Banker s Bible - Dublin Academy...2016/02/06 · Banker’s Bible "Nothing consoles and comforts like certainty does." ~Amit Kalantri, Indian Book Author. No part of this publication

6th Year

Maths

Higher Level

Banker’s Bible

"Nothing consoles and comforts like certainty does." ~Amit Kalantri, Indian Book Author.

No part of this publication may be copied, reproduced or transmitted in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from

The Dublin School of Grinds. (Notes Reference: 6-mat-h-Bankers Bible).

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NOTE: These courses are built on the fact that there are certain predicable trends that appear and reoccur over and over again in the State Examinations.

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NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY.

Timetable An extensive range of course options are available over a two-week period to cater for students’ timetable needs. Courses are held over the following weeks:

» Monday 21st March – Friday 25th March 2016 » Monday 28th March – Friday 1st April 2016

All Easter Revision Courses take place in The Talbot Hotel, Stillorgan (formerly known as The Stillorgan Park Hotel).

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6th Year Easter Revision CoursesSUBJECT LEVEL DATES TIME

Accounting H Monday 21st March – Friday 25th March 8:00am - 9:30am

Agricultural Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Applied Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am

Art History H Monday 28th March – Friday 1 April 8:00am - 9:30am

Biology Course A* H Monday 21st March – Friday 25th March 8:00am - 9:30am

Biology Course A* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Biology Course A* H Monday 28th March – Friday 1st April 10:00am - 11:30am

Biology Course B* H Monday 21st March – Friday 25th March 10:00am - 11:30am

Biology Course B* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

Biology Course B* H Monday 28th March – Friday 1st April 8:00am - 9:30am

Business H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Business H Monday 28th March – Friday 1st April 8:00am - 9:30am

Chemistry Course A* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Chemistry Course B* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Classical Studies H Monday 21st March – Friday 25th March 8:00am - 9:30am

Economics H Monday 21st March – Friday 25th March 8:00am - 9:30am

Economics H Monday 28th March – Friday 1st April 10:00am - 11:30am

English Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

English Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am

English Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

English Paper 2* H Monday 28th March – Friday 1st April 10:00am - 11:30am

English Paper 2* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

French H Monday 21st March – Friday 25th March 10:00am - 11:30am

French H Monday 28th March – Friday 1st April 8:00am - 9:30am

Geography H Monday 28th March – Friday 1st April 8:00am - 9:30am

Geography H Monday 28th March – Friday 1st April 10:00am - 11:30am

German H Monday 21st March – Friday 25th March 10:00am - 11:30am

History (Europe)* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

History (Ireland)* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Home Economics H Monday 21st March – Friday 25th March 10:00am - 11:30am

Irish H Monday 21st March – Friday 25th March 10:00am - 11:30am

Irish H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Maths Paper 1* H Monday 21st March – Friday 25th March 8:00am - 9:30am

Maths Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Maths Paper 1* H Monday 28th March – Friday 1st April 10:00am - 11:30am

Maths Paper 1* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Maths Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am

Maths Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm



Maths O Monday 21st March – Friday 25th March 8:00am - 9:30am

Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Physics H Monday 28th March – Friday 1st April 10:00am - 11:30am

Spanish H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

Spanish H Monday 28th March – Friday 1st April 10:00am - 11:30am

6th Year Oral Preparation CoursesSUBJECT LEVEL DATES TIME

French H Sunday 20th March 10:00am - 2:00pm

German H Saturday 26th March 10:00am - 2:00pm

Irish H Saturday 26th March 10:00am - 2:00pm

Spanish H Saturday 19th March 1:00pm - 5:00pm

5th Year Easter Revision CoursesSUBJECT LEVEL DATES TIME

Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am

English H Monday 28th March – Friday 1st April 4:00pm - 5:30pm

Note: 5th year students are welcome to attend any 6th year course as part of our buy 1 get 1 free offer.

3rd Year Easter Revision CoursesSUBJECT LEVEL DATES TIME

Business Studies H Monday 28th March – Friday 1st April 8:00am - 9:30am

English H Monday 21st March – Friday 25th March 8:00am - 9:30am

English H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

French H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Geography H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

German H Monday 21st March – Friday 25th March 8:00am - 9:30am

History H Monday 21st March – Friday 25th March 4:00pm - 5:30pm

Irish H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Maths H Monday 21st March – Friday 25th March 10:00am - 11:30am

Maths H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Maths H Monday 28th March – Friday 1st April 10:00am - 11:30am

Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Science H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

Spanish H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

2nd Year Easter Revision CoursesSUBJECT LEVEL DATES TIME

Maths H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY.

© The Dublin School of Grinds Page 1

The syllabus lists ‘official’ proofs that you can be asked to write in your Leaving Certificate. There are also ‘unofficial’ proofs, which are not listed on the syllabus but are examinable. This set of notes contains all the official and unofficial proofs. The examiner is likely to ask one or two of these, meaning this booklet is worth between 2% and 6% of your Leaving Certificate. Learn them off by heart. Free marks.

