**Year 13 Further Maths homework to Wednesday 1 November**

This worksheet is designed to give you one or two questions covering every important idea we’ve covered this term, mostly with a bit of explanation or revision notes before the question(s).

Every question is taken from the textbook so, if stuck, you can check with Solution Bank. (You can of course email me, too).

The worksheet will prepare you for the test, which you’re also asked to do outside class time, between now and 1 November. The test will cover exactly the same ground.

**Year 12 Further Maths homework to Wednesday 1 November**

Do ten questions from this question sheet:

Some of the questions you’ve done for your 18 October homework, and some you’ve done for your 20 October test. Please select some new ones.

Q.21 looks like there’s a typo in it, and they mean to describe the equation as:

x^{4} + ax^{3} + bx^{2} + cx + d = 0

I think Q.17 may have a typo in it, too. The square root involves a complicated surd. With the calculator method you can only get an approximate square root: you can’t see that the decimal you get on the calculator is in the way you can see that 1.414213562 is √2. On the other hand, maybe they only want an approximate answer in decimals: they don’t say: give the answer as an exact surd. I’d skip Q.17.

Other than that: choose questions nearer the end of the question sheet if you’re confident and want to push yourself, choose questions nearer the start if you’re feeling less confident.

I’ll post worked answers for all the questions here soon.

**Year 13 Further Maths homework to Wednesday 18 October**

Part B: Watch this video about Möbius transformations

and write about 50 words to put down on paper what you have learned from it.

Finish Activity 8 in your Möbius transformations workbook. Sketch the locus (path) of w for each of the three transformations listed if the locus (path) of z is |z|=1 (i.e. the circle with radius 1 and centre the origin). Preferably, use this applet to help you:

http://www.math.ucla.edu/~tao/java/Mobius.html

(The applet won’t run in Chrome, but it will in Firefox or Safari, as long as you update your Java installation).

Part C: Ex.3E Q.3 (roots of complex numbers). Ex.6F Q.3 (Maclaurin and Taylor series).

Part A: Bolaji – For what range of z is the Maclaurin series for ln(1+z) valid? For what range of x is the Maclaurin series for ln(1 + 2 cos x) valid?

Brenda: Complete Q.1 and 2 of Ex.3E. Do selected bits from the homework of the last couple of weeks which you’ve missed.

Genie: Complete Q.1 and 2 of Ex.3E. Do selected bits from the homework of the last couple of weeks which you’ve missed.

Michelle: No Part A.

Mohaned: Complete Q.1 and 2 of Ex.3E. Do selected bits from the homework of the last couple of weeks which you’ve missed.

Onesimus: No Part A.

Umut: No Part A.

**Year 12 Further Maths homework to Wednesday 18 October**

No Part A or Part C, just this worksheet:

Complex numbers homework for 18 October.

**Year 12 Further Maths homework to Wednesday 11 October**

Part A: Abass – redo Q.10 and 11 from p.23

Aniqa – redo Q.6, 7, 8 from p.23

Callum – complete p.23 Q.13

David – redo p.26 Q.5 and Q.7

Enoch – finish p.26 from booklet

Helen – no Part A

Howard – finish p.26 from booklet, Q.5-6-7. Find √(4+4i) by using the Classwiz calculator.

Jeffrey – redo Q.8 and 9 from page 23 of booklet.

Lara – p.23 Q.1, 2, 3, 8; p.26 Q.1, 4, 5.

Obi – redo p.23 Q.10 and p.26 Q.1

Mya – Find quadratic equations with roots:

a. 5 ± 7i

b. 7 ± 5i

Reece – Redo p.23 Q.10 and 11

No Part C this week.

Part B: complex numbers workbook p.28. Click here for an answer sheet. Please mark your own work before handing it in.

**Year 13 Further Maths homework to Wednesday 11 October**

No Part A this week (but Mohaned, Brenda, and Genie need to do a lot of catch-up)

Part B: Complete Ex.3E Q.1, 2 and 6

Write out the Edexcel proof of De Moivre’s Theorem (p.28 and 29 of the textbook) as concisely and briefly as you can (while still being satisfactory to an Edexcel examiner).

Complete Activity 6 in the Möbius transformations booklet

Part C: Ex.6F Q.2

**Year 12 Further Maths homework to Wednesday 4 October**

Please hand in your homework on Wednesday morning. Please mark Parts B and C before handing in your work: click here for answer sheet. Thanks to Mya for spotting the error in Q.5c on p.26 – the roots should be given as − 1 ± 9i.

Part B

Finish pages 23 and 26 of your complex numbers workbook, https://mathsmartinthomas.wordpress.com/2017/09/03/classwork-and-homework-for-as-further-maths-complex-numbers/

Part C

Do these calculations, using a Classwiz calculator or the online calculator at https://www.intmath.com/complex-numbers/convert-polar-rectangular-interactive.php. Use radians. “Cartesian form” means like a+b**i**. “Modulus-argument form” means as enlargements-and-rotations – which you can write equally correctly in any of these three ways, r cis θ or r (cos θ + i sin θ); or r ∠ θ

Part A

**Javonne**, **Jeffrey** – using a Classwiz calculator or the online calculator at https://www.intmath.com/complex-numbers/convert-polar-rectangular-interactive.php, find modulus (= enlargement = r) and argument (= angle = rotation = θ = ∠) **in radians** for these complex numbers. Check your answers against the answer sheet here.

**Abass**, **Aniqa**, **Reece** – Draw diagrams of these complex numbers (using a ruler). The angles are in radians.

3 cis π/4

4 cis (-π/3)

2 cis 4π/3

**David**, **Helen**, **Lara** – complete these. (Ask me or another student if you get stuck).

**Billie**, **Callum**, **Enoch**, **Howard**, **Mya**, **Obi** – no part C

**Year 13 Further Maths homework to Wednesday 4 October**

Part B: Finish textbook Ex.3D Q.1, 3, 4 and Review Exercises 1 Q. 31 and 42. (Please mark your work before handing it in).

Part C: Ex.6F from FP2 book, Q.1

Part A: **Tegan** – (a) Find a Maclaurin series for cos 2x (terms up to x^{4}) by a shortcut; (b) rework Review Exercises 2 Q.44 using what I’ve written in your book.

**Michelle** – complete Review Exercises 2 Q.37 and Q.44 using what I’ve written in your book

**Umut** – complete Review Exercises 2 Q.37

**Bolaji** – Work through Review Exercises 2 Q.44 using what I’ve written in your book.

**Onesimus** – rework Q.37 and Q.44 of Review Exercises 2 using my written comments in your book.

**Mohaned** – Review Exercises 2 Q.37 and 38

**Brenda** – complete the homework for 27/9/17 which you didn’t complete because of illness.

**Genie** – No Part A.

**Year 12 Further Maths homework to Wednesday 27 September**

No Part A or Part C this week, just Part B. Complete these calculations. Use either a Classwiz calculator or the online calculator at https://www.intmath.com/complex-numbers/convert-polar-rectangular-interactive.php. Draw a little diagram for each conversion, using a ruler. Give answers for arguments in radians, not degrees.

**Year 13 Further Maths homework to Wednesday 27 September**

No Part A or Part C this week, just Part B, which is to complete questions 36, 37, 38, 39, 43 and 44 from Review Exercises 2 in the FP2 textbook. Since your textbook and Solution Bank will tell you answers, please make sure to mark your work yourself before handing it in.

**Year 12 Further Maths homework to Wednesday 20 September**

Part A: click here: it’s different for each individual. None for Mya or Haymam.

Part B: cis θ is short for cos θ + i sin θ. Do these calculations:

Part C: Calculate these values

sin 0

cos 0

sin 30

cos 30

sin 60

cos 60

sin 90

cos 90

**Year 13 Further Maths homework to Wednesday 20 September**

**Part A:** to come, in individual notes to each of you.

**Part B:** finish pages 27 and 30 of the Maclaurin workbook.

Page 27

Exercise 6E, page 122. Q.2. The variable y satisfies

and at x=0

Find a series expansion for y in ascending powers of x up to and including the term in x^{3}

Q3. Given that y satisfies the differential equation

and that at x=0, y=2, find a series solution for y in ascending powers of x up to and including the term in x^{3}

Page 30

(Worked example 8 from textbook, p.113): Given that terms in x^{n} may be neglected for n>4, show that:

[Note: this is done in the book in a different way from how we did in class, but either way is fine]

Find the first few terms of a Maclaurin series for

Q.8 from textbook, p.114: Expand

in ascending powers of x as far as the term in x^{3} by considering the product of the expansions of sin x and (1−x)^{-2}

[Note: this means: write out the Maclaurin series for

sin x

and for

(1-x)^{−2}

up to the terms in x^{3}, and then do the multiplication of the two series as if they were polynomials.

When you multiply, the constant term is zero (because zero constant term in the series for sin x), and the coefficient of the x^{3} term is:

3 (from x term in first series times x^{2} term in second) − (from x^{3} term in first series times constant term in second)

You need to find the coefficients of the x and x^{2} terms.]

[Further note: the first answer on page 26 of your Maclaurin workbook is wrong, sorry, because I copied it from the textbook, but I also slightly changed the question when copying from the textbook. The answer we got in class is right. I’ve changed the pdf of the workbook on this website].

**Part C:** Differentiate these using the chain rule and the product rule

(1−x)^{−3}

(1−e^{x})^{−1}

x^{3}e^{x}

**Year 12 Further Maths homework to Wednesday 13 September**

**Part B**: finish off pages 5, 6, and 13 of your complex numbers workbook

**Part C**: do these surd calculations

**Year 13 Further Maths homework to Wednesday 13 September**

**Part B: finish off pages 22, 23, 24, 29 of the Maclaurin workbook, as below**

Your basic formulae for all this are

Calculate the Maclaurin series up to the x

^{5}term for

is also the solution to the differential equation

with y=1 and when x=0. Get the same Maclaurin series from the differential equation.

Calculate the Maclaurin series for

Use these approximations to calculate

and compare your answers with what you get from your calculator.

(Q.4 from textbook, p.111): Using the series expansions for

and

respectively, find, correct to 3 decimal places, the value of:

(a)

(b)

(Q.5 from textbook, p.111): Use Maclaurin’s expansion and differentiation to expand, in ascending powers of x up to and including the term in x

^{4}

(a)

(b)

(c)

[Note: means ]

**Part C**

Use the product rule and the chain rule to differentiate:

y = sin x . cos x

y = x . sin x

y = x^{2} . cos x

y = exp (x^{2})

**Year 13 (year 12 as was) Further Maths homework to Friday 21 July**

Abdal is test-driving and will present a report (on 21 July) about the Casio Classwiz calculator.

The rest of you have borrowed books, and will present a two-minute report on them (as far as you’ve read in them, anyway) on 21 July.

Ashley: Love and Math

Bolaji: How many socks

Brenda: From zero to infinity

Genie: 50 mathematical ideas

Michelle: The Black Swan

Mohaned: Love and Math

Onesimus: Adam Spencer’s Book of Numbers

Rashi: Statistical Methods for Economics

Tegan: Love and Math

Umut: Finding Moonshine

You can also have a look at what we did on Friday 23 June about Maclaurin series, and follow that investigation further, at this web page

**Year 13 Further Maths homework to Monday 21 June**

Please mark every past paper before handing it in. You can hand it in to either Mr Osborn or Mr Thomas, and Mr Thomas will check it and provide feedback and help.

**For Wednesday 24 May**

● FP3 practice paper A, FP3 practice paper B, in the main FP3 past-papers booklet

● FP2 June 2013 “withdrawn” paper and mark scheme

● FP2 “extra” paper 4 – https://mathsmartinthomas.wordpress.com/2017/04/14/extra-fp2-practice-papers/

**For Monday 5 June**

● FP3 sample paper, in the booklet

● FP3 “extra” paper A, from Booklet of extra FP3 practice papers (mark schemes for the questions which make up these papers are in your main FP3 past-papers booklet https://mathsmartinthomas.files.wordpress.com/2016/02/fp3_frost.pdf)

● FP2 “extra” papers 5 and 6 – https://mathsmartinthomas.wordpress.com/2017/04/14/extra-fp2-practice-papers/

(FP2 exam is Wednesday 7 June)

**For Wednesday 14 June**

● FP3 “extra” papers B, C, D, E from Booklet of extra FP3 practice papers (mark schemes for the questions which make up these papers are in your main FP3 past-papers booklet https://mathsmartinthomas.files.wordpress.com/2016/02/fp3_frost.pdf)

**For Wednesday 21 June**

● FP3 “extra” papers F, G, H, I from Booklet of extra FP3 practice papers (mark schemes for the questions which make up these papers are in your main FP3 past-papers booklet)

(FP3 exam is Monday 26 June)

**Other practice material**

● Other FP3 “extra” papers from booklet of extra FP3 practice papers (mark schemes for the questions which make up these papers are in your main FP3 past-papers booklet)

**Wednesday 10 May 2017**

Please mark your work before handing it in.

