×

5

6=

15

24=

5

8

Directed Numbers

BIDMAS

( ) 23 5 − 1 2 = 24

𝑥2 5 × 6 − 32 = 21

×÷ 10−4

2= 3

Indices (powers)

23 × 24 = 27 2−3 =1

23

25 ÷ 23 = 22 21 = 2

23 5 = 215 20 = 1

4

Geometry

Sequences

𝑦 = 𝑥2 9𝑥 − 22

𝑦 = 𝑥 11 𝑥 − 2

Graph Sketching

When

When

ce tre = 3, 1

𝑥,𝑦 𝑚 = 2 𝑟𝑎𝑑𝑖𝑢𝑠 = 4

𝑦 − 1 = 2 𝑥 − 3

𝑥 − 3 2 𝑦 − 1 2 = 16

Line

𝑦 = 2𝑥 − 5 or 2𝑥 − 𝑦 − 5 = 0

Circle

1, 3

5, 8

Normal (perpendicular line)

𝑚𝑖𝑑 𝑝𝑜𝑖𝑛𝑡 = 1 5

2,3 8

2

𝑑𝑖𝑠𝑡2 = 5 − 1 2 8 − 3 2

gradie t 𝑚1 =8 − 3

5 − 1=

5

4

𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑛𝑜𝑟𝑚𝑎𝑙 𝑚2 =−4

5

Arithmetic Sequence

First term: 𝑎,

Common difference: 𝑑

Number of terms: 𝑛

𝑛th term: 𝑈𝑛 = 𝑎 𝑛 − 1 𝑑

Sum to 𝑛 terms: 𝑆𝑛 =𝑛

2 2𝑎 𝑛 − 1 𝑑

𝑆𝑛 =𝑛

2 𝑎 𝑙

5

WS TRIGONOMETRY 1 ~ Reciprocal Trig

[C2 Revision and Introducing the Reciprocal Trig Functions]

Key words: Reciprocal, sec, cosec, cot

Folder Checklist

Please check you have the following in your folders. If you’re missing anything, get it!

What should I have? This ‘Starter pack’!

C3 Homework Pack

Summer Work – checked by your teacher

Dividers

Exercise A Revision of C2 Solving trig equations using identities

Solve the following equations

a) 1 c − 2 2 = 0 0 360

b) 2 2 − 3 c 2 = 0 −

c) 3600sin3cos1713 2 xxx

d) 3600cossinsin1 2 xxxx

Exercise B Solving simple equations with reciprocal trig functions

Solve the following equations for −180 180

a) c ec = 1 b) ec = −3

c) c t = 3 45 d)* ec = 2 c

e)* c ec 2 = 4 f)** 2 c t2 − c t − 5 = 0

Answers:

Ex A a) 60 , 180 , 300 b) −

2, 0 34 , 2 80 c) 48 2, 311 8 d) 90, 45, 225, 2 0,

Ex B a) 90 b) ±109 c) −164,16 2 d) ± 45, ±135

e) −1 3, −9 2, 24, 82 8 f) −152,−36 5, 28 4, 143

6

WS TRIGONOMETRY 2 ~ Reciprocal Trig

[Proofs and Equations with Reciprocal Trig Functions]

Key words: Reciprocal, sec, cosec, cot

Exercise A Proofs Prove the following identities:

1. c i ta ec 2.

1−

1−

2 ec

3. 1 ta 2 ec2 4. c ec2 c t2 1 2

1− 2

5. ec2 − c 2 i 2 1 ec2 6. ec2 c t2 − c 2 c t2

Exercise B Solving trigonometric equations

1. (a) Sketch on one pair of axes the graphs of

= c t =1

3 for 0

2

(b) Give a reason why the equation: c t =1

3

must have exactly one root in the interval 0

2

2. Solve the equation 5 3 ec = ta 2

giving all solutions to the nearest 0.1° in the interval 0 360

3. Solve the equation c ec = 3

giving all solutions in degrees to the nearest 0.1° in the interval 0 360

4. (a) Show that the equation ta 2 ec = 11

can be written as 2 − 12 = 0

where = ec

(b) Hence solve the equation

ta 2 ec = 11 giving all solutions to the nearest 0.1° in the interval 0 360

5. (a) Prove the identity:

(b) Hence, or otherwise, find all solutions in the interval 0 2 of the equation

giving your answers in terms of π.

6. (a) Prove that 2 is a factor of = 2 3 − 2 − 1 10

and factorise completely.