Contents Paper 1 1) Financial maths ....................................................................................................................................... 2 2) Number Systems .................................................................................................................................... 3 3) Complex numbers .................................................................................................................................. 7 4) Sequences & Series & Patterns ........................................................................................................ 8 Paper 2 5) Trigonometry .......................................................................................................................................... 9 6) Statistics .................................................................................................................................................. 13 7) Probability ............................................................................................................................................. 16 8) Geometry ................................................................................................................................................ 17 9) Co-ordinate Geometry of the Circle ............................................................................................ 22


1) Financial maths

The Examiner can ask you to derive the formula for a ‘mortgage repayment’ from page 31 of the Formulae and Tables Booklet:

Derivation: P = Loan amount A = Repayment amount each period t = Number of payment periods i = interest rate for the payment period as a decimal.

𝑃 =𝐴

(1 + 𝑖)1+

𝐴

(1 + 𝑖)2+

𝐴

(1 + 𝑖)3+ ⋯ +

𝐴

(1 + 𝑖)𝑡

This is a geometric series: 𝑆𝑛 =𝑎(1−𝑟𝑛)

1−𝑟

𝑎 =𝐴

(1 + 𝑖)1=

𝐴

1 + 𝑖

𝑟 =

𝐴(1 + 𝑖)2

𝐴(1 + 𝑖)1

=1

1 + 𝑖

𝑛 = 𝑡

=> 𝑃 =

𝐴 1 + 𝑖

(1 − (1

1 + 𝑖)

𝑡

)

1 −1

1 + 𝑖

𝑃 (1 −1

1 + 𝑖) =

𝐴

1 + 𝑖(1 − (

1

1 + 𝑖)

𝑡

)

𝑃 (1 + 𝑖 − 1

1 + 𝑖) =

𝐴

1 + 𝑖(1 −

1

(1 + 𝑖)𝑡)

𝑃 (𝑖

1 + 𝑖) (1 + 𝑖)

(1 −1

(1 + 𝑖)𝑡)= 𝐴

=> 𝐴 =𝑃𝑖

(1 + 𝑖)𝑡 − 1(1 + 𝑖)𝑡

𝐴 =𝑃𝑖(1 + 𝑖)𝑡

(1 + 𝑖)𝑡 − 1


2) Number Systems

Constructing a line of length √2: Step 1. Let the line segment 𝐴𝐵 be of length 1 unit.

Step 2. Construct a line 𝑚 perpendicular to [𝐴𝐵] at 𝐵.

Step 3. Construct a circle with centre 𝐵 and radius [𝐴𝐵].

Step 4. Mark the intersection, 𝐶, of the circle and 𝑚.


Step 5. Draw the line segment 𝐶𝐴.

=> |𝐶𝐴| = √2


Constructing a line of length √3: Step 1. Let the line segment 𝐴𝐵 be a length of 1 unit.

Step 2. Construct a circle with centre 𝐴 and radius length |𝐴𝐵|.

Step 3. Construct a circle with centre 𝐵 and radius length |𝐴𝐵|.

Step 4. Mark the intersection of the two circles as 𝐶 and 𝐷.

Step 5. Draw the line segment [𝐶𝐷].

|𝐶𝐷| = √3


Proof that √𝟐 is irrational

The Examiner can ask you to prove √2 is irrational (which is the same as “not rational”). Here is a nice shortened full mark version.

Step 1: Assume √2 is rational.

i.e. √2 can be written as 𝑎

𝑏 where a and b are relatively prime numbers.

Step 2: => 2 =𝑎2

𝑏2

=> 2𝑏2 = 𝑎2 => a is even Step 3: Let 𝑎 = 2𝑥 => 𝑎2 = 4𝑥2

=> 2𝑏2 = 4𝑥2

=> 𝑏2 = 2𝑥2 => 𝑏 is even. Step 4: This contradicts Step 1

=> √2 is not rational.


3) Complex numbers

You need to be able to prove De Moivre’s Theorem for 𝑛 ∈ 𝑁.

We do this using what’s called ‘Proof by induction’. Proof of De Moivre’s Theorem for 𝒏 ∈ 𝑵: [𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)]𝑛 = 𝑟𝑛(𝑐𝑜𝑠𝑛𝜃 + 𝑖𝑠𝑖𝑛𝑛𝜃) Step 1: Show true for 𝑛 = 1:

𝐿𝐻𝑆 = [𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)]1 𝑅𝐻𝑆 = 𝑟1(𝑐𝑜𝑠1𝜃 + 𝑖𝑠𝑖𝑛1𝜃) = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃) = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)

TRUE

Step 2: Assume true for 𝑛 = 𝑘:

[𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)]𝑘 = 𝑟𝑘(𝑐𝑜𝑠𝑘𝜃 + 𝑖𝑠𝑖𝑛𝑘𝜃) Step 3: Prove true for 𝑛 = 𝑘 + 1 𝐿𝐻𝑆 = [𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)]𝑘+1 𝑅𝐻𝑆 = 𝑟𝑘+1[cos (𝑘 + 1)𝜃 + 𝑖𝑠𝑖𝑛(𝑘 + 1)𝜃] = [𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)]𝑘[𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)]1 Using Step 2: = [𝑟𝑘(𝑐𝑜𝑠𝑘𝜃 + 𝑖𝑠𝑖𝑛𝑘𝜃)][𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)] = 𝑟𝑘+1(𝑐𝑜𝑠𝑘𝜃𝑐𝑜𝑠𝜃 + 𝑖𝑐𝑜𝑠𝑘𝜃𝑠𝑖𝑛𝜃 + 𝑖𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝑘𝜃 + 𝑖2𝑠𝑖𝑛𝑘𝜃𝑠𝑖𝑛𝜃] = 𝑟𝑘+1(𝑐𝑜𝑠𝑘𝜃𝑐𝑜𝑠𝜃 + 𝑖𝑐𝑜𝑠𝑘𝜃𝑠𝑖𝑛𝜃 + 𝑖𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝑘𝜃 − 𝑠𝑖𝑛𝑘𝜃𝑠𝑖𝑛𝜃) = 𝑟𝑘+1[(𝑐𝑜𝑠𝑘𝜃𝑐𝑜𝑠𝜃 − 𝑠𝑖𝑛𝑘𝜃𝑠𝑖𝑛𝜃) + 𝑖(𝑐𝑜𝑠𝑘𝜃𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝑘𝜃)] = 𝑟𝑘+1[𝑐𝑜𝑠(𝑘 + 1) 𝜃 + 𝑖𝑠𝑖𝑛(𝑘 + 1)𝜃] (Note: you don’t have to know how this line came about, just write it down. Notice that it’s the same as what we have on the right hand side). Step 4: True for 𝑛 = 𝑘 + 1, assuming true for 𝑛 = 𝑘 By induction, true for all 𝑛 ∈ 𝑁.


4) Sequences & Series & Patterns

Proof of the sum of a geometric series: 𝑆𝑛 =𝑎(1−𝑟𝑛)

1−𝑟

𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟2 + 𝑎𝑟3 + … … … + 𝑎𝑟𝑛−1.

𝑟𝑆𝑛 = 𝑎𝑟 + 𝑎𝑟2 + 𝑎𝑟3 + … … … + 𝑎𝑟𝑛−1 + 𝑎𝑟𝑛 .

𝑆𝑛 − 𝑟𝑆𝑛 = 𝑎 − 𝑎𝑟𝑛 .

(1 − 𝑟)𝑆𝑛 = 𝑎(1 − 𝑟𝑛).

⇒ 𝑆𝑛 =𝑎(1−𝑟𝑛)

1−𝑟.

Proof of the sum of a geometric series to infinity: 𝑆∞ = 𝑎

1−𝑟

𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟2 + 𝑎𝑟3 + … … … + 𝑎𝑟𝑛−1.

𝑟𝑆𝑛 = 𝑎𝑟 + 𝑎𝑟2 + 𝑎𝑟3 + … … … + 𝑎𝑟𝑛−1 + 𝑎𝑟𝑛 .

𝑆𝑛 − 𝑟𝑆𝑛 = 𝑎 − 𝑎𝑟𝑛 .

(1 − 𝑟)𝑆𝑛 = 𝑎(1 − 𝑟𝑛).

⇒ 𝑆𝑛 =𝑎(1 − 𝑟𝑛)

1 − 𝑟

⇒ 𝑆∞ = 𝑎(1−𝑟∞)

1−𝑟

If |r| < 1 ⇒ 𝑟∞ = o

⇒ 𝑆∞ = 𝑎

1−𝑟

Prove that 2nd difference is 2a for quadratic sequence

Given the general term of a quadratic sequence is:

𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐 , where 𝑎, 𝑏, 𝑐 ε 𝑅 and 𝑎 ≠ 0, prove that the second difference for the sequence is 2𝑎.

𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐 𝑇𝑛+1 = 𝑎(𝑛 + 1)2 + 𝑏(𝑛 + 1) + 𝑐 = 𝑎(𝑛2 + 2𝑛 + 1) + 𝑏𝑛 + 𝑏 + 𝑐 = 𝑎𝑛2 + 2𝑎𝑛 + 𝑎 + 𝑏𝑛 + 𝑏 + 𝑐 𝑇𝑛+2 = 𝑎(𝑛 + 2)2 + 𝑏(𝑛 + 2) + 𝑐 = 𝑎(𝑛2 + 4𝑛 + 4) + 𝑏𝑛 + 2𝑏 + 𝑐 = 𝑎𝑛2 + 4𝑎𝑛 + 4𝑎 + 𝑏𝑛 + 2𝑏 + 𝑐

[𝑎𝑛2 + 𝑏𝑛 + 𝑐], [𝑎𝑛2 + 2𝑎𝑛 + 𝑎 + 𝑏𝑛 + 𝑏 + 𝑐], [𝑎𝑛2 + 4𝑎𝑛 + 4𝑎 + 𝑏𝑛 + 2𝑏 + 𝑐]

2𝑎𝑛 + 𝑎 + 𝑏 2𝑎𝑛 + 3𝑎 + 𝑏

2𝑎


5) Trigonometry Area of Segment

The diagram shows a circle of centre 𝑂 and radius 𝑟. 𝑃 and 𝑄 are points on the circle and |∠𝑃𝑂𝑄| = 𝜃, (in radians). The chord [𝑃𝑄] divides the circle into a minor segment (shaded region) and a major segment.