**Year 12 Further Maths**

• Mark the work you did on Friday on the M2 work-energy past-paper questions (there are mark schemes in the booklet, and examiners’ reports)

• Do two “F1 IAL” past papers, preferably the 2017 and 2016 ones in the booklet of F1 IAL papers

**Year 13 Further Maths**

• FP3 past paper June 2013 R

• FP2 “extra” paper 3

• FP2 “extra” paper 4

**OR** instead of the second FP2 “extra” paper

• Work through the vectors practice sheet (pdf). If you’re not sure about an answer, check the textbook or email me. Then take another copy of the practice sheet, and work through the questions again, without looking at your first effort. Repeat until the basic facts about 3-dimensional vectors are lodged in your long-term memory.

**Wednesday 3 May 2017**

**Year 12 Further Maths**

Finish marking the M2 statics past-paper questions you did in class

Do:

FP1 past paper January 2014 I

FP1 past paper June 2014 R

**Year 13 Further Maths**

FP3 past paper June 2014 R

FP2 “extra” paper 1

FP2 “extra” paper 2

Plus: everyone bar Victor and Wei Kong can see gaps in the list of past papers you’ve done. Start to fill in the gaps by doing an extra

paper?https://docs.google.com/spreadsheets/d/16rciWX8H06UOfZ40rg1SLzM5-l74E9tqi2TaMVTg9tE

**Wednesday 26 April 2017**

**Year 12 Further Maths**

Mark the FP1 past paper January 2012 you did in class

Do:

FP1 past paper January 2013

FP1 past paper June 2013

**Year 13 Further Maths**

FP3 past paper June 2015

F2 IAL past paper June 2014

F2 IAL specimen paper (or, if you’ve done that, one of the past papers already covered that you have missed)

**Wednesday 19 April**

The homework is all past papers. Please follow the advice at:

https://mathsmartinthomas.wordpress.com/2016/02/19/six-points-about-doing-past-papers/

If mark schemes for these papers are not in your booklets, you can find them at:

http://www.physicsandmathstutor.com/a-level-maths-papers

**Year 12 Further Maths**

FP1 January 2011

FP1 June 2011

FP1 June 2012

available online here: https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_frost1.pdf

**Year 13 Further Maths**

F2 IAL June 2016

F2 IAL June 2015

available online here: https://mathsmartinthomas.files.wordpress.com/2016/02/fp2-ial-papers1.pdf

FP3 Past Paper A

FP3 Past Paper B

available online here: https://mathsmartinthomas.files.wordpress.com/2016/02/fp3_frost.pdf

**Wednesday 29 March**

Please mark your work before handing it in.

**Year 12 Further Maths**

Past paper: June 2010 FP1. Thanks to Vinh, who did this paper for 22 March, and should do January 2010 for 29 March, for pointing out a typo. Q.3 should read:

f(x) = x^{3} − ^{7}⁄_{x} + 2

not

f(x) = x^{3} − ^{7}⁄_{2} + 2

Also: please finish the statics question you were on in class. No extra questions. Just finish the one you were on.

Writing maths: do the “writing maths” worksheet

https://mathsmartinthomas.files.wordpress.com/2017/03/170305writingmaths-homework.pdf

This is an exercise in rewriting answers in which the working is correct, but the writing-out is confusing. You’re asked to rewrite, especially addding a few words and diagrams, so as to make the answers clear, concise, and convincing.

**Year 13 Further Maths**

Past paper: June 2009 FP2

Writing maths: do the “writing maths” worksheet

https://mathsmartinthomas.files.wordpress.com/2017/03/170305writingmaths-homework.pdf

Writing maths and conics: write out *one* of the solutions to Ex.2G Q.4 part (b) (assuming part (a)) as clearly and concisely as you can. Here are my writings-out of the three solutions: see if you can improve on them.

Write out, as clearly and concisely as you can, *two* proofs of the fact that if A and B are foci of an ellipse (S′ and S in the diagram below)

and P is a point on an ellipse, than PA+PB=2a

One, by using the fact that every point P on the ellipse has

distance from the focus = e × distance from the directrix

as shown in Example 16, p.37 in the textbook.

Two, by putting

P = (a cos θ, b sin θ), A = (-ae, 0), B = (ae, 0), b^{2} = a^{2} (1 − e^{2})

and doing the algebra.

**Wednesday 22 March**

**Year 12 Further Maths**

Statics: Complete Q.21-24 of Review Questions (p.155-6). You should have done most of these in class. Worked answers are available at https://mathsmartinthomas.wordpress.com/2016/02/05/m2-statics-worked-answers-for-q-21-27-of-review-exercises/ or on Solution Bank. David and Danny will do Q.25-28, and of course if anyone else is up for doing more, that’ll be good.

Past paper: January 2010 FP1

**Year 13 Further Maths**

FP3 conics: Complete the booklet of FP3 conics past-paper questions: https://mathsmartinthomas.files.wordpress.com/2017/03/fp3conics-q-1.pdf

FP2 past paper: June 2010

**Wednesday 15 March**

**Year 13 Further Maths**

Please mark your work before handing it in.

FP3 conics: Read

https://mathsmartinthomas.wordpress.com/2017/03/08/fp3-conics-ellipses-and-hyperbolas/

and

Complete Ex.2A Q.1-2, Ex.2B Q.1-5, Ex.2C Q.2, Ex.2D Q.1-2.

FP2 past paper: June 2011

**Year 12 Further Maths**

Statics: complete Ex. 5B (p.136) Q. 1, 2, 3, 4, 5, 8.

Draw a large diagram, with a ruler, for every problem.

Danny and David are doing Review Exercises 2 Q.21-25 instead. Y12 students who feel confident can attempt Ex.5E Q.14, or even those Review Exercises if they like. Worked answers for those Review Exercises at https://mathsmartinthomas.files.wordpress.com/2016/02/statics-q-21-27.pdf

FP1 past paper: January 2009

**Wednesday 8 March**

**Year 12 Further Maths**

June 2009 FP1 past paper from your book of past papers: https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_frost.pdf

If you can, arrange to do it together with your “partner” or another student. If not, consult your “partner” whenever you get stuck.

Rashi and Ashley

Vinh and Peter

Onesimus and Brenda

Tegan and Mohaned

Hannah and Genie

Abdal and Bolaji

Michelle and Umut

**Year 13 Further Maths**

Finish off these vectors examples

Ex.5B Q.6, 7, 8, 9, 12 (areas)

Ex.5C Q.1, 2, 3, 5, 6 (volumes)

Angle between planes: Ex.5F Q.4,5; Ex.5G Q.11. Angle between lines Ex.5G Q.16

Angle between line and plane: Ex.5F Q.8; Ex.5G Q.12

Distance from point to plane: Ex.5F Q.9, Ex.5G Q.6, Q.13 (distance between a line parallel to a plane, and the plane, is the same as the distance from any point – might as well be the “** a**” point – on the line to that plane); Review Exercises 2 Q.16 p.195

Distance between parallel planes: Ex.5F Q.10

Distance between skew lines: Ex.5F Q.11, Ex.5G Q.1, 2, 3

Distance from point to line, and distance between parallel lines: Ex.5F Q.12; Review Exercises 2 Q.19 p.196

June 2012 FP2 past paper

**Homework for 1 March**

Please mark your work before handing it in.

**Year 12 Further Maths**

Complete these questions on work, energy, and power: Ex.3D Q.10, 14, 15. That’s all you have to do, but if you want more practice before the test, do Ex.3E Q.12, 13, 15 and Review Exercise page 96 Q. 43, 44.

Past paper: FP1 May 2015, from your booklet of past papers.

Revised “partners” for doing past papers

Rashi and Ashley

Vinh and Peter

Onesimus and Brenda

Tegan and Mohaned

Hannah and Genie

Abdal and Bolaji

Michelle and Umut

**Year 13 Further Maths**

Complete: Eigenvalues, eigenvectors, diagonalisation

Review Exercises 2 from textbook, Q. 36, 37, 38, 40

and make sure you’ve done Ex.6G Q.1-3 if you haven’t already

FP2 past paper this week: June 2013

**Homework for 22 February**

Please mark your work before handing it in.

**Year 12 Further Maths**

1. Newton-Raphson

Finish Ex.2C Q.6, Ex.2D Q.4, Ex.2D Q.8 using 2.4 as your first guess for the solution

2. Parabolas and hyperbolas

Review Exercises 1 Q. 57, 60, 61, 62

3. Work-energy

Finish Ex.3B Q.1, 2

Ex.3C Q.1, 2, 5, 6, 15, 17

4. Past paper

FP1 May 2016, from your booklet of past papers.

Quick answer sheet in the booklet

Worked answers here: https://mathsmartinthomas.files.wordpress.com/2016/05/fp1-may2016-worked-answers.pdf

Mark scheme here: https://mathsmartinthomas.files.wordpress.com/2016/05/1605_fp1_may_2016_mark_scheme-1.pdf

**Year 13 Further Maths**

1. Vectors

Finish

Ex.5A Q.1 (g) to (j); Q.3; Q.4

Ex.5D Q.1

Ex.5E Q.6

Ex.5F Q.3

Ex.5G Q.9 (a), (b)

Review Exercises 2 from textbook, Q.1, 5, 9 (a), (b), (c), 10

2. Eigenvalues, eigenvectors, diagonalisation

Review Exercises 2 from textbook, Q. 36, 37, 38, 40

and make sure you’ve done Ex.6G Q.1-3 if you haven’t already (there’ll be questions just like that in the mock)

3. Two FP2 past papers

**June 2014**

The one which starts with: Q.1 (a) Express ^{2}⁄_{(r+2)(r+4)} in partial fractions.

Mark scheme at: http://pmt.physicsandmathstutor.com/download/Maths/A-level/FP2/Papers-Edexcel/June%202014%20MS%20-%20FP2%20Edexcel.pdf

**June 2013 (R)**

Which starts: Q.1. A transformation T from the z-plane to the w-plane….

Mark scheme at: http://pmt.physicsandmathstutor.com/download/Maths/A-level/FP2/Papers-Edexcel/June%202013%20(R)%20MS%20-%20FP2%20Edexcel.pdf

**Homework for 8 February**

Please mark your work before handing it in.

**Year 13 Further Maths**

Eigenvalues, eigenvectors, diagonalisation

Complete Ex.6F (all, up to Q.10) and Ex.6G Q.1-3

Past paper to do this week: June 2014 (I) (i.e. not June 2014 (R), which you’ve already done, or plain June 2014)

Victor and Wei Kong have done the past paper already. Others:

Monday: Nii

Tuesday: Jetmir

Wednesday: Tobi, Taija

Friday: Alex and Conner

**Year 12 Further Maths**

Finish off the parabola and hyperbola past-paper questions. These questions are more difficult than you’re used, so don’t hesitate to ask other students, or email me, if you get stuck.

Click here to get the worksheet with the past-paper questions

**Homework for 1 February**

Please mark your work before handing it in.

**Year 12 Further Maths**

Parabolas and hyperbolas: finish off

Ex.3C Q.1-4

Ex.3D Q.1-3

Ex.3E Q.1-4

**Year 13 Further Maths**

Finish off Ex.6D Q.4-7 (if you haven’t already)

Ex.6E Q.1, 2, 3, 6

Past paper to do this week: June 2015

Monday: Nii; Victor and Wei Kong

Tuesday: Jetmir

Wednesday: Tobi, Taija

Friday: Alex and Conner

Your booklet contains basic answers for the June 2015, but only basic answers: full June 2015 mark scheme here

**Homework for 25 January**

**Year 12 Further Maths**

One: finish off any curve-sketches you were doing in the lesson but couldn’t finish because we went on to the next bit of work. The formulas we were sketching are all at:

https://mathsmartinthomas.wordpress.com/2014/08/22/classwork-and-homework-for-fp1-lessons-21-26/

Two: finish off questions 4 to 7 of exercise 3A. (Don’t bother with Q.1-3).