(b) Hence, or otherwise, solve the equation 2c ec3 − c ec2 − 1 c ec 10 = 0,

giving all solutions in degrees, to the nearest 0.5°, in the interval 0 360

xxx

2tan21cosec

1

1cosec

1

,61cosec

1

1cosec

1

xx

7

Exercise C Exam Questions 1. [June 2008 Q5]

(a) Given that sin2 θ + cos

2 θ ≡ 1, show that 1 + cot

2 θ ≡ cosec

2 θ . (2)

(b) Solve, for 0 θ < 180°, the equation 2 cot2 θ – 9 cosec θ = 3,

giving your answers to 1 decimal place. (6)

2. [Jan 2012 Q5]

Solve, for 0 180°,

2 cot2 3 = 7 cosec 3 – 5.

Give your answers in degrees to 1 decimal place. (10)

Answers

Ex B 1.(a)

(b) Graphs intersect once between 0 and π/2

2. 76.5˚, 141.2˚, 218.8˚, 283.5˚ 3. 67.5˚, 292.5˚

4. (a) Proof (b) 70.5˚, 104.5˚, 255.5˚, 289.5˚

5. (a) Proof (b)

2 4 5, , ,

3 3 3 3

6. (a) −2 = 0 = 2 2 − 1 − 5

(b) 11.5 ˚, 168.5 ˚, 210 ˚, 330 ˚

Ex C

1. (b) 11.5 ,168.5

2. 6.5, 53.5, 126.5, 173.5

0.5 1 1.5 2

–1

1

2

3

4

5

6

7

x

y

8

WS TRIGONOMETRY 3 ~ Compound Angle Formula

Key words compound angle formula, double angle formula

i = i − = −

c = − c − =

ta =

1 −

ta − = −

1

Exercise A

1. Express the following as a single sin, cosine or tangent

a) i 15 c 20 c 15 i 20

b) c 130 c 80 − i 130 i 80

c) 76− 45

1 76 45

c) 2 c i 40

2. Prove the following identities

a) 60 − 60

b)

ta ta

c)

1 c t c t

d) c c − c 2 − i 2

3. Solve 0 360 the following equations

a) 3 c = 2 i 60

b) c 25 i 65 = 1

c) ta − 45 = 6 ta

d) c = c 60

Exercise B

1. [Jan 07] (a) By writing sin 3 as sin (2 + ), show that

sin 3 = 3 sin – 4 sin3 . (5)

(b) Given that sin = 4

3, find the exact value of sin 3 . (2)

2. [Jan 09] (a) (i) By writing 3θ = (2θ + θ), show that

sin 3θ = 3 sin θ – 4 sin3 θ. (4)

(ii) Hence, or otherwise, for 0 < θ < 3

, solve

9

8 sin3

θ – 6 sin θ + 1 = 0.

Give your answers in terms of π.

(5)

(b) Using sin (θ – ) = sin θ cos – cos θ sin , or otherwise, show that

sin 15 = 4

1(6 – 2). (4)

Extension Express each of the following as the sum or difference of trigonometric

functions (see addition rules in formulae book)

a) 2 i 30 c 10

b) c 49 i 25

c) 2 c i 40

Answers ExA

1a) i 35 b) c 210 c) ta 31

3a) 51.7, 231.7 b) 56.5, 303.5 c) 153.4, 161.6, 333.4, 341.6

d) 150, 330

Ex B

1) 9 3

16

2) (a) (ii) 18

,

18

5

Extension a) i 40 i 20 (b) 1

2 i 4 −

1

2 i 24 c) i 2 40 i 40

10

WS TRIGONOMETRY 4 ~ Double Angle Formula

Exercise A

1. Write the following expressions as a single ratio

a) 2 i 10 c 10 b) c 240 − i 240 c) 1 − 2 i 2 25 d) 8

8

2. Write the following in their simplest form

a) c 2 3 − i 2 3 b) 6 i 2 c 2 c) 4 i 2 c 2

d) i 4 − 2 i 2 c 2 c 4

3. Prove the following identities

a) 2

= c − i b)

−

2 2 i −

c) 1− 2

2 ta d) − 2 c t 2 c 2

4. Solve the following equations

a) i 2 = i 0 2 b) c 2 = 1 − −180 180

c) 3 c 2 = 2 c 2 0 2 d) 2 ta 2 ta = 3 0 2

e) 4 ta = ta 2 0 360

Exercise B Exam questions

1. Given that

tan ° = p, where p is a constant, p 1,

use standard trigonometric identities, to find in terms of p,

(a) tan 2 °, (2)

(b) cos °, (2)

(c) cot ( – 45)°. (2)

Write each answer in its simplest form.