Show that the area of the minor segment is 1

2𝑟2(𝜃 − 𝑆𝑖𝑛𝜃).

= 1

2𝑟2𝜃 −

1

2𝑎𝑏𝑆𝑖𝑛𝐶 [Both from page 9 of log tables]

= 1

2𝑟2𝜃 −

1

2(𝑟)(𝑟)𝑆𝑖𝑛𝜃

= 1

2𝑟2𝜃 −

1

2𝑟2𝑆𝑖𝑛𝜃

= 1

2𝑟2 (𝜃 − 𝑆𝑖𝑛𝜃)


Derivation of a trigonometry formula There are 8 formulae the Examiner can ask you to derive:

1) To prove 𝐶𝑜𝑠2𝐴 + 𝑆𝑖𝑛2𝐴 = 1

From the unit circle: (𝐶𝑜𝑠)2𝐴 + (𝑆𝑖𝑛)2𝐴 = 12

𝐶𝑜𝑠2𝐴 + 𝑆𝑖𝑛2𝐴 = 1

2) To prove The Sine Rule:

𝑎

𝑆𝑖𝑛𝐴=

𝑏

𝑆𝑖𝑛𝐵=

𝑐

𝑆𝑖𝑛𝐶

Case 1: A is acute

𝑆𝑖𝑛𝐴 =𝑦

𝑏 𝑆𝑖𝑛𝐵 =

𝑦

𝑎

𝑏𝑆𝑖𝑛𝐴 = 𝑦 𝑎𝑆𝑖𝑛𝐵 = 𝑦

𝑎𝑆𝑖𝑛𝐵 = 𝑏𝑆𝑖𝑛𝐴

𝑎𝑆𝑖𝑛𝐵

𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵=

𝑏𝑆𝑖𝑛𝐴

𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵

𝑎

𝑆𝑖𝑛𝐴=

𝑏

𝑆𝑖𝑛𝐵

Case 2: A is obtuse

𝐸 = 180° − 𝐴

𝑆𝑖𝑛𝐸 = 𝑆𝑖𝑛(180° − 𝐴) = 𝑆𝑖𝑛𝐴

𝑆𝑖𝑛𝐴 =𝑦

𝑏 𝑆𝑖𝑛𝐵 =

𝑦

𝑎

𝑏𝑆𝑖𝑛𝐴 = 𝑦 𝑎𝑆𝑖𝑛𝐵 = 𝑦

𝑎𝑆𝑖𝑛𝐵 = 𝑏𝑆𝑖𝑛𝐴

𝑎𝑆𝑖𝑛𝐵

𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵=

𝑏𝑆𝑖𝑛𝐴

𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵

𝑎

𝑆𝑖𝑛𝐴=

𝑏

𝑆𝑖𝑛𝐵

Similarly we can show 𝒂

𝑺𝒊𝒏𝑨=

𝒄

𝑺𝒊𝒏𝑪

Therefore: 𝒂

𝑺𝒊𝒏𝑨=

𝒃

𝑺𝒊𝒏𝑩=

𝒄

𝑺𝒊𝒏𝑪

1

1 -1

-1

1

0

A

CosA

SinA

A B

b a

y

A B

b

a y

E


3) To prove: The Cosine Rule: 𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄𝑪𝒐𝒔𝑨

Place the triangle with angle A placed at the origin and side c placed along the X-axis. Draw a circle of radius b around the origin. As this circle is b times larger than the unit circle, the x and y coordinates will also be b times larger. The coordinates of the triangle are (0, 0), (c, 0) and (bCosA, bSinA).

Use the distance formula to find an expression for the length a.

𝑎 = √(𝑏𝐶𝑜𝑠𝐴 − 𝑐)2 + (𝑏𝑆𝑖𝑛𝐴 − 0)2 𝑎2 = 𝑏2𝐶𝑜𝑠2𝐴 − 2𝑏𝑐𝐶𝑜𝑠𝐴 + 𝑐2 + 𝑏2𝑆𝑖𝑛2𝐴 = 𝑏2(𝐶𝑜𝑠2𝐴 + 𝑆𝑖𝑛2𝐴) + 𝑐2 − 2𝑏𝑐𝐶𝑜𝑠𝐴 = 𝑏2(1) + 𝑐2 − 2𝑏𝑐𝐶𝑜𝑠𝐴 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐𝐶𝑜𝑠𝐴

4) To Prove: 𝑪𝒐𝒔(𝑨 − 𝑩) = 𝑪𝒐𝒔𝑨 𝑪𝒐𝒔𝑩 + 𝑺𝒊𝒏𝑨 𝑺𝒊𝒏𝑩

Let 𝑃(𝐶𝑜𝑠𝐴, 𝑆𝑖𝑛𝐴) and 𝑄(𝐶𝑜𝑠𝐵, 𝑆𝑖𝑛𝐵) be two points on a unit circle:

Using the distance formula:

𝑃𝑄 = √(𝐶𝑜𝑠𝐴 − 𝐶𝑜𝑠𝐵)2 + (𝑆𝑖𝑛𝐴 − 𝑆𝑖𝑛𝐵)2

|𝑃𝑄|2 = (𝐶𝑜𝑠𝐴 − 𝐶𝑜𝑠𝐵)2 + (𝑆𝑖𝑛𝐴 − 𝑆𝑖𝑛𝐵)2

= 𝐶𝑜𝑠2𝐴 − 2𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 + 𝐶𝑜𝑠2𝐵 + 𝑆𝑖𝑛2𝐴 − 2𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵 + 𝑆𝑖𝑛2𝐵

= (𝐶𝑜𝑠2𝐴 + 𝑆𝑖𝑛2𝐴) + (𝐶𝑜𝑠2𝐵 + 𝑆𝑖𝑛2𝐵) − 2(𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵)

= 2 − 2(𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵) (1)

Using the Cosine formula on ΔOPQ:

|𝑃𝑄|2 = 12 + 12 − 2(1)(1)𝐶𝑜𝑠(𝐴 − 𝐵)

= 2 − 2𝐶𝑜𝑠(𝐴 − 𝐵) (2)

Equating (1) and (2):

2 − 2𝐶𝑜𝑠(𝐴 − 𝐵) = 2 − 2(𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵)

𝐶𝑜𝑠(𝐴 − 𝐵) = 𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵

O

b

c

a

(bCosA, bSinA)

(c, 0)

A

O

1

1

P(CosA, SinA)

B A A-B

Q(CosB, SinB)


5) To prove: 𝐶𝑜𝑠(𝐴 + 𝐵) = 𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 − 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵

Well: 𝐶𝑜𝑠(𝐴 − 𝐵) = 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵 … this is from Proof 4. You don’t have to prove Proof 4 here

however .

Now, replace B with –B:

𝐶𝑜𝑠(𝐴 − (−𝐵)) = 𝐶𝑜𝑠𝐴 𝐶𝑜𝑠(−𝐵) + 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛(−𝐵)

𝐶𝑜𝑠(𝐴 + 𝐵) = 𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴 (−𝑆𝑖𝑛𝐵)

𝐶𝑜𝑠(𝐴 + 𝐵) = 𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 − 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵

6) To Prove: 𝐶𝑜𝑠2𝐴 = 𝐶𝑜𝑠2𝐴 − 𝑆𝑖𝑛2𝐴

Well: 𝐶𝑜𝑠(𝐴 + 𝐵) = 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 − 𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵 … this is from Proof 5. You don’t have to prove Proof 5 here

however .

Now replace B with A:

𝐶𝑜𝑠(𝐴 + 𝐴) = 𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐴 − 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐴

𝐶𝑜𝑠2𝐴 = 𝐶𝑜𝑠2𝐴 − 𝑆𝑖𝑛2𝐴

7) To Prove: 𝑆𝑖𝑛(𝐴 + 𝐵) = 𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐵 + 𝐶𝑜𝑠𝐴 𝑆𝑖𝑛𝐵

Well: 𝐶𝑜𝑠(𝐴 − 𝐵) = 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵 … this is from Proof 4. You don’t have to prove Proof 4 here

however .

Now, replace A with (90˚−A):

𝐶𝑜𝑠(90° − 𝐴 − 𝐵) = 𝐶𝑜𝑠(90° − 𝐴)𝐶𝑜𝑠𝐵 + 𝑆𝑖𝑛(90° − 𝐴)𝑆𝑖𝑛𝐵

𝐶𝑜𝑠(90° − (𝐴 + 𝐵)) = 𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐵 + 𝐶𝑜𝑠𝐴 𝑆𝑖𝑛𝐵

𝑆𝑖𝑛(𝐴 + 𝐵) = 𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐵 + 𝐶𝑜𝑠𝐴 𝑆𝑖𝑛𝐵

8) To prove: 𝑇𝑎𝑛(𝐴 + 𝐵) =𝑇𝑎𝑛𝐴+𝑇𝑎𝑛𝐵

1−𝑇𝑎𝑛𝐴 𝑇𝑎𝑛𝐵

Well 𝑇𝑎𝑛(𝐴 + 𝐵) =𝑆𝑖𝑛(𝐴+𝐵)

𝐶𝑜𝑠(𝐴+𝐵)

Now, using proofs 5 and 7 (you don’t have to prove these here however ).

=𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐵 + 𝐶𝑜𝑠𝐴 𝑆𝑖𝑛𝐵

𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵 − 𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵

=

𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐵𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵

+𝐶𝑜𝑠𝐴 𝑆𝑖𝑛𝐵𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵

𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵

+𝑆𝑖𝑛𝐴 𝑆𝑖𝑛𝐵𝐶𝑜𝑠𝐴 𝐶𝑜𝑠𝐵

=𝑇𝑎𝑛𝐴 + 𝑇𝑎𝑛𝐵

1 − 𝑇𝑎𝑛𝐴 𝑇𝑎𝑛𝐵


6) Statistics Types of Studies: Observational study: An observational study is a study in which a researcher observes behaviour without influence.