Three: rework all the questions from your mock paper where you made mistakes or bombed out.

The paper, and the mark scheme, and worked answers, are at:

**Year 13 Further Maths**

Determinants, inverses, and multiplication of 3×3 matrices: please do Q.22, 24, 26 (parts a and b only) and 27 (parts a and b only) from the second Review Exercises in the FP3 book.

Past-paper work: this week you are doing the FP2 Mock Paper from your past-paper booklet.

Tuesday: Jetmir

Wednesday: Taija

Thursday: Victor

Friday: Nii, Alex, Conner, Tobi, Wei Kong

It’s your job to make sure you have the paper and you organise a place to do it quietly, giving yourself 90 minutes.

It’s also your job to mark your paper once you’ve done it and to leave the marked paper in my blue tray or with me to check.

**Homework for 18 January**

Please mark your own work before handing it in.

**Year 12 Further Maths**

1. Consolidation of the work we did today on different ways of picturing complex numbers

These names can be divided into two groups: some describe different ways of looking at one particular quantity, and some of them different ways of looking at a different quantity.

List the two groups

modulus

argument

magnitude

direction

angle

enlargement

rotation

square root of determinant

2. Examples to get used to the ideas we learned today

Q.24 p.66

Q.26 p.66

Ex.4E Q.3 (you’ve done this before, but you may now find it easier)

Ex.1I Q.9 (you need to switch into geometry here: it’s a question Y12 students usually find difficult)

3. Consolidation on transforming triangles with matrices

Ex.4I Q.1 and 2

(You can use my worked answer to question 7(ii) from the mock to help you. All the questions of this sort are pretty much the same once you are used to them.)

**Year 13 Further Maths**

**Polar coordinates**

Finish Q.62 from Review Exercises 2, p.153-4

Finish whatever other question from the Review Exercises you were last doing in class

Do one more Review Exercise question on polar coordinates (probably Q.58, p.153, if you got up to Q.56 in class, Q.55 or 56, p.152, if you didn’t)

**FP2 past papers:** do or redo these papers, giving yourself 90 minutes in as near to exam conditions as you can manage, and marking your own work before handing it in.

Tobi: June 2014(R)

Nii: June 2014 (R)

Jetmir: to bring me his practice papers on 13 January, and to re-do one of them (of his choice)

Conner: June 2014 (R)

Wei-Kong: Practice Paper B

Alex: June 2014 (R)

Victor: June 2014 (R)

Taija: Practice Paper A

As from next week, 18-19 January, all of you will stay late on either Thursday or Friday to do an FP2 past paper.

Thursday: Jetmir, Victor

Friday: Nii, Alex, Conner, Tobi, Wei Kong

To be decided: Taija

**Homework for 11 January**

Please mark your own work before handing it in.

**Year 12 Further Maths**

Your choice of homework: a past paper from the past-papers booklet (skipping the Newton-Raphson, parabolas, and hyperbolas questions), or whatever you prefer to practise to get ready for the mocks. Whatever you do, *please mark it yourself* and *bring it to me to check* on Wednesday.

**Year 13 Further Maths**

Polar coordinates: finish off

Ex.7C Q.1-2

Ex.7D Q.1-9

Ex.7E Q.2

In addition, each of you has work yet to complete from the holiday assignment:

Practice Paper A, and mark scheme

Practice Paper B, and mark scheme

June 2014 (R) paper and mark scheme.

See my comments on the 4 January homework for what each of you, individually, should do.

**Homework for 4 January**

**Year 13 Further Maths**

First re-do the test you did on 15 December.

Check your work against the mark scheme and my worked answers, spot where you went wrong, then do this same paper again straight away at the start of the holidays, and mark it yourself.

(Then we’ll do the exact same paper again at the start of the January term, before going on to polar coordinates).

Paper at:

Worked answers at:

Mark scheme at:

Over the rest of the holidays, please do these three papers from your FP2 past-papers pack:

Practice Paper A

Practice Paper B

June 2014 (R)

These are also available online at:

Practice Paper A, and mark scheme

Practice Paper B, and mark scheme

June 2014 (R) paper and mark scheme.

Give yourself 90 minutes for each one. Skip the polar coordinates questions, since we haven’t covered that yet.

Then give yourself another 90 minutes to:

- do any problems you ran out of time on
- check the answers against the mark scheme, and mark your work. Calculate your percentage and write in into your FP2 past-papers pack.
- using the mark scheme, work out how to do any problems you got stuck on, or made mistakes in
- email me at mthomas@cityacademy.co.uk for help if and when you’re still baffled

**Year 12 Further Maths**

Please do these three made-up “practice papers”. Give yourself 90 minutes for each one.

Then give yourself another 90 minutes to:

- do any problems you ran out of time on
- check the answers against the back of the book, and mark your work. Give yourself 4 for each completely correct answer, 2 for half-correct, 1 for quarter-correct, etc., so you get a total mark of out 40
- using Solution Bank, work out how to do any problems you got stuck on, or made mistakes in
- email me at mthomas@cityacademy.co.uk for help if and when you’re still baffled

The second 90 minutes is as important as the first 90 minutes. Of course, if you get all the questions completely right straight off, then the “second 90 minutes” may be less than 90 minutes.

That makes 9 hours’ FP1 work altogether, out of the 408 hours of the Xmas holidays. Please hand in the work to me in the first week of the January term.

**Practice paper 1**

Review Exercises 1. Q.5 – complex numbers

Exercise 1I, Q.6 – complex numbers

Review Exercises 1. Q.34 – numerical methods

Review Exercises 2. Q.3 – matrices

Q.14 – matrices

Q.22 – series

Q.39 – induction (series)

Q.54 – induction (divisibility)

Q.45 – induction (sequences)

Q.50 – induction (matrices)

**Practice paper 2**

Review Exercises 1. Q.7 – complex numbers

Exercise 1I, Q.4 – complex numbers

Review Exercises 1. Q.45 parts a and c – numerical methods

Review Exercises 2. Q.5 – matrices

Q.13 – matrices

Q.23 – series

Q.41 – induction (series)

Q.55 – induction (divisibility)

Q.48 – induction (sequences)

Exercise 6E, Q.2 – induction (matrices)

**Practice paper 3**

Ex.1I, Q.7 – complex numbers

Review Exercises 1. Q.1 – complex numbers

Ex.2D Q.6a and b – numerical methods

Review Exercises 2. Q.6 – matrices

Q.19 – matrices

Q.24 – series

Q.43 – induction (series)

Q.46 – induction (divisibility)

Ex.6E Q.5(a) – induction (matrices)

Ex.6E Q.4 – induction (sequences)

Homework for 13-14 December and work for “study day” 9 December

**Year 13 Further Maths**

Finish off these pages in the “self-scaffolding” worksheet, following up from the mock exam, which we worked on today, Thursday 8th.

De Moivre (Q.6 from exam, plus 47 and 61)

Complex roots (Q.1 from exam, 52 and 41)

1st order differential equation with integrating factor (Q.3 from exam, 37 and 6)

Mobius transformations (Q.5 from exam, 35 and 9)

Differences and series (Q.4 from exam, 12 and 17)

Inequalities (Q.2 from exam, 5 and 18)

That will leave us Q.8 (2nd order differential equation) and Q.7 (Taylor series) to do next Thursday morning.

To help you, there are worked answers on this site: click here for the worked answers

Please also do the following question:

QUESTION

4. (i) p dx/dt + qx = r, where p, q and r are constants.

Given that x = 0 when t = 0,

(a) find x in terms of t

(b) find the limiting value of x as t tends to infinity.

The answer to the question is:

(a) x = (r/q) [1 – exp(-qt/p)] assuming neither p and q is zero

(b) x tends to r/q as t tends to infinity (assuming that p and q are both positive or both negative)

**Year 12 Further Maths**

Homework for 13-14 December and work for “study day” 9 December

Finish Ex.1B Q.7-12

Ex.1C Q.3-4, 11, 16, 17

Ex.1F Q.1, 2, 3

Year 12 has a study day on 9 December 2016, so please spend the time you would have spent in Further Maths class:

1. Studying these notes about using formulas for sums of series

and doing Q. 2 and 4 of Ex.5F

2. Doing this FP1 “practice paper” compiled from Review Exercise questions in the textbook.

Then give yourself some more time to:

- do any problems you ran out of time on
- check the answers against the back of the book, and mark your work. Give yourself 4 for each completely correct answer, 2 for half-correct, 1 for quarter-correct, etc., so you get a total mark of out 40
- using Solution Bank, work out how to do any problems you got stuck on, or made mistakes in
- email me at mthomas@cityacademy.co.uk for help if and when you’re still baffled

**Practice paper**

Review Exercises 1. Q.4 – complex numbers

Q.14 – complex numbers

Q.31 – numerical methods

Review Exercises 2. Q.2 -matrices

Q.7 – matrices

Q.21 – series

Q.40 – induction (series)

Q.53 – induction (divisibility)

Q.44 – induction (sequences)

Q.47 – induction (matrices)

**Homework for Tue-Wed 29-30 Nov 2016**

**Year 13 Further Maths**:

Work through your Thursday test paper with the mark scheme and my comments, and do the extra practice recommended in the comments: see

Test paper, mark scheme, comments

Then work through your mock paper, done on 28 November, with the mark scheme.

FP2 paper and mark scheme for 28 November 2016 mock

**Year 12 Further Maths**

Work through your test paper from Friday 25 November with the mark scheme. Redo it so that you have answered every question perfectly.

**Homework for Tue-Wed 22-23 Nov 2016**

**Year 13 Further Maths**:

Please make sure it’s in my blue tray in 2B7 by close of school on Tuesday, or Wednesday before tutor group at latest. Please mark your own work before handing it in.

Second-order differential equations – substitutions and boundary conditions. Ex.5G Q.14

Taylor and Maclaurin series. Ex. 5E, Q.1 and 2

And: this practice paper for the mocks.

**Year 12 Further Maths**

Inverting matrices: complete Ex.4G Q.1-7 and all of Ex.4H

Induction proofs with matrices: complete Ex.6D

(you’ll have done some of this in class), plus

this practice sheet on radians (*only the first page* of the sheet handed out in class)

**Homework for Tue-Wed 15-16 Nov 2016**

**Year 12 Further Maths**:

Practise matrix multiplying with Exercise 4C, page 81, Q.2-8

Practise matrix addition with Ex 4A, p.75, Q.3, and Ex 4B, p.77, Q. 1-4. (You may want to look at p.76 to help with Ex 4B, and p.74 to help with Ex 4A)

Practise determinants with Ex 4I Q.4

Do the new factorisation practice sheet (click here to get it) – unless you were perfect with last week’s factorisation practice, in which case skip this and do another question of your choice from Ex.4I.

**Year 13 Further Maths**:

Differential equations. Ex.4E Q.6,7,11. Ex.5A Q.5,6. Ex.5B Q.5,6

**Homework for Tue-Wed 8-9 Nov 2016**

**Year 12 Further Maths**:

1. Heading: Matrices

Finish Ex.4E.

2. Heading: Common factors practice

Take out the best common factor in each of the expressions on:

Do the work on the sheet, then glue it into your book. Tick each answer if you’re confident it’s right, mark with ? if doubtful.

**Year 13 Further Maths**:

Heading: First order differential equations

Finish Ex.4B Q.1-6

**Homework for Tue-Wed 1-2 Nov 2016**

**Year 12 Further Maths**: Watch this video of Tim Gowers talking to CoLA students about mathematical proof

The password is citymaths1.

A few things in the subtitles need to be fixed in the next few days, but you should still get the drift even if those things haven’t yet been fixed when you watch the video.

Read the booklet about mathematical proof

Booklet about mathematical proof

Write out at least one mathematical proof of a claim in the booklet or a claim which Tim Gowers mentions in his talk.

**Year 13 Further Maths**: Watch this video about differential equations

Video about differential equations

Work through the test paper you did on Thursday (in your red book), and prove to yourself that you can do all the questions in it completely and accurately.

I will post worked answers to help you, and scan your marked paper and email it to you.