2. Solve, for 0 ≤ x ≤ 270°, the equation

tan 2 tan 502

1 tan 2 tan 50

x

x

Give your answers in degrees to 2 decimal places. (6)

i 2 = 2 c i

2 =2

1 − ta 2

c 2 = c 2 − i 2

c 2 = 1 − 2 i 2

c 2 = 2 c 2 − 1

11

3 [jan 06 q7].

(a) Show that

(i) xx

x

sincos

2cos

cos x – sin x, x (n –

41 ), n ℤ, (2)

(ii) 21 (cos 2x – sin 2x) cos

2 x – cos x sin x –

21 . (3)

(b) Hence, or otherwise, show that the equation

cos 2

1

sincos

2cos

can be written as

sin 2 = cos 2. (3)

(c) Solve, for 0 < 2,

sin 2 = cos 2,

giving your answers in terms of . (4)

Answers

1a) i 20 b) c 80 c) c 50 d) 1

2 i 16

2a) c 6 b) 3 i 4 c) i 22 d) c 2 2

4a) 0,

3, ,

5

3, 2 b) ±38 c)

6,5

6,7

6,11

6 d) 0.579, 0.816, 3.721,

5.704

e) 0, 35.3, 144.7, 180, 215.3, 324.7

Exercise B

1a) 2

1− 2 b) 1

√1 2 c)

1

1−

2) awrt 6.72 ,96.72 ,186.72x

3) (c) = 8

13,

8

9,

8

5,

8

12

WS EXPONENTIAL FUNCTIONS

Exercise A

1) Without using a calculator, write down the value of

a. 7

b. 0 5 16

c. 3 − 5

d. 2 3

e. 5

f. 1

2

g. 3 −4

2) Calculate the value of

a. =3

2

b. 2 = 12

c. 0 2 = −1

d. 2 = 36

e. − = −1

6

f. 3 = 2

3) Solve the following equations giving an exact answer in terms of or natural

logarithms

a. = 13

b. 3 −2 − 5 = 0

c. 1

2 − 3 = 1

d. (1

3 2) =

7

2

e. 6 − 2 = −11

f. 5 −10 3 = 20

4) Sketching the following curves on the same diagram, giving the coordinates of any

intersection with the coordinate axis. What can you observe from your sketches?

a. = and =

b. = 2 and = − 2

c. = 3 and = 3

d. = 5 and =

5

5) C3 June 2007 Q1

Find the exact solutions to the equations

a. 3 = 6 (2)

b. 3 − = 4 (4)

1 = 0

=

= 1

= 2 18281 If = then =

If = then =

13

Section B - Applications

6) A quantity R is decreasing such that at time t

= 120 −0 05 a. Find the value of when = 12

b. Find the value of when = 2

7) A radioactive substance is decaying such that its mass, grams, at a time years

after initial observation is given by = 320 , where is a constant.

Given that when = 15, = 100,

a. Find the value of ,

b. Find the time, to the nearest year, it takes for the mass of the substance to be

halved.

8) A quantity is increasing such that at time = 0

.

Given that at time = 20, = 3000 and that at time = 40, = 4250

a. Find the values of the constants 0 and to 3 significant figures.

b. Find the value of when = 5000

9) [C3 January 2011 Q4]

Joan brings a cup of hot tea into a room and places the cup on a table. At time

minutes after Joan places the cup on the table, the temperature, °C, of the tea is

modelled by the equation

= 20 − where and are positive constants.

Given that the initial temperature of the tea was 90 °C,

a. Find the value of A. (2)

The tea takes 5 minutes to decrease in temperature from 90 °C to 55 °C.

b. Show that =1

5 2 (3)

Section C – Functions

10) [Jan 2006 Q8]

The functions and are defined by

2 2 , ℝ

2 , ℝ

a. Prove that the composite function is

4 4 , ℝ (4)

b. Sketch the curve with equation = , and show the coordinates of the point

where it cuts the -axis. (1)

c. Write down the range of (1)

14

11) [Jan 2007 Q6]

The function is defined by

4 − 2 , 2 and ℝ

a. Show that the inverse function of is defined by

−1 2 −1

2 ,

and write down the domain of −1 (4)

b. Write down the range of −1. (1)

c. Sketch the graph of = −1 ).

State the coordinates of intersection with the and axes. (4)

12) [Jan 2010 Q9]

i. Find the exact solutions to the equations

a. 3 − = 5, (3)

b. 3 7 2 = 15, (5)

ii. The functions and are defined by

= 2 3, ℝ,

= − 1 , ℝ, 1.

a. Find −1 and state its domain. (4)

b. Find and state its range. (3)

Extension Questions

a. Oxford MAT Test 2015

How many distinct solutions does the following equation have?

g 2 2 4 − 5 2 − 6 3 = 2 (i) None (ii) 1 (iii) 2 (iv) 4 (v) Infinitely many

b. Positive integers and satisfy the condition g2 g2 ( g2 21000 ) = 0

Find the sum of all possible values of

c. Oxford Physics Aptitude Test 2015

If = for 0 and = − 2 for 0, sketch the function and its first, second and third derivatives. (8 marks)

d. Find the sum of the series 1, − , −2 , −3 For what values of will this series

converge?

e. Why was Euler a good dancer?