Example: Traffic on a road. Advantage: Can be inexpensive. Disadvantage: Reliance on subjective measurement (ie: the researcher may be biased).

Designed experiment: A designed experiment is an experiment in which a treatment is applied and the researcher observes the effects. Example: Testing of drugs.

Advantage: Researcher can control variables. Disadvantage: Problem of ethics. Sample survey: A sample survey is a study that obtains data from a sample of a population in order to estimate attributes of the population.

Example: Opinion poll on government parties. Advantage: Can be inexpensive.

Disadvantage: May include sampling error.

Census -v- Sample Survey A census is a survey of the whole population in question.

A sample survey is a study that obtains data from a sample of a population in order to estimate attributes of the

population.

Population/Sampling frame/Sampling units The population is the entire group being studied, eg. students of UCD.

(Note: The word population doesn’t just mean the number of people in a country)

The sampling frame is the actual list of sampling units from which the sample is selected, eg. Those on the

secretaries computer in the science building.

The sampling units is the set of elements chosen from the sampling frame, eg. The actual ones chosen to be

surveyed.

Population parameter -v- Sample statistic A population parameter is a numerical measure from the entire population.

A sample statistic is a numerical measure from a sample of the population.

Sampling Distribution: A sampling distribution is a distibution of statistics obtained through a large number of same size samples drawn

from a specific population.


Types of sample surveys: Simple Random Sampling: A simple random sample gives each member of the population an equal chance of being chosen. Example: Random names can be pulled from a hat. Advantage: Can be highly representative. Disadvantage: Expensive, as those sampled may be spread over a large area. Stratified Random Sampling: The population is divided into subgroups, so that subjects within each subgroup have a common characteristic. Then a simple random sample is drawn from each subgroup according to their proportion of the population, to make the full sample. Example: Which team is going to win the All-Ireland this year? Here we would break the country into county subgroups and take a simple random sample from each. Advantage: Eliminates potential bias. Disadvantage: May be difficult in identifying appropriate strata. Systematic Random Sampling: This is random sampling with a system. From the sampling frame, a starting point is chosen at random, and thereafter at regular intervals. Example: Suppose you want to sample 8 houses from a street of 120 houses. 120/8=15, so every 15th house is chosen after a random starting point between 1 and 15. If the random starting point is 11, then the houses selected are 11, 26, 41, 56, 71, 86, 101, and 116. Advantage: Easier to conduct than a simple random sample. Disadvantage: The system may interact with some hidden pattern in the population, e.g. every third house along the street might always be the middle one of a terrace of three. Cluster Sampling: The population is divided into clusters, and some of these are then chosen at random. Then units are chosen within each cluster. Example: Households in the same street. Advantage: Saving of travelling time, and consequent reduction in cost. Disadvantage: Units close to each other may be very similar and so less likely to represent the whole population. Quota Sampling: In quota sampling the selection of the sample is made by the interviewer, who has been given quotas to fill from specified sub-groups of the population. Example: An interviewer may be told to sample 50 females between the age of 45 and 60. Advantage: Quick and cheap to organize. Disadvantage: Because the sample is non-random, there could be bias. Convenience Sampling: This is a non-probablity sampling method where subjects are chosen in the most convenient way. Example: Just ask family members. Advantage: Quick and cheap. Disadvantage: Could be very biased. Other Definitions:

1. Descriptive statistics: use of graphs, tables, charts and various measurements/calculations to organise and summarise information.

2. Inferential statistics: a sample or portion of the population is taken and then conclusions are made about the entire population.

3. Variable: characteristic of interest in each element of the sample or population.

4. Observations: value of a variable for ONE PARTICULAR element of the sample.

5. Data set: value of all observations of a variable for the elements of the sample.

6. Data capture: process by which data is transferred from paper copy eg: questionnaire to an electronic file eg: a computer.

7. Primary data: data collected by the organisation/person who is going to use it.


8. Secondary data: data already available not collected by the person/organisation who is going to use it. (newspapers, books, historical, records, internet)

9. Univariate data: only one item of information is collected from each member of a group eg: height or annual salary or eye colour.

10. Bivariate data: two items of information are collected from each member of a group eg: height and weight of a person or starting salary and number of years in education.

11. Outlier: very high or very. low value not typical of the other values in the data.

12. Explanatory variable: this is the controlled variable whose effects on the response variable we wish to study.

13. Response variable: this is the variable whose changes we wish to study.

14. Control group: split a group into 2 parts to test a drug. Both think they are getting the drug but the second group are getting an inactive substance called a placebo. This group is called the control group.

15. Quantitative data (numerical): data that can be counted or measured. 2 types exist. Discrete: only have certain values in a given range eg: number of goals in a match, shoe size,

no. of tails when tossing a coin 30 times. Continuous: can take any value in a given range. eg: height, weight, time, rainfall

measurements.