**Homework for Tue-Wed 11-12 October**

**Year 12 Further Maths**: Choose one of these three paths:

- If you’ve only just started on divisibility, finish the workbook pages on proofs by induction with divisibility
- If you’re well on with divisibility, do Q.11,12,13 from Ex.6B in the textbook
- If you’ve gone on to chapter 5 (series), finish Q.1-6 in Ex.5E

**Year 13 Further Maths**: Finish Ex.3E (complex roots) and Ex.1A Q.5-9 (inequalities).

**Homework for Tue-Wed 4-5 October**

**Year 12 Further Maths**: Finish the proofs by induction with series up to Q.1,2,3 on p.16 in the workbook. (Q.4 and 5 on p.16 you’ve done already, so you can skip them). If you have time, go on to the proofs by induction with divisibility on page 17 and the following pages. If you have mislaid the workbook or not taken it home, you can download it at https://mathsmartinthomas.files.wordpress.com/2016/09/workbook-induction3.pdf

Exceptions: Vinh, Peter, Tegan, Bolaji: please do as much as you can of the divisibility examples in the proof-by-induction workbook.

Daniel, David: please do Q.5, 6, 8 of Ex.6A from the textbook.

Michelle, Abdal: just do as much as you can from the workbook, starting from where Onesimus and Rashi and Ashley helped you to get up to on Friday.

**Year 13 Further Maths**: Using De Moivre. Finish off Q.2, 3, 5, 6, 7 of Ex.3D (p.36); then, if you can (the questions involve a lot of working, but you’ll get quicker with practice) Q.1, Q.2, Q.3 a, b, c, Q.4, of Ex.3I (p.61).

**Homework for Wed 28 September**

**Year 12 Further Maths**: Your homework is to finish off the examples we worked on in class about proof with sequences:

1. Exam questions on page 9 of the workbook

2. Questions 1 to 4 from p.132-3 of the textbook (Exercise 6C)

3. Q.6 from p.132-3 of the textbook (Exercise 6C)

You don’t have to do Q.7 and 8 of Exercise 6C

As always, email me if you come across problems or difficulties. How do you mark your own work? Tick each proof if you’re happy that it is a convincing argument to prove the answer correct; mark it with a question-mark if you’re not sure.

**Year 13 Further Maths**: From your “Further complex numbers: exercises and answers” booklet, do:

1. “Exercise on exp and ln functions”, ninth page in the booklet (with “41” on bottom right corner of photocopy)

2. “Exercise 3A”, eleventh page in the booklet (with “23” on bottom right corner of photocopy)

3. More Möbius questions if I’ve asked for them in my comments on your last homework

**Homework for Wednesday 21 September**

Please make sure it’s in my blue tray in 2B7 by close of school on Tuesday, or Wednesday before tutor group at latest. Please mark your own work before handing it in.

**Year 13 Further Maths**: do three more questions (or more if you think you need more practice to get good) by the geometric method from the booklet of FP2 past paper questions

FP2 past-paper questions (Möbius)

Use the worked answers to check your work.

You may choose to skip Q.6 and Q.9 at this point. Q. 6 (“The point P represents a complex number z on an Argand diagram, where |z+1|=|2z-1|…”) is very much designed to be done by the algebraic method, and the geometric method is not specially faster for that one. Q.9 (“The transformation T from the z-plane, where x=x+iy…”) is written to *demand* you do it by the algebraic method (though you can check it by the geometric method, and it’s much quicker).

**Year 12 Further Maths**

Finish pages 1-8 in the induction workbook: click here for induction workbook

**Homework for Wednesday 14 September**

Please make sure it’s in my blue tray in 2B7 by close of school on Tuesday.

**Year 13 Further Maths**

Do (by the geometric method) Q.1 to 5 from the booklet of FP2 past paper questions

FP2 past-paper questions (Möbius)

Click here for worked answers to check your work

**Year 12 Further Maths**

Finish the “Algebra skills for proof by induction” questions, 1 to 4 on pages 1 and 2 of your workbook.

Click here for induction workbook

Check your work against the answers given in the workbook.

If you think you need more help or more practice to get good with this, use

https://mathsmartinthomas.wordpress.com/2015/10/23/taking-out-common-factors/

**Homework to be done for the start of term, 8 September 2016**

**Year 13 FP2 Further Maths**:

1. Re-read your “Möbius transformations” colour-printed booklet. (Download a revised edition here if you don’t have your printed copy to hand).

2. Read the proofs at https://mathsmartinthomas.wordpress.com/2015/08/18/images-of-loci-of-complex-numbers/

3. *Re-do* Activity 8 – Example 36, and questions 13 and 15, on the 7th page of the booklet – by the geometric method. (You’ve already done them by the algebraic method).

**Homework for 7 July 2016**

**Year 13 (formerly Year 12) Further Maths**: From your “Transformations of the complex plane” colour-printed booklet (click here to get it online):

do the exercises on the 7th page by the algebraic method. (We will study how to do them by the geometric method next week).

Heading on your work (you should have left your book with me for marking, and so you’re doing the homework on loose-leaf paper): Algebraic method: examples.

That is “Example 36”

“Q.13”

“Q.15”

All these are examples where an ordinary circle (not a line) is transformed into an ordinary circle (not a line).

Click here for the worked answer from the book for example 36. The answer for Q.13 (from Ex.3H in book) is circle with centre (9/2, −3/8) and radius 15/8; the answer for Q.15 (from Ex.3I in the book) is circle with centre (−2,−4) and radius 2√5.

Steps in the algebraic method:

# Change subject of equation so that it’s “the wrong way round”, i.e. if you have a known locus in the z-plane and want to find the w-locus, change the subject of the equation to “z=….”

# Write w=u+iv (so u=Re(w), v=Im(w)) and put the equation in terms of u and v

# Use what you know about z (it might be |z|=2 if the z-locus is a circle, centre the origin, radius 2; or Re(z)=Im(z) if the z-locus is the line y=x; or Re(z)=Im(z)≥0 if the z-locus is the half-line arg(z)=π/4) to get an equation for u and v

# If that equation has expressions like |(u−1)+i(v+1)| in it, square both sides to get an equation without |…| signs in terms of u and v

# Expand, collect terms, simplify

# Fiddle with the equation for u and v to get it in a standard circle or line form

Click here for some worked examples.

**Homework for 28 June**

**Year 13 (formerly Year 12) Further Maths**: From your “Further Complex Numbers” exercises booklet (click here to get it online), do as many as you can of Exercise 3H Q.1, 3, 5, 6. Heading in your books: Transformations of the complex plane.

Work through these notes, and you’ll find it all not so difficult.

Ex.3H Q.1 (a). “The circle |z|=1” means that you draw in the z-plane a circle of all the complex numbers z which have modulus (=”size”=distance from the origin) equal to 1. That is, a circle, centre the origin, radius 1.

Then the transformation w=z+4+3i means that the image in the w-plane is a similar circle, shifted 4 units to the right, 3 units up.

(b) “The half-line arg(z)= π/2” means that you draw in the z-plane the line which is all the complex numbers at an anticlockwise angle of π/2(=90 degrees) to the positive real axis. That’s just the positive imaginary axis. It’s called a half-line… well, because it’s half a line.

In the w-plane you draw a similar half-line, but shifted 4 units to the right, 3 units up.

(c) No mystery about how to draw the line y=x. It’s a full line 45 degrees anticlockwise from the positive real axis. In the w-plane you draw a similar half-line, but shifted 4 units to the right, 3 units up.

Never mind about Q.2. You can skip that.

Questions 3 and 5 are just the same sort of thing as question 1.

Question 5 looks a bit scarier, but it isn’t really. “The circle |z−1|=3” is just a circle in the complex plane with centre 1 and radius 3. “The half-line arg(z−1+i)=π/4” is a half line starting at 1−i and going up and to the right at an angle of π/4 (45 degrees) anticlockwise from the positive real axis.

Question 6 is the only one which introduces something more complicated than enlargement and translation.

w=1/z. You work that out in algebra by changing the subject of the equation.

z=1/w

Then when |z|=2, |1/w|=2, so 1/|w|=2, so |w|=½. So in the w-plane you draw a circle with centre the origin and radius ½.

When arg(z)=π/4, Re(z)=Im(z) and both are positive. So Re(1/w)=Im(1/w).

Call the real and imaginary parts of w, u and v, so w=u+iv

Then 1/w=1/(u+iv)=(u−iv)/(u^{2}+v^{2})

So Re(1/w)=Im(1/w) means u=−v. And if Re(1/w) and Im(1/w) are both positive, u is positive.

In the w-plane, the up-and-down axis (what you would usually call the “y-axis”) is the v-axis, and the across axis (what you would usually call the “x-axis”) is the u-axis.

So, draw the line u=−v, or rather, the bit of it for which u is positive.

**Homework for Tuesday 17 May**

**Year 13 Maths**: a C3 or C4 paper of your choice.

**Year 13 Further Maths**: no FP2 or FP3 homework. Use the time for M3 practice.

**Year 12 Further Maths**

Last-ever FP1 homework: FP1 past papers February 2010 and January 2013 (except that Nii, who has already done February 2010, should do Practice Paper B)

Next Monday, 16th: if you’re not in a revision session for another subject, come to your usual lesson time with Mr Osborn and do FP1 past paper January 2011

Next Thursday, 19th: we’ll have a two-hour revision lesson, periods 5 to 8, but we’ll make it as low-key and relaxed as possible. After school on Thursday, don’t do any more revision. Do something enjoyable which has nothing to do with maths, get a good night’s sleep, and be relaxed and confident for Friday morning.

Sheet summarising everything you need to know for FP1: https://mathsmartinthomas.files.wordpress.com/2015/04/fp1do-dont41.pdf

Or almost everything. Practise with your calculator to make sure you know Rose’s method for converting results from Pol( , ) from decimals (like r=1.414214, θ=0.785398), to exact form (sqrt 2, π/4). https://mathsmartinthomas.wordpress.com/2016/03/24/using-the-s-d-key-with-pol-and-rec/

You may also find these video clips on this website useful:

**Homework for Tuesday 10 May 2016**

**Year 12 Further Maths**

FP1 June 2009 paper: https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_june_2009.pdf.

Please check your work against the mark scheme: https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_june_2009_ms.doc

**Year 13 Further Maths**: an FP2 or FP3 past paper of your choice.

**Year 13 Maths**: a C3 or C4 paper of your choice.

**Homework for Tuesday 3 May 2016**

**Year 12 Further Maths**: Parabola and hyperbola questions from recent IAL papers.

“Scaffolded” version (with hints): https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_recent_ial_paperssplit-and-merged-2.pdf

Unscaffolded version: https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_recent_ial_papers-split-and-merged.pdf

Worked solutions: https://mathsmartinthomas.files.wordpress.com/2016/03/ial_workedanswers.pdf

**Year 13 Further Maths**: an FP2 or FP3 past paper of your choice, plus the “check-your-answers” activity: https://mathsmartinthomas.files.wordpress.com/2015/05/fp3_checks.pdf

**Year 13 Maths**: a C3 or C4 question of your choice, except, for Arif and Matthew, six of the Review Exercises from the book on vectors.

**Homework for Tuesday 26 April 2016**

The *first* bit of homework, for everyone, is to work through the comments I’ve written on your previous homework, and make another effort to complete questions you had trouble with (or failed to attempt) first time round. Then:

**Year 13 Maths**

Do six questions of your choice from the trig past-paper questions booklet.

https://mathsmartinthomas.files.wordpress.com/2015/12/c3_trigonometry1.pdf

If you’re confident about that, do:

Ex.7E Q.8, 9, 10c, 11a, from the C3 book (p.127-8). These are on the “factor formulae”, which are in the formula book and in the textbook. They rarely come up in the exam, but, who knows, this year might be the year.

Or if you feel super-confident about all of trig, do another C3 or C4 past paper of your choice.

**Year 12 Further Maths**

Then do six further questions from the Review Exercises in the textbook on parabolas and hyperbolas.

**Year 13 Further Maths**

1. Do a check-the-answers on the FP3 mock paper and the FP2 2014 paper at:

https://mathsmartinthomas.files.wordpress.com/2015/05/fp3_checks.pdf

showing your reasoning in each check.

2. Do an FP2 or FP3 past paper of your choice. If you find that too much, make sure that at least you do the check-the-answers.