15

Answers

1a) 7 (b) 4 (c) 3

5 d) 9 (e) 5 (f) -2 (g) -12

2a) 3

2 (b) 6 (c) -5 (d) 6 (e) -6 (f) 81

3a) = 13 (b) =1

3 5 2 (c) = 8 (d) = 3

2 − 2 (e) = 17

2 (f) =

1

10( −

17

5)

4) Use graph drawing software such as Geogebra or Omnigraph to check your answers

5)a) = 2 (b) = 0, 3

6)a) = 66 (b) = 82

7)a) = −0 0 5 (b) = 9 years

8)a) = 0 01 4, 0 = 2120 b) t = 49.3 9)-12) Use mark-schemes which can be found online.

Extension a) 2 (b) 881 (c) Use graph drawing software (d)

−1, 0 (e) He had natural

logarithm!

16

WS DIFFERENTIATION 1~ Trigonometric and Exponential

Functions

Exercise A

1. For = find a)

b)

2

2

2. For = c find a) b)

3. For → find

4. For = −1 find a) b)

Exercise B

1a. Find the normal to the curve = i =

4

b. Where does this normal meet the x axis? This is the point B.

c. Give another x coordinate for which the curve has the same gradient as it does at =

4 .

d. The tangent to g(x) at =

2 meets the x axis at C. Find the coordinates of C

e. The point where the tangent and normal meet is point A, find the area of the triangle

ABC.

2a. Find the tangent (T) to the curve → , ℝ when = 2

b. Find the normal (N) to the curve = at the point = 3. Give your answer in the form

= 0 where a, b and c are given in exact form. c. Find the x - coordinate of the point where T and N meet. Give your answer in exact

form. (hint: a diagram will help!)

3. Using differentiation find the stationary points in the interval 0, 2 and determine the nature of the stationary point for

a. = i b. = c

4. Using what you know about the relationship between integration and differentiation

What is the area between the curve = c and the x axis between 0 2

→

c = − i

i = c − 1 = − 1

→

17

Answers

Part A

1a) c b) − i 2a) − i b) −c 3) 4a) 1

b) −

1

2

Continued Overleaf…

Part B

1a) 2 2 −

2− 1 = 0 b)

2

4, 0 c)

9

4 - think of period of sine! d) (

4− 1 , 0)

e) 3 2

8

2a) = 2 − 2 b) 3 − 9 − 3 = 0 c) 9 3 2

2 3

3a)

2 maximum,

3

2 minimum b) 0 2 − −

4) 4 units2

18

Review Exercise for 1-1’s Questions based on Jun 2014 (R) C3 paper

1. Express

2

3 1 6

2 3 2 3 4 9x x x

as a single fraction in its simplest form. (4)

2. A curve C has equation y = e4x

+ x4 + 8x + 5.

(a) Show that the x coordinate of any turning point of C satisfies the equation

x3 = –2 – e

4x

(3)

(b) On a pair of axes, sketch, on a single diagram, the curves with equations

(i) y = x3,

(ii) y = –2 – e4x

On your diagram give the coordinates of the points where each curve crosses the y-axis

and state the equation of any asymptotes.

(4)

(c) Explain how your diagram illustrates that the equation x3 = –2 – e

4x has only one root.

(1)

3. (i) (a) Show that 2 tan x – cot x = 5 cosec x may be written in the form

a cos2 x + b cos x + c = 0

stating the values of the constants a, b and c.

(4)

(b) Hence solve, for 0 ≤ x < 2π, the equation

2 tan x – cot x = 5 cosec x

giving your answers to 3 significant figures.

(4)

(ii) Show that

tan θ + cot θ ≡ λ cosec 2θ, 2

n , n

stating the value of the constant λ.

(4)

19

4. The function f is defined by

f : 2 2e xx k , x , k is a positive constant.

(a) State the range of f.

(1)

(b) Find f –1

and state its domain.

(3)

The function g is defined by

g : ln 2x x , x > 0

(c) Solve the equation

g(x) + g(x2) + g(x

3) = 6

giving your answer in its simplest form.