16. Qualitative data (categorical): data that cannot be counted or measured or answered in numbers. 2 types exist.

Ordinal: data has an obvious order (pain level: none/low/moderate/severe, or restaurant service: poor/good/excellent, or exam results: A/B/C/D/E/F/NG).

Nominal data: data that cannot be ordered i.e. blood type A, B, O, AB or hair colour or favourite football team.

17. Questionnaires: set of questions designed to obtain data from a population.

18. Respondent: People who answer questionnaires.

19. Good qualities of a questionnaire: (a) be as brief as possible (b) use clear and simple language. (c) not cause embarrassment or offend (d) no leading questions (e) accommodate all possible answers (f) contains tick boxes

20. Margin of error: The margin of error is the maximum value of the radius of the 95% confidence interval.


7) Probability The fundamental principle of counting:

If one event has 𝑚 possible outcomes and a second event has 𝑛 possible outcomes, then the total number of possible outcomes is 𝑚 × 𝑛.

A sample space is the set of all possible outcomes of an experiment.

Mutually exclusive events: 𝐴 ∩ 𝐵 = 0

Two events are mutually exclusive if they cannot occur at the same time. eg. Rolling a dice and getting an even number and the number 5.

Independent events: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵) Two events are independent if the outcome of one does not affect the other . eg. Rolling a dice and tossing a coin.

4 requirements for Bernoulli:

1) There is a finite number of trials.

2) There are only two outcomes - success and failure.

3) Trials are independent of each other.

4) The probability of success is the same for each trial.


8) Geometry Theorems: You need to be able to prove Theorem 11, Theorem 12 and Theorem 13.

Theorem 11 – If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal. Given: Three parallel lines 𝑙, 𝑚 and 𝑛, intersecting the trasversal 𝑡 at the points 𝐴, 𝐵 and 𝐶 such that |𝐴𝐵| = |𝐵𝐶|. Another transversal, 𝑘, intersects the lines at 𝐷, 𝐸 and 𝐹 To Prove: |𝐷𝐸| = |𝐸𝐹| Construction: Through 𝐸, construct a line parallel to 𝑡 and intersecting 𝑙 at the point 𝑃 and at the point 𝑄. Proof: 𝑃𝐸𝐵𝐴 and 𝐸𝑄𝐶𝐵 are parallelograms

Then |𝑃𝐸| = |𝐴𝐵| and |𝐸𝑄| = |𝐵𝐶|…. Opposite sides of a parallelogram

But |𝐴𝐵| = |𝐵𝐶|

So, |𝑃𝐸| = |𝐸𝑄| (Side)

|⦟𝑃𝐸𝐷| = |⦟𝐹𝐸𝑄|…… Vertically opposite angles (Angle)

|⦟𝐷𝑃𝐸| = |⦟𝐹𝑄𝐸|…… Alternate angles (Angle)

∴ ∆𝐷𝐸𝑃 and ∆𝐹𝐸𝑄 are congruent (ASA)

∴ |𝐷𝐸| = |𝐸𝐹|

Q.E.D.

D

E

F

A

B

C Q

P

t k

l

m

n


Theorem 12 – Let 𝐴𝐵𝐶 be a triangle. If a line 𝐼 is parallel to 𝐵𝐶 and cuts |𝐴𝐵| in the ratio 𝑠: 𝑡, then it also cuts |𝐴𝐶| in the same ratio Given: A triangle 𝐴𝐵𝐶 and a line 𝑋𝑌 parallel to 𝐵𝐶 which cuts |𝐴𝐵| in the ratio 𝑠: 𝑡 To Prove: |𝐴𝑌|: |𝑌𝐶| = 𝑠: 𝑡 Construction: Divide |𝐴𝑋| into 𝑠 equal parts and |𝑋𝐵| into 𝑡 equal parts. Through each point of division, draw a line parallel to 𝐵𝐶 Proof: According to Theorem 11, the parallel lines cut off segments of equal length along |𝐴𝐶|. Let 𝑘 be the length of each of these equal segments.

|𝐴𝑌| = 𝑠𝑘 and |𝑌𝐶| = 𝑡𝑘

|𝐴𝑌| ∶ |𝑌𝐶| = 𝑠𝑘 ∶ 𝑡𝑘 = 𝑠 ∶ 𝑡

Q.E.D.