**Homework for Tuesday 19 April 2016**

The *first* bit of homework, for everyone, is to work through the comments I’ve written on your previous homework, and make another effort to complete questions you had trouble with (or failed to attempt) first time round. Then:

**Year 12 Further Maths**

Do FP1 “Practice Paper A” (I gave it to you today).

https://mathsmartinthomas.files.wordpress.com/2016/02/fp1-practice-papers.pdf

**Year 13 Further Maths**

Do at least Questions 17 to 12 (working backwards) of the FP3 textbook “Review Exercises” on vectors (pages 194-5 of the book).

**Year 13 Maths**

Do another C3 or C4 past paper of your choice, but *not* the C4 June 2015 paper, which I gave you today, 14 April, but you will do in class with Ms Varney on Monday.

**Homework for Tuesday 12 April 2016**

**Year 12 Further Maths**

The four recent “International A level” papers in this booklet: https://mathsmartinthomas.files.wordpress.com/2016/02/recent_ial_papers.pdf.

You can find mark schemes online at https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_recent_ial_ms.pdf.

Click here for worked answers to all questions.

**Year 13 Further Maths**

FP2 papers June 2009 and June 2010, available at: https://mathsmartinthomas.files.wordpress.com/2016/02/fp2-2009-2013.pdf. Mark schemes: https://mathsmartinthomas.files.wordpress.com/2016/02/fp2_2009-2013-markschemes.pdf.

FP3 “practice papers” A and B, at https://mathsmartinthomas.files.wordpress.com/2015/03/fp3-2009-edexcel-practice-a-and-b.pdf with mark schemes.

**Year 13 Maths**

For everyone except Arif and Matthew:

Two “C3/4” “International A level” papers:

1. The “Specimen” paper you already have in this booklet – https://mathsmartinthomas.files.wordpress.com/2016/02/c4_ial_june2014_spec.pdf, with mark scheme at https://mathsmartinthomas.files.wordpress.com/2016/02/c4_ial_june2014_ms.pdf

2. The January 2016 paper in this booklet: https://mathsmartinthomas.files.wordpress.com/2016/02/c34_ial_recent.pdf. Mark scheme here: https://mathsmartinthomas.files.wordpress.com/2016/02/c34-ial-recent-ms.pdf.

(We’ll do the June 2015 and January 2015 “IAL” papers later. If you’re up for doing some additional past papers for practice over the holidays, which is fine, please select not those ones, but some others from the range at https://mathsmartinthomas.wordpress.com/2016/02/19/c4-past-papers-and-mark-schemes/).

For Arif and Matthew:

Week starting 28 March – get fluent with exp and ln

https://mathsmartinthomas.files.wordpress.com/2015/12/c3exponential-log-1.pdf

Differentiating exp and ln

https://mathsmartinthomas.files.wordpress.com/2015/12/c3differentiation.pdf

Week starting 4 April – get fluent with differentiation (implicit differentiation, and the water-flowing-out-of-a-tank-type differential equation questions)

https://mathsmartinthomas.files.wordpress.com/2016/02/c4-differentiation-2.pdf

**Homework for 22 March 2016**

Please hand this work in on Tuesday 22 March. Please mark your work before you hand it in. Please also sort out your personal folder of maths booklets and papers, and get your list of the past papers you’ve done and the marks you got for them up to date.

**Year 12 Further Maths**

Do FP1 past paper June 2011. (If you’ve already done that one, do “Practice Paper A” instead).

The June 2011 paper is in your printed booklet of past papers, or available online.

A mark scheme for it is in your printed booklet of past-papers, or available online.

Worked answers for the June 2011 paper are online at https://mathsmartinthomas.files.wordpress.com/2016/02/fp1_june2011_workedanswers.pdf.

Practice Paper A is in this booklet: https://mathsmartinthomas.files.wordpress.com/2014/12/fp1_a-g.pdf, and the mark scheme is at https://mathsmartinthomas.files.wordpress.com/2014/12/fp1_a-g_ms.pdf.

**Year 13 Further Maths**

Do at least questions 13 to 20 on vectors in the booklet you have:

https://mathsmartinthomas.files.wordpress.com/2014/12/fp3-matrices-vectors-examwizard.pdf. Email me if and when you get stuck on questions.

**Year 13 Maths**

For everyone except Arif and Matthew:

Finish the June 2014 C4 “International A level” paper.

https://mathsmartinthomas.files.wordpress.com/2016/02/c4_ial_june2014_spec.pdf

The mark scheme is in your booklet. https://mathsmartinthomas.files.wordpress.com/2016/02/c4_ial_june2014_ms.pdf.

For Arif and Matthew:

Finish the “Partial Fractions and Binomial Expansion” booklet:

https://mathsmartinthomas.files.wordpress.com/2015/02/c4-partials-binom.pdf

**Homework for 15 March 2016**

**Year 12 Further Maths:** do the FP1 June 2013 paper or February 2010 paper. Worked answers at https://mathsmartinthomas.wordpress.com/2016/02/19/fp1-collections-of-past-paper-questions/. Plus statics Q.34-35, p.158-9, from the M2 book, and work-energy Q.45 and 48, p.96, from the M2 book.

**Year 13 Further Maths:** do an FP2 paper and an FP3 paper of your choice.

**Year 13 Maths:** do a C3 and a C4 past paper of your choice.

**Homework for 1 March 2016**

Please mark your work yourself before handing it in. Please hand it in on Tuesday morning 1 March.

More homework than usual these weeks because of the mock exams coming up.

**Year 12 Further Maths:** For 1 March: do the FP1 June 2014 paper plus statics Q.36-40, p.159-60, from the M2 book.

For week starting 7 March: do the FP1 June 2013 paper or February 2010 paper. Worked answers at https://mathsmartinthomas.wordpress.com/2016/02/19/fp1-collections-of-past-paper-questions/. Plus statics Q.34-35, p.158-9, from the M2 book, and work-energy Q.45 and 48, p.96, from the M2 book.

**Year 13 Further Maths:** For 1 March: do at least eight of the questions on 3-D vectors from Ex.5G in the FP3 book, and finish the worksheet of Maclaurin-Taylor past paper questions.

For week starting 7 March: do an FP2 paper and an FP3 paper of your choice.

**Year 13 Maths:** For 1 March: do a C4 past paper of your choice.

For week starting 7 March: do a C3 and a C4 past paper of your choice.

Homework for 23 February 2016.

**C3/C4**: Do a C4 past paper from the booklet of past papers: https://mathsmartinthomas.wordpress.com/2015/03/20/c4-past-papers-and-mark-schemes/

The June 2014 paper, unless you have a strong reason for doing a different one.

You will probably find the paper difficult at this stage. That’s all right. You’ll get there. Mark your own work before handing it in to me.

**M2/ FP1**: As preparation for the catch-up session on the Thursday of half-term you are asked to work through the FP1 paper you did at the start of this term and write out answers to all the questions, using my worked answers and my notes about your papers.

The aim of that activity is to help you prove to yourself you *can* do all those questions. Bring your reworked paper along with you on Thursday.

**FP3**: Four questions of your choice from Ex.5G – using vectors for geometry in 3D.

Homework for 9 February 2016.

**C3/C4**: The *first* thing is to catch up on your student responses to the homework, and any work requested by me in marking your previous homework. Then do this **C3 test paper**: click here to download pdf. Give yourself exactly 90 minutes to do it, under as near exam conditions as you can manage. Then mark your work against the mark scheme: click here to download pdf. You’re doing a C3 resit next Tuesday or Wednesday (whenever on Tuesday or Wednesday is best for you: see Mr Osborn to get the paper, and put your finished work in my blue box in the maths office).

**M2**: The *first* thing is to catch up on your student responses to the homework, and any work requested by me in marking your previous homework. In order to give you time to do that, it’s a short homework request this week: just Q.24 and 25 of the **Statics** on p.156-7 of the M2 book. If you were on the trip on Friday, you may find that difficult, so please start this homework early and give yourself time to ask for help when you’re stuck.

**FP3**: The *first* thing is to catch up on your student responses to the homework, and any work requested by me in marking your previous homework. In order to give you time to do that, it’s a short homework request this week: just these **Cross-Product** problems – Ex.5A Q.1 h, i, j; Ex.5B, Q.1; Ex.5C, Q.1.

Homework for 2 February 2016.

**M2**: Work-energy and Statics

Work-energy: p.95, Q.44, 50

Statics: finish working through Example 2 on page 134. (You should read both methods, but need only do one).

Then: Ex. 5B (p.136) Q. 1, 2, 3

Draw a large diagram, with a ruler, for every problem!

**C4**: Parameters, areas, volumes

Finish Ex.6L, Q. 15, 17, 2, 23, 27, 28, 34

**FP3**: Diagonalising matrices; multiplying vectors

Multiplying vectors:

Work through the first and the third videos at:

https://mathsmartinthomas.wordpress.com/2015/12/23/quaternions-video-clips/

i.e.

and

Work through the “Dot-product dating” and “dot-product shopping” and “work, in mechanics” exercises in the booklet “Notes for FP3 matrices and vectors”.

Fill in the table “Differences between dot product and cross product” in that booklet.

Diagonalising: Ex.6G Q.8

Homework for 26 January 2016.

**M2**: Work, energy, power

Review Exercises, p.95 and following – Q.41, 42, 43; 47, 48, 49

**C4**: Integration (all methods) from Ex.6L

Questions about techniques of integration only: 25 – parts; 29 – subst, parts; 31 – parts; 13 – subst, parts, partial frac. Questions involving only techniques of integration and separation of variables to solve differential equations: 5 – parts; 8 – parts; 16 – standard results; 18 – subst, trig

**FP3**: Eigenvalues and eigenvectors

Finish off Ex.6F

Homework for 19 January 2016.

**FP3**: How 3×3 matrices transform planes and lines in 3D

Ex 6D (p.160) Q.1, 4, 5, 7

**M2**: Work-energy principle

Ex 3C (p.76) Q.15-18

**C4**: Integration using trig identities, and integration by substitution

- “Baby” substitutions: Ex 6E Q.1
- Fiercer substitutions: p.120 Q.7 (a) and (b) only; p.124-5 Q.18 (a) only and Q.29 (a), (c), (d) [not (b)]
- Trig identities: Ex 6C Q.2 (a), (d), (g) if you haven’t done them in class

Homework for 12 January 2016. Please mark your work yourself before handing it in. Please hand it in on Tuesday morning 12 January.

**FP3**: Matrices.

1. Watch the first four videos at https://mathsmartinthomas.wordpress.com/2015/12/23/inverting-matrices-using-the-casio-991es-and-using-rref-video-clip/.

2. Ex.6A, Q.2a, 4a

3. Ex.6C, Q.1,2. In all the parts of question 1, do the inversion **both** by the method in the book and by RREF. For question 1 parts a and b, find the determinant by row echelon format.

In Question 2, use whichever method you find best.

**FP1**: Parabolas and hyperbolas, plus follow-up

1. Do the follow-up I asked you, individually, to do from your holiday homework. If you’re at all unclear about what that is, just do another FP1 past paper.

2. Do questions 2 and 3 from the booklet of past-paper questions on parabolas and hyperbolas

Or (if you prefer, since you have answers with the questions) two questions from this question sheet on parabolas and hyperbolas:

Question sheet with worked answers

**C4**: Integration from standard results, plus follow-up

1. Do the follow-up I asked you, individually, to do from your holiday homework. If you’re at all unclear about what that is, just do another C3 past paper.

2. Ex 6B Q.3a, e, i, and Q4a

**Homework for 5 January 2016**

**Note for everyone: how to do past papers**

Give yourself 90 minutes for each paper, in as near exam conditions as you can manage. Have a formula book to hand – it’s available on the web.

After you’ve done 90 minutes, check your work against the mark scheme. Mark it. Give yourself a realistic estimate of how well you’re doing. Learn from your mistakes: puzzle out how to do questions you got wrong or didn’t find time to get to.

After you have marked and corrected your paper, please scan it and email it to me.

**FP1**:

Almost everyone did better than in the mid-term exam, and some people much better.

We aim for everyone to get at least a C in the mock exam second week back, and at least six of you to get a A.

On the evidence of the test you did on 11 December, that’s possible, if you can all make an effort to fix weak areas in your learning and to increase your confidence. But only with that effort.

1. I have written up, and posted on this site, detailed comments on the test you did on 11 December; and send each of you an individual email about your individual paper.

The test paper itself is available on this site.

Please work through the test paper, convincing yourself that you *can* do all the questions, identifying the mistakes you made in the actual test, learning from them.