(4)

(d) Find fg(x), giving your answer in its simplest form. (2)

(e) Find, in terms of the constant k, the solution of the equation

fg(x) = 2k2

(2)

Answers

1

2

3 a =3, b = 5, c = -2 x = 1.23 or 5.05 = 2

4 2f ( )x k 1 2 21

f ( ) ln( ),2

x x k x k c)

2 d) 4 2 e)

2

20

WS DIFFERENTIATION 2 ~ Chain Rule

21

22

WS DIFFERENTIATION 3 ~ Product Rule

Formula (to remember):

For )()( xvxuy , dx

dvu

dx

duv

dx

dy

Exercise A

1) Find the value of f '(x) at the value of x indicated in each case.

(a) f(x) = (5x 4) ln 3x, x = 3

1 (b) f(x) = 32

1

21 xx , x = 4

1

(c) f(x) = sin2xcos

3x, x =

6

2) (a) Find the coordinates of any stationary points on each curve.

(i) y = x2(2x 3)

4 (ii) y = 2 + x

2e

−4x

(b) Find the coordinates of any stationary points on the curve given in the interval

y = 2 sec x − tan x 0 ≤ x ≤ 2π.

3) Find an equation for the tangent to each curve at the point on the curve with the given

x-coordinate.

(a) y = 3x2e

x, x = 1 (b) y = (4x 1) ln 2x, x =

2

1

4) Find an equation for the normal to each curve at the point on the curve with the given

x-coordinate. Give your answers in the form ax + by + c = 0, where a, b and c are integers.

(a) y = x ln (3x 5), x = 2 (b) 4 xxy , x = 8

(c) y = 3 + x cos 2x, at the point where it crosses the y-axis.

Exercise B

1)

The diagram shows part of the curve with equation 2xxey and the tangent to the curve

at the point P with x-coordinate 1.

23

(a) Find an equation for the tangent to the curve at P.

(b) Show that the area of the triangle bounded by this tangent and the coordinate axes is

e3

2.

2) A curve has the equation y = e−x sin x.

(a) Find dx

dyand

2

2

dx

yd

(b) Find the exact coordinates of the stationary points of the curve in the interval

-π ≤ x ≤ π and determine their nature.

3) A curve has the equation y = cosec (x − 6

) and crosses the y-axis at the point P.

(a) Find an equation for the normal to the curve at P.

The point Q on the curve has x-coordinate 3

.

(b) Find an equation for the tangent to the curve at Q.

The normal to the curve at P and the tangent to the curve at Q intersect at the point R.

(c) Show that the x-coordinate of R is given by 13

438 .

Answers

ExA

1) (a) -7 (b) 8

5 (c)

32

9

2) (a) (0,0), (2

1,4), (

2

3,0) (b) (0,2), (

2

1,2 +

4

1e

-2) (c) (6

, 3 ) , (

6

5,- 3 )

3) (a) y = 3e(3x – 2) (b) y = 2x – 1

4) (a) 026 yx (b) 0724 yx (c) y = 3 – x

Ex B

1) (a) y = e(3x – 2)

2) (a) xxe x sincos , xe x cos2 (b) (

4

3 , 4

3

2

1

e ) , (4

, 4

2

1

e )

3) (a) y = 6

3x – 2 (b) 0326336 yx

24

WS DIFFERENTIATION 4 ~ Quotient Rule

(

) =

−

2

If =

where and are functions of ,

then

=

−

2

Section A

1. Differentiate each of the following with respect to and simplify your answers.

i. 3

2 −3 ii.

2 iii. −1 2

iv.

v.

3

vi.

3

3

2. Find

in each of the following cases

i. = 2 1 3

ii. =

2

iii. =

3 iv. =3 1

−2

3. Find the coordinates of any stationary point on each curve.

i. = 2

3− ii. =

3 −1 iii. =

2

3 iv. =

2−3

2

4. Differentiate =2

−3 using i. the product rule ii. the quotient rule

Section B

5. Show using the quotient rule that

ta = ec2

6) Show that

c t = − 2

7) Prove using the quotient rule the results for

and

. How could you check your results

using a different method?

8) By writing

= −1 prove the quotient rule formula.

Section C

C3 Solomon Paper K – Q5

9) Differentiate each of the following with respect to and simplify your answers.

a. c t 2 (2)

b. 2 − (3)

c.

3 2 (4)

25

C3 January 2011 Q7

The curve has equation

=3 i 2

2 c 2

(a) Show that

=

6 i 2 4 c 2 2

2 c 2 2

(4)

(b) Find an equation of the tangent to at the point on where =

2

Write your answer in the form = , where and are exact constants.

(4)

Answers

1i. 2 4 −9

2 −3 2 ii.

−2

3 iii. −1 2 −

2 iv.

2−

2 32

v. 3 3

2 vi. −

3 3 3

3 2

2i.