Y X

A

B C

Y X

A

B C

k


Theorem 13 – If two triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are similar, then their sides are proportional, in order: |𝐴𝐵|

|𝐷𝐸|=

|𝐵𝐶|

|𝐸𝐹|=

|𝐴𝐶|

|𝐷𝐹|

Given: The triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 in which |⦟1| = |⦟4|, |⦟2| = |⦟5| and |⦟3| = |⦟6|

To Prove: |𝐴𝐵|

|𝐷𝐸=

|𝐵𝐶|

|𝐸𝐹|=

|𝐴𝐶|

|𝐷𝐹|

Construction: Mark the point 𝑋 on [𝐴𝐵] such that |𝐴𝑋| = |𝐷𝐸|. Mark the point 𝑌 on [𝐴𝐶] such that |𝐴𝑌| = |𝐷𝐹| Join 𝑋𝑌 Proof: The triangles 𝐴𝑋𝑌 and 𝐷𝐸𝐹 are congruent (SAS)

∴ |⦟𝐴𝑋𝑌| = |⦟𝐷𝐸𝐹|…..Corresponding angles ∴ |⦟𝐴𝑋𝑌| = |⦟𝐴𝐵𝐶| ∴ 𝑋𝑌||𝐵𝐶

∴|𝐴𝐵|

|𝐴𝑋|=

|𝐴𝐶|

|𝐴𝑌| ….. A line parallel to one side divides the other side in the same ratio (Theorem 12)

∴|𝐴𝐵|

|𝐷𝐸|=

|𝐴𝐶|

|𝐷𝐹|

Similarly, it can be proven that ∴|𝐴𝐵|

|𝐷𝐸|=

|𝐵𝐶|

|𝐸𝐹|

∴|𝐴𝐵|

|𝐷𝐸|=

|𝐵𝐶|

|𝐸𝐹|=

|𝐴𝐶|

|𝐷𝐹|

Q.E.D.

A

B C

D

E F

1

2 3 5 6

4

A

B C

D

E F

X Y

1

2 3

5 6

5 6

4


Definitions: You need to know 9 geometry definitions

1) Theorem: A theorem is a rule that has been proved following a certain number of logical steps or by using a previous theorem or axiom that you already know. eg. The angles in a triangle add up to 180°

2) Axiom:

An axiom is a rule or statement accepted without any proof. eg. There are 360° in a full circle.

3) Corollary

A corollary is a statement that follows from a previous term. eg. One theorem states that in a parallelogram, opposite sides are equal and opposite angles are equal. A corollary of this is that a diagonal divides a parallelogram into two congruent triangles.

4) Converse The converse of a theorem is the reverse of a theorem.

eg. Theorem: If there are two equal angles in a triangle the triangle is isosceles. Converse: If a triangle is isosceles, there are two equal angles in the triangle. Sometimes a converse isn’t true. eg. Theorem: In a square, opposite sides are equal. Converse: If opposite sides are equal, then it is a square. This is not true, as it could be a rectangle!

5) Implies

To imply something is to use something we have proved previously. The symbol for implies is => or ∴

eg. 𝒂 + 𝒃 =𝟏𝟓

𝟑

𝒂 + 𝒃 = 𝟓

6) Proof

A proof is a series of logical steps we use to prove a theorem. eg. (You will only get asked this definition if you are asked to do one of the Theorems. Therefore you will use the theorem as your example for the proof).

7) Proof by Contradiction A proof by contradiction is a proof where we make an assumption, then prove it is false by using valid axioms. Alternatively you can state something is false and prove yourself wrong, meaning it is true. Example: Prove that a triangle cannot have 2 right angles. Assume a triangle CAN have 2 right angles. Now, the sum of the interior angles of a triangle is 180. ∴ 90 + 90 + 𝑐 = 180.

180 + 𝑐 = 180

𝑐 = 0. This implies that the third angle has to be 0. However, if the third angle is 0, this means that all 3 points of the triangle are collinear (in a straight line). Therefore, by contradiction a triangle can't have 2 right angles.


8) Is Equivalent to This means that something has the same value or measure as, or corresponds to, something else.

Eg: 5

2 has the same value as 0.4

9) If and only if

This means that one thing is true (or false) only if another thing is true (or false). Eg: A parallelogram is a rhombus if, and only if, all four of its sides have the same lengths. Constructions:

At Leaving Cert Higher Level there are 22 constructions. These must be learned off by heart. The best way to do this is by looking at a video of the constructions being done. These are available on the Dublin School of Grinds website: www.dublinschoolofgrinds.ie/constructions

http://www.dublinschoolofgrinds.ie/constructions


9) Co-ordinate Geometry of the Circle Prove that 𝑔2 = 𝑐 when x-axis is tangent

From diagram: radius = | − 𝑓|

From log tables: radius = √𝑔2 + 𝑓2 − 𝑐

=> √𝑔2 + 𝑓2 − 𝑐 = | − 𝑓|

Square both sides: 𝑔2 + 𝑓2 − 𝑐 = 𝑓2 =>𝑔2 = 𝑐 Prove that 𝑓2 = 𝑐 when y-axis is tangent

From diagram: radius = | − 𝑔|

From log tables: radius = √𝑔2 + 𝑓2 − 𝑐

=> √𝑔2 + 𝑓2 − 𝑐 = | − 𝑔|

Square both sides: => 𝑔2 + 𝑓2 − 𝑐 = 𝑔2 => 𝑓2 = 𝑐

Documents

Banker s Bible - Dublin Academy...2016/02/06 · Banker’s Bible "Nothing consoles and comforts like certainty does." ~Amit Kalantri, Indian Book Author. No part of this publication