2. On Nii’s suggestion, I’m collecting and making some video clips about bits of FP1, especially the bits we’ve done differently from the textbook.

They’re at https://mathsmartinthomas.wordpress.com/category/video-clips/

Watch them with a pen and paper to hand, working through the examples discussed in the videos, pausing or rewinding from time to time to make sure you’re confident.

3. Please do three more FP1 past papers. You have a booklet of past papers. If you’ve mislaid the booklet, it’s online (with mark schemes) on this website. Check out some trickier FP1 questions here: if you feel confident that you’ve covered similar questions in the FP1 past papers you’ve done, fine; if not, please work through those trickier questions (solutions are provided).

**FP2**

1. Please work through the test paper you did on 11 December. Purpose: to convince yourself that you *can* do all the questions, identify the mistakes you made in the actual test, and learn from them.

I’ve posted the test paper and the mark scheme on this website.

2. Please do at least three more FP2 past papers.

You can find past papers on this website. Do each paper by giving yourself 90 minutes for it, in as near exam conditions as you can manage. Have a formula book to hand – it’s on the web. You can find collections of past-paper questions by topic on this website.

3. Please work through the rest of the chapter on Maclaurin and Taylor.

The hard thing to understand here is the concept of expressing all functions, all well-behaved functions anyway, as infinite series. We’ve covered that and you need to consolidate it by working through the other bits in the book.

Use the exercises in the book, but do questions only from the beginning of each exercise and from the Review Exercises. The book contains some questions on Maclaurin and Taylor which are significantly more complicated than anything you’ll meet in the exam.

Beyond a certain level, it’s best to practise this topic by directly doing actual past-paper questions: there’s a selection on this site.

Usually in FP2 there is only a half-question or one-third-question on Maclaurin and Taylor, and that a pretty simple one.

**C3**

1. I will write up for each of you an individual email about your individual 11 December test paper. Please read that email.

Almost everyone did worse than I expected in that test paper. But no-one did so badly as to look hopeless. The odd thing is, generally you got the difficult things right and the easy things wrong.

That’s the sort of problem that is fixed by practice. With enough practice, I think everyone can get at least a C, and the majority can get an A. But only with enough practice…

2. Please do at least three more C3 past papers.

You can find past papers at:

http://www.physicsandmathstutor.com/a-level-maths-papers/c3-edexcel/

**Homework for 8 December 2015**

**Everyone: please first look at the marking of your last homework. Make sure you’ve written a “student response” (in red). If you’re asked to do something extra, do that first.**

And, with every week’s homework, *please mark it yourself*, *before you hand it in*.

**FP2**

FP2 June 2013 withdrawn paper (except Q.1(c) and Q.3(c), which you won’t yet be able to do because you haven’t yet studied Maclaurin and Taylor series). Click here for a mark scheme. Please give yourself 90 minutes to do the paper, and mark your work against the mark scheme.

**FP1**: FP1 June 2014, from this collection, which also includes a mark scheme for the paper. Skip questions 6 and 8, which you won’t be able to do because you’ve haven’t yet studied parametrics. Please mark your work against the mark scheme.

**C3**: One more C3 past paper. That is probably June 2007 for most of you, but anyway, the next past paper you haven’t done. Please mark your work against the mark scheme. Click here to get the past papers and mark schemes online.

**Homework for 1 December 2015**

**C3**

Heading for this homework: Past papers

Mark the June 2005 past paper you did in class. Give yourself exactly 90 minutes to do the January 2006, and mark that. Give me both marked papers on Tuesday 1 December, please.

**FP2**

Heading for this homework: Second order differential equations

Ex. 5F Q.1,2; Ex.5G Q.1, 3, 6; Review Exercises p.147 Q.17,18

**FP1**

Heading for this homework: Matrices

Complete (you’ll have done some in class) the following:

matrix addition – Ex.4A Q.2 and Ex.4B Q.1

matrix multiplication Ex.4C Q.2(b), 7, 8

seeing what matrices represent geometrically Ex.4D Q.5 and Ex.4E Q.1-3

determinants Ex.4I Q.4

**Homework for 24 November 2015**

**C3**

Heading for this homework: Differentiation

Three more questions from differentiation past-paper questions.

**FP2**

Heading for this homework: Second order differential equations

Ex.5C, Q.1, 3, 5, 7, 9

**FP1**

Heading for this homework: Mopping up complex numbers

Do Ex.1I, p.30, Q.5, 7, 8, 9. (Also Ex.1G Q.1, 2, 9 if you didn’t finish them in class).

These are about as difficult as you’ll get for FP1 complex numbers questions. Year 12 students generally find Q.9 difficult. But all it needs is a little thinking. You can also email me for help when you get stuck.

**Homework for 17 November 2015**

**FP2**

Heading for homework: First-order differential equations

Ex.4C Q.8 and 9 and Ex.4D Q.1-3

Because our lesson on Thursday was cut short, please read the relevant pages in the textbook carefully and work through Example 10 in the book before starting your homework.

e^{∫Pdx} can also be written as exp [∫Pdx]. Quite often you can guess the integrating factor without having to go through the exp [∫Pdx] formula.

**C3**

Heading for homework: Past-paper questions on trigonometry

Do *three* more questions from the past-paper booklet I gave you: C3 past-paper questions on trigonometry. You can check your answers against the mark schemes.

**FP1**

Heading for homework: Solving cubics

Find all three roots of the cubics in questions 1 to 8. (We went through Q.1 to 3 in class. I’ve worked Q.1 below as an example).

Facts to use:

If z = x+iy = r cis θ, the conjugate of z is z* = x–iy = r cis (–θ)

- [1] If z is a root of a cubic or a quartic (an equation with z
^{4}), then z* is also a root - [2] Sum of roots = − coefficient of second highest power in the equation (coefficient of z
^{2}if it’s a cubic, coefficient of z^{3}if it’s a quartic) - [3] Product of roots = − constant term in the equation if it’s a cubic

Product of roots = constant term in the equation if it’s a quartic

Using these facts (actually, just using the first two):

1. If 1+i is a root of z^{3}−5z^{2}+8z−6=0, find the other two roots

By fact [1], 1−i is also a root

By fact [2], the sum of the roots is 5 (because the coefficient of z^{2} is −5)

So, if the third root is a, a+1+i+1−i=5. Therefore a=3.

The three roots are 1+i, 1−i, and 3.

2. If i is a root of z^{3}−2z^{2}+z−2=0, find the other two roots

3. If 1+i is a root of z^{3}−2z+4=0, find the other two roots

4. If 2 cis (2π/3) is a root of z^{3}−8=0, find the other two roots

5. If 1+2i is a root of z^{3}−3z^{2}+7z−5=0, find the other two roots

6. If 3+2i is a root of z^{3}−6z^{2}+13z=0, find the other two roots

7. If 3−2i is a root of z^{3}−9z^{2}+31z−39=0, find the other two roots

8. If 3−2i is a root of z^{3}−10z^{2}+37z−52=0, find the other two roots

9. If 3 cis (2π/3) is a root of z^{3}−27=0, find the other two roots

You can check your answers by clicking here.

If you get stuck, email me at mthomas@cityacademy.co.uk or ask a Sherpa. Don’t read the textbook. It won’t help. For more on this see the “don’t do as the book says” notes.

**Homework for 10 Nov 2015**

**FP2**

Heading for homework: First-order differential equations

Ex.4B Q.1-6

**C3**

Heading for homework: Past-paper questions on trigonometry

Do *three* more questions from the past-paper booklet I gave you: C3 past-paper questions on trigonometry. You can check your answers against the mark schemes.

**FP1**

Heading for homework: Adding, multiplying, dividing, and converting complex numbers

Finish off these sections of Ex.1a, 1b, 1c, 1e

1e, Q.1-8: convert from (x+iy) rectangular form into (r,θ) polar form, using Pol on your calculator. “Modulus” is just the “r” of the complex number written in polar form, and “argument” is just the “θ” of the number written in polar form.

1a, Q.1-10: add, in x+iy form

1b, Q.1-10: multiply, in x+iy form (it’s like multiplying in algebra, as if i were a variable, except that you then simplify further by replacing i^{2} by −1)

1c, Q.3-10, divide, in x+iy form (it’s like dividing with surds. You multiply both top and bottom of the division sum by the conjugate of the bottom number)

**Homework for 3 Nov**

**C3**

Heading for homework: Re-run of C3 test

Purpose of this homework: To help you prove to yourself that you really can do all the questions in the C3 test; and the ones you did right first time, you can do even more concisely and neatly.

**FP1**

Heading for homework: Re-run of FP1 test

Purpose of this homework: To help you prove to yourself that you really can do all the questions in the FP1 test; and the ones you did right first time, you can do even more concisely and neatly.

**FP2**

Heading for homework: Re-run of FP2 test

Purpose of this homework: To help you prove to yourself that you really can do all the questions in the FP2 test; and the ones you did right first time, you can do even more concisely and neatly.

**Homework for Tuesday 20 October 2015**

**Everyone: please first look at the marking of your last homework. Make sure you’ve written a “student response” (in red). If you’re asked to do something extra, do that first.**

And, with every week’s homework, *please mark it yourself*, against the back of the book (if you can), *before you hand it in*.

**FP1: series**

Ex.5E Q.1 and 2

Ex.5F Q.5 and 6

**C3: trigonometry**

Ex.7A Q.1, 2, 3, and 4 (a), (c), and (d) only

**FP2: mopping up**

Do all of the following questions that you haven’t already done for revision.

Polar coordinates: p.153-4 Q.62

Inequalities: p.9 Q.5, Q.7

Series: p.16 Q.7

Complex numbers: p.68 Q.30, Q.32, Q.33

**Homework for Tuesday 13 October 2015**

**Everyone: please first look at the marking of your last homework. Make sure you’ve written a “student response” (in red). If you’re asked to do something extra, do that first.**

And, with every week’s homework, *please mark it yourself*, against the back of the book (if you can), *before you hand it in*.

This week’s homework is all geared to revision for the tests on 15 and 16 October. If you get stuck, take advantage of the catch-up sessions: Tuesday P1, Thursday P1, or Friday P1, or before tutor group on Tuesday, Thursday, or Friday. Just come to the maths office.

**FP1: revision for test on 16 October**

It’s a lot of homework this week!

First, do p.40 Q.1 and Q.6 (a) and (b) [not (c)] to make sure you’re good on interval bisection and linear interpolation.

Second, do this sheet on basic algebra facts you use a lot in induction proofs (and with series): click here for worksheet.

Third, do a few induction proofs, if necessary by looking at standard examples and copying step-by-step, so that you can manage them at least on a *“getting used to it”* basis, if not yet quite “understanding”.

Use this model to write proofs that:

- if f(n)=4
^{2n−1}+5^{2n−1}, then for all n, 9 | f(n); and - if f(n)=5
^{2n−1}+6^{2n−1}, then for all n, 11 | f(n).

Even if you do it by mechanically copying out and replacing “3” in the model by “4”, and “4” in the model by “5”, etc., it should help you get sufficiently *used to* divisibility proofs to manage in the test on Friday 16 October.

9 | f(n) is short for: 9 divides exactly into f(n)

Use this model induction proof with sequences to do Q.44 on p.141.

Use this model proof with series to do Q.40 on p.141.

If you get stuck, email me and say where you’ve got stuck. You must be able at least to start all these induction proofs, if only just by copying the model proof.

**C3: a bit of trigonometry and revision for test.** Ex.6F Q.2. Also make sure you’ve finished Ex.6D Q.1-2 and Ex.6E Q.1 at least. Apart from that, *choose your own homework* to prepare you for the test on Friday 16 October. The test is on chapters 1, 2, 3, 4, and 8 (not chapter 5, and not the trigonometry we’ve done in chapters 6 and 7). I can tell you that the test doesn’t include anything super-complicated on differentiation, so if you’re confident with Ex.6A you should be all right for the test. Part of your homework is also to *name which topics you want to have time in our revision lesson on Thursday 15th*.

**FP2: revision for test on Thursday 15 October**. Choose your own homework. The test will cover inequalities (ch.1), series (ch.2), De Moivre (ch.3), and polar coordinates. Best choose a few examples from each area and identify where you need to do more work. Email me if you get stuck. Please make sure to hand in your work on Tuesday, so that I can get an idea of what you’re doing with revision, and where you’ve having difficulty.

We will have 35-40 minutes revision time on Thursday before the test.