=

2 1 2 8 1

2 ii.

=

2 2

2 iii.

=

3−3 3 3

3 2 iv.

=

3 −13

2 −1 32

3i. 0,0 6, −12 ii. (8

15,5

3

3) iii. (

2,

2

3 ) iv. −1,−2 −3,−6

4. −6 − 3 2

26

WS NUMERICAL METHODS

Keywords interval, root, change of sign, iterative method,

Exercise A

Show that there is a root ( ) of the equation = 0 in the given interval.

1 = 3 − 2, 1, 1 5 2 = − 2, 0 6, 0

3 = 5 c − 3 , 0 5, 1 4 = 2 5 − 3 2, 1 1, 1 2

5 = 2 5, −6,−5 6 = 4 − 1 2 , 0 4, 0 5

Exercise B

For each equation show that the root is correct to the given level of accuracy.

1 2 = 0, = 0 t 1 d 2 4 − = 0, = −0 82 t 2 d

3 5 c 2 − 2 = 0, = 0 32 t 3d 4 5 = 4 3 − 3, = 1 885 t 3d

5 a On the same set of axes, sketch the graphs of = 3 and = 4 − .

b Hence, show that the equation 3 − 4 = 0 has exactly one root.

c Show that the root lies in the interval (1, 1.5).

Exercise C

For each equation, shoe that it can be arranged into the given iterative form and state the

values of and . Use 0 to find 1, 2 and 3 to 3 decimal places.

1 2 −1 − 6 = 0 1 = 1 0 = 1

2 2

c − 3 = 0 1 =

− 0 = 0 8

3 2 3 − 6 − 11 = 0 1 = √

0 = 2

4 15 3 − 4 = 0 1 = 0 = −2 5

Exercise D – Exam Questions

1 [C3 Jun 2010 Q3]

f(x) = 4 cosec x − 4x +1, where x is in radians.

(a) Show that there is a root α of f(x) = 0 in the interval [1.2, 1.3].

(2)

(b) Show that the equation f(x) = 0 can be written in the form

x = xsin

1 +

4

1

(2)

27

(c) Use the iterative formula

xn+ 1 = nxsin

1 +

4

1, x0 = 1.25,

to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.

(3)

(d) By considering the change of sign of f(x) in a suitable interval, verify that α = 1.291

correct to 3 decimal places.

(2)

2 [C3 Jun 2014 Q6]

Figure 2

Figure 2 shows a sketch of part of the curve with equation

2 312cos 3 2

2y x x x

The curve crosses the x-axis at the point Q and has a minimum turning point at R.

(a) Show that the x coordinate of Q lies between 2.1 and 2.2.

(2)

(b) Show that the x coordinate of R is a solution of the equation

22 11 sin

3 2x x x

(4)

Using the iterative formula

21

2 11 sin

3 2n n nx x x

, x0 = 1.3

(c) find the values of x1 and x2 to 3 decimal places. (2)

Answers

Exercise C

1 =1

2, = 6, 1 = 1 661, 2 = 1 650, 3 = 1 646 2 = 2, = 3, 1 = 0 868, 2 = 0 850, 3 = 0 855

3 = 3, = 5 5, 1 = 2 398, 2 = 2 301, 3 = 2 322 4 =4

15, = −3, 1 = −2 48 , 2 = −2 485, 3 = −2 484

Exercise D

1 c 1 = 1 3038, 2 = 1 286 , 3 = 1 291 2 c 1 = 1 284, 2 = 1 2 6,

28

WS MULTIPLE TRANSFORMATIONS

Exercise A – write in words what the following transformations represent

1. 2. 3. 4. 5. − 6. −

Exercise B Using online graphical calculators

Go to www.desmos.com/calculator or use Geogebra. You can also download the apps to your

phone for free. 1. Plot the following curves: a) = 3

b) = 3 − 2 3

c) = − 3 − 2

Write down the types of transformations that take you from graph a to graphs b and c.

2. Plot the following curves: a) =

b) = 2 −

c) = 4 3

d) = − − 1

Write down the types of transformations that take you from graph a to graphs b,c and d.

Write down the equations of any asymptotes in the graphs of a,b,c and d.

3. Plot the following curves: a) = i

b) = i (

2)

c) = − i −

2

d) = i 2 − 2

Write down the types of transformations that take you from graph a to graphs b,c and d.

What do you notice about graphs b and c? Write down another equation which gives you

the same graph.

4. Experiment with some graphs of your own.

Exercise C – Drawing graphs

On paper draw the following curves. Include any intercepts with the axes and any

asymptotes. Check on Desmos or Geogebra that your graph is correct.