**Homework for Tuesday 6 October 2015**

**FP1: proof by induction**: “Scaffolded” induction proofs with series and “scaffolded” induction proofs with divisibility.

**FP2 polar coordinates:** Do at least four questions from this collection of past-paper questions on polar coordinates. (You can mark your own work against the mark scheme provided).

**C3 trigonometry:** Ex.6C Q.1-4

**Homework for Tuesday 29 September 2015**

**FP1: proof by induction:** Do at least six questions from Ex.6A (induction with series)- from the sheet with Ex.6A and 6B (induction). (Don’t do any Ex.6B questions for now). And (if you’ve not finished it already) finish the sheet of re-run past exam questions on induction with sequences. If you finish those with time to spare, do as many as you can of the sheet of past-paper questions on induction with series

**FP2 polar coordinates:** Review questions (p.153) nos. 59, 60, 61, 62

**C3 numerical methods:** Ex.4C Q.1, 2, 3, 4

**Homework for Tuesday 22 September 2015**

And, with every week’s homework, *please mark it yourself*, against the back of the book (if you can), *before you hand it in*.

Then:

**FP1: proof by induction:** Finish the collection of past exam questions on mathematical induction with sequences: Past exam questions on induction with sequences and (if you haven’t finished it yet) the sheet of “extra” induction problems with sequences: Extra induction problems with sequences

**FP2 polar coordinates:** Ex.7D Q. 3, 4, 6, 7

**C3 Exponentials and logs:** Ex.3C Q.1(a) and (f) only, Q.2(a) and (f) only, Q.3, Q.4.

**Homework for Tuesday 15 September 2015**

**C3: Exponentials and logs** Finish Ex.3A

**FP1: proof by induction** Read the booklet on “Mathematical Induction and the Lurgy”.

Write another example, beside diseases, forest fires, collapsing buildings, and dominoes, of “mathematical induction” in everyday life.

Do proofs of claims 1 and 2 in that booklet (page 7). You don’t have to attempt Claim 3, or the three examples on page 6.

Please *don’t *start reading the textbook, or doing examples from it.

**FP2: polar coordinates** Ex.7C, questions 1 to 6, part (c) only of each question. You can use http://www.webmath.com/polar.html to help. Make sure you set the range big enough.

**Homework for Tuesday 8 September 2015**

**C3: Algebraic fractions** Exercise 1E, Q.1 (a) and (d), Q.2 (a) and (d), Q.3-6.

**FP1: proof** Write proofs of two claims chosen from A3 to A5,

B1, B2, or B4 to B8, in the booklet

Lesson_and_homework_on_proof; and read the whole booklet through. You can check your answers at https://mathsmartinthomas.files.wordpress.com/2014/08/answers_for_booklet_on_proof.pdf.

**FP2: polar coordinates** With diagrams, show what happens to the graph in polar coordinates:

r = p + cos θ

as p moves from p=0 to p=2.

You can use http://www.webmath.com/polar.html to help, especially if you missed the lesson on Thursday 3 September.

Sketch the graphs of

r = sin 2θ

r = sin 5θ

r = cos 2θ

θ = π/4

θ = π/2

You can use http://www.webmath.com/polar.html to help.

You can check your answers against the textbook. Remember that the textbook misses out the parts of these graphs where r is negative, so your answer may be right but a bit different from the textbook.

**Homework for Tuesday 14 July**

Lou-Lou and Debora: do as much as you can of Ex.1C (on inequalities) in the FP2 book.

Everyone else: do these six problems from the website “I want to study engineering”. Write your working in your book as well as your answer!

http://i-want-to-study-engineering.org/q/children/

http://i-want-to-study-engineering.org/q/mental_arithmetic/

http://i-want-to-study-engineering.org/q/graph_sketching3/ (this looks hard, but is the easiest of the graph-sketching questions)

http://i-want-to-study-engineering.org/q/graph_sketching1/

http://i-want-to-study-engineering.org/q/graph_sketching2/ (can be done more easily than suggested method on website)

http://i-want-to-study-engineering.org/q/cubic_equation/ (harder)

**Homework for Tuesday 7 July**

Do these six problems from the website “I want to study engineering”.

If you get stuck, the website offers help and then a full solution. So you must be able to get the answers one way or another. Obviously, first try to do the problems without using the website help. Please mark your work before you hand it in, giving yourself extra credit if you got a solution without using the website help.

http://i-want-to-study-engineering.org/q/cogs/

http://i-want-to-study-engineering.org/q/construction_worker1/

http://i-want-to-study-engineering.org/q/bouncy_ball/

http://i-want-to-study-engineering.org/q/electric_heater/

http://i-want-to-study-engineering.org/q/cards_and_logic/

http://i-want-to-study-engineering.org/q/arcs_inside_a_square/

(this last one is harder but doesn’t demand any knowledge you don’t have)

**Homework for Wednesday 1 July**

Please put your homework in my blue tray in the maths office by end of school on Wednesday 1 July.

Do these six problems from the website “I want to study engineering”.

Don’t use the “problem generator” on that website, because that may give you a problem that you need year 13 knowledge for. You can do all these six problems with year 12 knowledge and some thinking.

If you get stuck, the website offers help and then a full solution. So you must be able to get the answers one way or another. Obviously, first try to do the problems without using the website help. Please mark your work before you hand it in, giving yourself extra credit if you got a solution without using the website help.

From Sharif, Mugisha, Hamse, Juan, Dion, and Joan I also want to see what you’ve done on the MAT symmetry problems and the MAT simple special cases problems. Congratulations to Lou-Lou on her work on the symmetry problems.

Aircraft drink

Beakers

Coconuts

Bacteria

Arc

Clock

**Homework for Tuesday 5 May**

Please give your homework to me at Assembly on 5 May.

Homework now is all past papers. *Marking your work, using the mark scheme, is part of the homework!* All the booklets of past papers and mark schemes you need are on this website, if you’ve mislaid your paper copies.

**Year 13 Further Maths**: Two more FP3 or S2 past papers of your choice from the papers at:

**Year 12 Further Maths**: *Either* six questions of your choice from the collection of FP1 past-paper questions on complex numbers, *or* two FP1 papers of your choice from:

**Year 12 Further Maths (Lou-Lou and Debora)**: to be decided.

**Homework for Tuesday 28 April**

Please give your homework to me at Assembly on 28 April.

Homework now is all past papers. *Marking your work, using the mark scheme, is part of the homework!* All the booklets of past papers and mark schemes you need are on this website, if you’ve mislaid your paper copies.

**Year 13 Further Maths**: Two more FP3 or S2 past papers of your choice from the papers at:

**Year 12 Further Maths**: Two FP1 papers of your choice from:

**Year 12 Further Maths (Lou-Lou and Debora)**: Four more questions, of your choice, from the booklet of past-paper questions by topic on parametrics.

https://mathsmartinthomas.wordpress.com/2015/03/20/fp1-collections-of-past-paper-questions/

**Year 13 maths**: Two past papers of your choice, D1 or C4, from those on the website.

https://mathsmartinthomas.wordpress.com/2015/03/20/c4-past-papers-and-mark-schemes/

**Homework for Tuesday 21 April**

I have a hospital appointment on 21 April, so won’t be in school that day. Please give your homework to Mr Osborn or put it in my blue tray in the maths office. I’ll be in school late morning on 22 April.

Homework now is all past papers. *Marking your work, using the mark scheme, is part of the homework!* All the booklets of past papers and mark schemes you need are on this website, if you’ve mislaid your paper copies.

**Year 13 Further Maths**: Two more FP3 past papers of your choice from the papers at:

https://mathsmartinthomas.wordpress.com/2015/03/20/fp3-past-papers/

**Year 12 Further Maths**: An FP1 paper of your choice from:

https://mathsmartinthomas.wordpress.com/2015/03/20/fp1-collections-of-past-paper-questions/

**Year 12 Further Maths (Lou-Lou and Debora)**: Four questions, of your choice, from the booklets of past-paper questions by topic. Lou-Lou, for you that would be four questions on parametrics. Debora: we’ll discuss on Friday what’s best.

https://mathsmartinthomas.wordpress.com/2015/03/20/fp1-collections-of-past-paper-questions/

**Year 13 maths**: A D1 or C4 past paper of your choice from those on the website.

https://mathsmartinthomas.wordpress.com/2015/03/20/c4-past-papers-and-mark-schemes/

**Homework for Easter, due in on Tuesday 14 April**

Because exams are so close, you need to do homework over the holiday. Even on papers where you did very well on the mocks, practice makes the difference between getting your A* or A, or making a few slips (even though you know all the basics) and getting a B.

*Marking your work, using the mark scheme, is part of the homework!* All the booklets of past papers and mark schemes you need are on this website, if you’ve mislaid your paper copies.

You can email me at mthomas@cityacademy.co.uk any time if you’re stuck even after looking at the mark scheme.

**Year 13 Further Maths**: Three FP3 past papers of your choice from the papers at:

https://mathsmartinthomas.wordpress.com/2015/03/20/fp3-past-papers/

**Year 12 Further Maths**: M2 Solomon paper C; and FP1 papers June 2013 and June 2010

https://mathsmartinthomas.wordpress.com/2015/03/20/m2-mechanics-collections-of-past-paper-questions/

https://mathsmartinthomas.wordpress.com/2015/03/20/fp1-collections-of-past-paper-questions/

**Year 12 Further Maths (Lou-Lou and Debora)**: Two FP1 past papers of your choice

https://mathsmartinthomas.wordpress.com/2015/03/20/fp1-collections-of-past-paper-questions/

**Year 13 maths**: One more D1 past paper (June 2013 for most of you, an earlier one if you’re already done that); and two C4 past papers, of your choice from those on the website.

https://mathsmartinthomas.wordpress.com/2015/03/20/c4-past-papers-and-mark-schemes/

**Homework week 9 term 2, due in on Tuesday 24 March**

*Marking your work, using the mark scheme, is part of the homework!*. All the booklets of past papers you need are on this website, if you’ve mislaid your paper copies.

**Year 13 Further Maths**: The most recent S2 past paper you have not yet done

**Year 12 Further Maths**: M2 Solomon paper B

**Year 12 Further Maths (Lou-Lou and Debora)**: FP1 paper June 2014 (R)

**Year 13 maths**: D1 past paper from June 2014 (R)

**Homework week 8 term 2, due in on Tuesday 17 March**

*Marking your work, using the mark scheme, is part of the homework!*

**Year 13 Further Maths**: S2 paper June 2014, plus more questions from the booklet of S2 questions on binomial, Normal, and Poisson distributions

**Year 12 Further Maths**: M2 Solomon paper A, plus more statics questions marked E from Revision Exercises in the textbook

**Year 12 Further Maths (Lou-Lou and Debora)**: FP1 paper June 2014

**Year 13 maths**: D1 past paper from June 2014, plus more questions from the booklets of past questions on linear programming and on matchings.

**Homework week 6 term 2, to be done over half-term and due in on Tuesday 24 February**

**Year 13 Further Maths**: S2 “practice” paper from pages 131-2 of FP3 textbook

FP3 “practice” paper from pages 200-2 of FP3 textbook

**Year 12 Further Maths**: FP1 past paper from June 2009

M2 “practice” paper from pages 161-3 of M2 textbook

*plus* (for everyone other than Juan and Joan) the FP1 “practice” paper from pages 142-4 of the FP1 textbook

**Year 13 maths**: D1 past paper from January 2009

C4 “practice” paper from pages 142-4 of C4 textbook

When you get stuck on questions, as you will, don’t give up.

1. Think. Use the chapter summaries from the textbooks to remind you of the methods you can use. Try different approaches.

2. Help each other if you can

3. Look up the mark scheme (on the web)

4. If it’s the practice paper from the textbook, look up SolutionBank http://www.physicsandmathstutor.com/a-level-maths/solution-banks-edexcel-heinemann-textbooks/

5. Email me, any time over half-term, for help.

**Homework week 5 term 2, due Tuesday 10 February; and work for Friday 6 February, when you’re not in school**

**M2:** Work for Friday: Do the FP1 practice exam paper on p.142-4 of the FP1 book. Check your answers against the back of the book

Homework: work, energy, and power. Ex. 3D (p.80) Q. 3, 4, 7, 10, 14

Remember the work-energy principle *not* the way it is stated in the book, but as:

PE at start + KE at start + work put in by person or engine – work done against friction = KE at end + PE at end

“Work done against friction” is really another way of saying “energy lost through friction”

Work = force×distance moved in the direction of the force

Power = force×speed, or work per second.