1. = − 2 2 6

2. = − 4

3. = 3 3 4

4. = − 4 3 8

5. = − − 3

6. = −c

7. = ta (

2) 1

8. = ec − − 1

29

Exercise D – Exam practice

1. [Jan 2010 6.]

The graph shows a sketch of the graph of y = f (x).

The graph intersects the y-axis at the point (0, 1) and the point A(2, 3) is the maximum

turning point.

Sketch, on separate axes, the graphs of

(i) y = f(–x) + 1,

(ii) y = f(x + 2) + 3,

(iii) y = 2f(2x) .

On each sketch, show the coordinates of the point at which your graph intersects the y-axis

and the coordinates of the point to which A is transformed. (9)

2. [Solomon Functions Worksheet F]

Sketch the curve with equation = 2 − 2 i for x in the interval 0 2 . Label on

your sketch the coordinates of any maximum or minimum points and any points where

the curve meets the coordinate axis.

Answers

Ex A 1. Horizontal translation of -a units 2. Vertical translation of a units

3. Horizontal stretch of scale factor 1/a 4. Vertical stretch of scale factor a 5. Reflection in the y-axis 6. Reflection in the x-axis Ex B

1. b. Horizontal translation of 2 units, then vertical stretch of scale factor 3 c. Reflection in the x-axis, then vertical translation of -2 units 2. a. asymptote: x-axis or y=0 b. transformation: reflection in the y-axis and stretch of scale factor 2 asymptote: x-axis or y=0 c. transformation: horizontal translation of -3 units and stretch of scale factor 4 asymptote: x-axis or y=0 d. transformation: reflection in the y-axis and the x-axis, then vertical transformation of 1 unit asymptote: y=1 3. b. horizontal translation of –pi/2 units

c. horizontal translation of pi/2 units and reflection in the x-axis d. horizontal stretch of scale factor 1/2 and vertical translation of -2 units *graphs b and c are the same. Another equation which gives the same graph is y=cos x

Ex D – See Mark-scheme – C3 January 2010 Q6

30

WS MODULUS FUNCTION ~ Sketching Graphs Complete on a separate sheet of paper and show clear working.

Mark using the answers below.

Key Words: Modulus Function, Absolute Value, Magnitude, Transformations

1) Evaluate the following:

a) − 2 b) 6 c) 2 − 4 2 d) −3 e) − − f) − − − 3

2) True or False?

Which of the following equations are valid?

a) | −22 | = |− 22 |

b) − 3 2 = −3 2

c) −22 = − 22 d) 2 3 = 2 3 e) 2 − 5 = 2 − 5

3) True or False?

Which of the following identities are true for all values of ?

a) 4 4 b) −3 −3 c) −10 10 d) 2 2 2 2

e) 2 − 2

4) Sketch each of the following graphs, labelling any intersections with axes, and

asymptotes if necessary.

a) = 2 − 3

b) = −2 − 6

c) = | 2 − 5 − 14|

d) = 2 − 8 15

e) = 2 4 9 (tricky! – HINT: complete the square)

f) = , − 2 2

g) = |3

| , 0

h) = − 4 2 10 16

31

5) Sketch each of the following graphs, labelling any intersections with axes, and

asymptotes if necessary.

a) = 5 15

b) = − − 2

c) = 2 − 4 − 5

d) = 2 − 16 63

e) = 2

f) = 3

g) = , − 2 2

h) = 5

, 0

i) = 3 − 9 2 26 − 24

j) = 3 3 2 2

5) Solve each of the following equations. Make sure you sketch the graphs to do this, in

order to ensure you find ALL solutions, and do not include any incorrect ones!

a) 5 = 3

b) 10 − 2 = 1

c) 2 = −1

d) 2 = − 5

e) |4

| = − 3

Exam Practice a) C3 January 2012

Figure 1

Figure 1 shows the graph of equation y = f(x).

The points P (– 3, 0) and Q (2, – 4) are stationary points on the graph.

HINT Look at the original cubic. Try

f(2) for this.

What does this tell you? How

does this help?

32

Sketch, on separate diagrams, the graphs of

(a) y = 3f(x + 2),

(3)

(b) y = f(x). (3)

On each diagram, show the coordinates of any stationary points.

b) C3 June 2015

Given that

f(x) = 2ex – 5, x ℝ,

(a) sketch, on separate diagrams, the curve with equation

(i) y = f (x),

(ii) y = |f (x)|.

On each diagram, show the coordinates of each point at which the curve meets or cuts the

axes and state the equation of the asymptote. (6)

(b) Deduce the set of values of x for which f (x) = |f (x)|. (1)

(c) Find the exact solutions of the equation |f (x)| = 2. (3)

Answers

1)

a) 2 b) 6 c) 0 d) 4 e) 2 f) 2

2) a) T b) T c) F d) T e) F 3)

a) T b) F c) T d) T e) F 6)

a) =3

5 , = −

3

5

b) =9

2 , =

11

2

c) no solutions

d) =3

2

e) = −1 , = 4

4) and 5)

Check your sketches online or with a mobile app/graphical calculator, such as Desmos: Link: https://www.desmos.com/calculator

App: Available via the Chrome app store, or via the iTunes app store.

Exam Practise Solutions (A)

For more detailed answers, check online.

Exam Practise Solutions (B)

For more detailed answers, check online.

b) (5

2) c) = (

7

2) , = (

3

2)

a) i)

a) ii)

a) b)

c)

34

WS TRIGONOMETRY ~ Applications of Compound Angle Formulae

i ± = ± c ± =

Read pages 71-74 and have a look at these examples.

Exercise A

1) Express the following in the form i (Find 0, and 0<

2 )

a. 4 i 3 c = i b. i − 3 c = i −

c. 5 c 4 i = c − d. 8c 2 − 15 i 2 = c 2

2) Sketch the following graphs and mark on the coordinates of any maximum or minimum points.

a. = 2 i − 45 b. = 3 c 30

c. = 5 i (

3) d. = 12 c ( −

12)

e. Use one of these graphs to help solve the equation 5 i (

3) = 4, − ,

3) For the following functions find the max value of the function and the value for which it occurs

a. = 10 i 52 4 b. = 6c − 24 5 c. = −3 c 55

4) Solve the following equations

a. 5 i + 12 c = 4, − , b. 8 c 15 i = 10, 0, 360

c. 6 i - 3 c = −2 , 0, 2

5a. Express 12 c 5 i in the form i , 0

2

b. Hence find the maximum and minimum values of

i. 24 c 10 i ii. 12 c 5 i 2

iii. 12 c 5 i 3 iv. 1

15 12 5

vi. 1

12 5 2 100 vii.

−12 −5 1

12 5 14

Exercise B

1 [C3 June 2014 Q9]

(a) Express 2 i − 4 c in the form – , where and are constants, 0 and 0 2

Give the value of to 3 decimal places. (3)

Given = 4 5(2 i 3 – 4 c 3 )2 Find

(b) (i) the maximum value of ,

(ii) the smallest value of , for 0 , at which this maximum value occurs. (3)

Find

35

(c) (i) the minimum value of , (ii) the largest value of , for 0 , at which this minimum value occurs. (3)

C3 (R) Paper – 2014

Figure 1

Figure 1 shows the curve , with equation = 6 2 5 for 0 2

(a) Express 6 2 5 in the form – , where and are constants with

0 and 0

2. Give your value of to 3 decimal places. (3)

(b) Find the coordinates of the points on the graph where the curve C crosses the coordinate axes. (3)

A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in

May with a recording of 18 hours, and continues until her final recording 52 weeks later.

She models her results with the continuous function given by

= 12 6 c (2

52) 2 5 i (

2

52), 0 52

where is the number of hours of daylight and is the number of weeks since her first recording.

Use this function to find

(c) the maximum and minimum values of predicted by the model, (3)

(d) the values for when = 16, giving your answers to the nearest whole number. (6)

Answers

1a) 5 i 0 643 b) 2 i −

3 c) 41 c − 0 6 5 d) 1 c 2 1 08

2) see desmos – free app available!

3a) 10 = 3 6 b) 6 = 24 5 c) 3 = 215

4a) = 0 8623, 1 6538 b) = 69 49, 321 55 c) = 1 616, 4 105 5a) 13 i 1 1 6

(b)i) = 26, = −26 (ii) = 169, = 0 (iii) = 219 , = −219

(iv) =1

2, =

1

28 (vi) =

1

100, =

1

269 (vii) = 14, = −

4

9

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 –cosec2 x

cosec x –cosec x cot x

IMPORTANT INFORMATION

Maths and Computer Science Teachers: room email

Ceinwen Hilton 232 ceinwen.hilton@candi.ac.uk

Elliot Henchy 232 elliot.henchy@candi.ac.uk

Flo Oakley 232 flo.oakley@candi.ac.uk

Greg Jefferys 218 greg.jefferys@candi.ac.uk

Dan Nelson 214 daniel.nelson@candi.ac.uk

Najm Anwar 214 najm.anwar@candi.ac.uk

Nadya de Villiers 214 nadya.devilliers@candi.ac.uk

Vijay Goswami 214 vijay.goswami@candi.ac.uk

Website

Please take some time to visit our website: www.candimaths.uk

Homework Work outside lessons should take 4+ 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.

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