**C4:** Work for Friday: Do another six questions from Ex.4F (differentiation).

Homework: binomial expansion. Finish Ex.3A.

**FP3:** Work for Friday: Vector geometry. Finish Ex.5G and do Review Exercise 2 Q.1-6 (p.193).

Homework: Eigenvalues and eigenvectors – finish Ex.6F.

**Week 4 term 2, due Tuesday 3 February**

**M2:** Work and energy. Ex.3C Q.5,6,15,17

If you can’t do these, then look for some other questions in Ex.3C you can do. Or email me for help.

Remember the work-energy principle *not* the way it is stated in the book, but as:

PE at start + KE at start + work put in by person or engine – work done against friction = KE at end + PE at end

“Work done against friction” is really another way of saying “energy lost through friction”

Work = force X distance moved in the direction of the force

**C4:** Differentiation. Do six questions of your choice from Ex.4F

**FP3:** vector geometry. Do as many questions as you can from Ex.5G, but at least six.

**FP1** (Lou-Lou and Debora): Do a selection of questions from the complex numbers chapter – about adding, multiplying and dividing complex numbers and about converting between polar (modulus-argument) and rectangular (real-imaginary) form – but *not* about getting roots to equations. (On that, I’ll explain a simpler method than the one in the book, next week).

**FP1** (Dandison): Do the FP1 past paper I gave you.

**Week 3 term 2, due Tuesday 27 January**

**FP1 (main class)**: Parametrics. Review exercises p.70-1, finish off as many as you can of Q.57-66, up to no.62 at least. (Sharif, Dion, Mugisha: you should do at least up to Q.6 of Ex.3F, and at least three of the review exercises on p.70-1.

**C4**: Consolidate vectors and start on differentiation. Do three more questions, of your choice, from Ex.5K; and do Ex.4B Q.1 and 2.

**FP3**: Using vectors for three-dimensional geometry. Line where two planes meet; angle between a line and a plane; angle between two planes; distance from a point to a plane. Do whatever you have not already done from Ex.5F Q.1-9.

**FP1 (Lou-Lou and Debora and Dandison)**: do Ex.3A Q.2 and 3, and a selection of your choice from Q.4.

**Week 2 term 2, due Tuesday 20 January**

**C4**: Vectors. Ex.5H Q.3, 4a, 4b. Ex.5I Q.1,2. Ex.5J Q.1,2

**FP3**: Vectors in 3D and cross-product. Ex.5D Q. 1c, 3a, 3b. Ex.5E Q.1c, 3a, 3b, 3c. Ex.5F Q.1a, 1b, 1c, 2a, 2b, 2c.

**FP1 (ordinary class)**: Parametrics. Finish Ex.3F, at least Q.1-6.

**FP1** (Lou-Lou and Debora and Dandison): Choose some questions on series (Lou-Lou) and induction (Deborah). Do enough questions from the matrices chapter to be confident with it (Dandison).

**Week 1 term 2, due Tuesday 13 January**

C4: Vectors. Finish off Ex.5B and 5C (using displacement vectors and position vectors to do problems in geometry of triangles).

FP3: Matrices. Do as many questions on inverting matrices as you think you need to be certain you know how to do it. (You decide how many

that is). Next week we continue on cross product and dot product of vectors.

FP1 (ordinary class): Parametrics. Finish Ex.3A Q.1 and 2 (drawing the graphs) and Ex.3E p.59, Q.1 to 4 (equations of tangents and normals). The main part of each question is just doing what we did in class about finding dy/dx and the equations of tangents and normals. The last bits of Q.3 and Q.4 need a bit more thought.

FP1 (Lou-Lou and Debora): Debora to do enough questions on induction with sequences and series to be confident you know how to do them. Lou-Lou to read the chapter on series and do examples from the book to practice.

**Week 13, due Tuesday 9 December**

**FP1**: Practise matrix multiplying with Exercise 4C, page 81, Q.2-7, and practise matrix addition with Ex 4A, p.75, Q.2, and Ex 4B, p.77, Q. 1-4. (You may want to look at p.76 to help with Ex 4B). Inverting matrices: Ex.4G Q.3, and Ex.4H Q.1-2.

**D1**: Linear programming: Ex.6E Q.1-2

**S2**: Finish the two S2 past papers, May 2013 and January 2013

**Week 12, due Tuesday 2 December**

**FP1**: Linear transformations and determinants. Ex.4E Q.1,2,3; Ex.4I Q.1(b) only, Q.4 (a),(b),(c) only

**D1**: Linear programming: Ex.6C Q.1-3

**S2**: Hypothesis testing: complete and write up the “how random can you get?” survey. This is the assignment sheet with the tallies typed in, and this is the maths on Benford’s Law. See also: these further notes on Benford’s Law.

**Week 11, due Tuesday 25 November**

**FP1**: In your book, work through the test paper again using the mark scheme and the worked answers and trying to find and fix all the mistakes you made in the actual exam. The mark scheme and worked answers will be available here. If you can’t follow the mark scheme and worked answers for some questions, then email me or ask another student.

**D1**: In your book, work through the exam paper again using the mark scheme and trying to find and fix all the mistakes you made in the actual exam. The mark scheme is available here. If you can’t see how the mark scheme gets the answer it does for a question, then email me or ask another student.

**S2**: In your book, work through the exam paper again using the mark scheme and trying to find and fix all the mistakes you made in the actual exam. The mark scheme is available here. If you can’t see how the mark scheme gets the answer it does for a question, then email me or ask another student.

**Week 10, due Tuesday 18 November**

Please mark your own work against the back of the book (if you can) before you hand it in. Year 13 students: please leave your books in my blue tray in the maths office on Monday or early Tuesday morning. The blue tray is on the workbench, at the far corner of the office from the door where you come in, under a sign saying “Homework for Martin Thomas” and a photo of my daughters.

**D1**: Schedules; matching. Finish Ex.5H Q.1-3. Ex.7B Q.1,2,3. When you do revision for the exam, take note of these “don’t do as the book says” notes.

**S2**: Hypothesis testing; continuous random variables. Ex.7D Q.1 and 2, plus four questions from Ex.3E in addition to what you’ve done in class. Plus, use the revision exercises in the book to revise for the exam.

**FP1**: Ex.1H, p.29, Q.4, 5, 7, 9. In Q.5 look out for what Mugisha noticed, that you must change the coefficient of x^{3} to 1 before using the rules about sum and product of roots. In other words, you change the equation to x^{3}−2x^{2}−(5/2)x−(3/2)=0 before using the rules.

Plus, use the practice sheet to catch up on the areas where you were weak in the test on 17 October. Write your work in your book.

Download pdf of practice induction and series problems..

Please mark your own work against the back of the book (if you can) before you hand it in.

**Week 9, due Tuesday 11 November**

**D1**: Critical path analysis and Gantt charts: Ex.5E, Q.1,2; Ex.5F Q.1; Ex.5I, Q.1, 2, 3.a to e; Extra: Ex.5I, Q.4.a to d, 5.a to c

**S2**: Hypothesis testing. Ex.7B Q.1-3, 8-10, 11-12.

**FP1**: Complex numbers. Finish Ex.1H, leaving out Q.8.

Plus, use the practice sheet to catch up on the areas where you were weak in the test on 17 October. Write your work in your book.

Download pdf of practice induction and series problems..

**Week 8, due Tuesday 4 November**

Please mark your own work against the back of the book (if you can) before you hand it in.

**All**: read Tim Gowers’s chapter on proof as a preparation for the sessions on Monday 3 November.

**D1**: Chinese postman algorithm, Ex.4C Q.4,5,6; p.81 Q.18, p.85 Q.29

**FP1**: Newton-Raphson, Ex.2C Q.1-7, but for each problem just guess an approximate solution and use Newton-Raphson just *once* to improve the guess. Your answers will be different from the answers in the back of the book.

Plus, use this practice sheet to catch up on the areas where you were weak in the test on 17 October. Write your work in your book.

**S2**: Hypothesis testing, Ex.7A, Q.1-7

**Week 7: no homework (half-term)**

**Week 6, due Tuesday 14/10/14**

Please mark your own work against the back of the book before you hand it in.

D1: Chinese postman algorithm, Ex.4C Q.1,2,3

FP1: Do the remaining series question:

Show, using the formulae for (sum from r=1 to r=n of r) and (sum from r=1 to r=n of r^{2}), that for all positive integers n:

sum from r=1 to r=n of (3r−1)^{2} equals ½n(6n^{2}+3n−1)

And for each of the following equations, show that the equation has a root in the range mentioned, and do two interval bisections to get a range of ¼ for the answer.

1. f(x)=x^{2}−8=0 in range [2,3]

2. f(x)=½x−(1/x)=0 in range [1,2]

3. f(x)=6x−3^{x}=0 in range [2,3]

S2: Words to use about sampling. Review Exercise, p.127, Q.6, and p.128 Q.14, page 129 Q.17. Exam practice paper p.131, Q.1

**Week 5, due Tuesday 07/10/14**

Please mark your own work against the back of the book, or the answer sheet, before you hand it in.

D1: Prim’s algorithm from distance matrix, and Dijkstra’s algorithm – Ex.3E Q.3, 4, 7, 8 *(I’ll distribute worksheets for these: please stick them in your book)*

S2: Continuous uniform distribution: Ex.4C Q.1, 2, 3 *plus: If you’re still not sure about the normal approximation, do a few more examples from that chapter*

FP1: Series: Ex.5C Q.1, 2, 3 *plus: try some problems from this sheet of relatively easy induction examples as practice if you think you need to consolidate induction*

**Week 4, due Tuesday 30/09/14**

D1: Kruskal’s and Prim’s algorithms. Ex.3A Q.1(b) and (c), Q.2; Ex.3B Q.1(b) and (c), Q.2

S2: Binomial approximation to normal. Ex.5B Q.1-4. You’ll need to do the continuity correction: check how to do it from the book.

FP1: Induction problems with series. Ex.6A Q.3-6

**Week 3, due Tuesday 23/09/14**

FP1: Proof by induction. Do another six questions from the sheet of past exam questions.

fp1-sequence-and-divisibility-induction-pastq

S2: Binomial and Poisson distributions. Ex.2F Q.6-10.

You should also **think** (for Thursday) about what hypotheses we might test about the patterns when we ask students and teachers to write down a list of 20 numbers between 1 and 999999 chosen at random.

D1: Bin-packing: Ex.1E Q.3-5. Graphs and trees:

Draw a weighted graph (network) representing the following vertices

A. The main entrance of CoLA

B. The entrance to the library

C. The entrance to the hall

D. The maths office

E. The Sixth Form Study Area. (For this purpose regard the whole of the Sixth Form Study Area as a single point, and assume you have arrived at that same single point whether you come by the South staircase or the West staircase).

You should include 15 edges:

– the routes between the main entrance, the library, and the hall across the ground floor of CoLA;

– routes via the West and North staircases from all three to the maths office;

– routes via the West and South staircases from all three to the Sixth Form Study Centre.

Measure them by pacing them out, and for this purpose assume one pace equals one metre.

**Extra** Read https://mathsmartinthomas.wordpress.com/2014/08/23/bin-packing-algorithms/ and write a short summary of it in your notebook.

**Week 2, due Tuesday 16/09/14**

S2: Binomial and Poisson distributions. Review exercises p.64 Q.1,2 , Ex.1E Q.9, Ex.2E Q.1-4.

D1: More on Quicksort and Bubble Sort. Binary search. Ex.1F Q.3. Ex.1C Q.3,4; Ex.1D, Q.5; p.81 Q.15.

FP1: Proof by induction. Ex.6C Q.3-5. Review exercises, p.141 Q. 44, 45, 48 (u_{n} type problems)

**Week 1, due Tuesday 09/09/14**

S2: Binomial distribution

Ex.1B Q.1-3

Ex.1C Q.1-3

Ex.1D Q.1-3

Ex.1E, Q.4, Q.7

Optional extra: Review exercises, p.64 Q.4, Q.8

D1: Quicksort and Bubble Sort.

Ex.1C Q.1,2

Ex.1F Q.2

Ex.1C Q.3,4

FP1: Write proofs of two claims chosen from A1 to A5, or B1 to B8, in the booklet

Lesson_and_homework_on_proof