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Cambridge Math AS level

Teacher: Carlos Sotuyo.
Email: csotuyo@imathesis.com & csotuyo@hotmail.com


If you change the way you look at things, the things you look at change.

Max Plank


If you can't explain it simply, you don't understand it well enough.
Albert Einstein

Any fool can know. The point is to understand.
Albert, again.



A & AS Level : Mathematics (9709) Syllabus.

Hialeah Gardens High School
Jack Espinosa, Cambridge coordinator website.

Textbooks at Cambridge.org webpage.

Advanced Level Mathematics: Pure Mathematics 1
ISBN: 9780521530118 at amazon.com

Advanced Level Mathematics: Pure Mathematics 2 & 3
ISBN: 978-0521530125 at amazon.com



Welcome to UNIVERSITY of CAMBRIDGE
International Examinations

Running Cambridge Exams.

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Selected Questions.
Suggested solutions by C. Sotuyo:

Pure Mathematics 1 by Hugh Neill and Douglas Quadling, Cambridge University Press, 2002.

Miscellaneous exercises 1, page 15:
1. Show that the triangle formed by the points (-2,5), (1, 3) and (5, 9) is right-angled.

Miscellaneous exercises 4, page 62:
4. For what values of k does the equation have a repeated root?

Exercises 6D, page 87:
12.
A curve has equation , where a is a constant. Find the equations of the tangents to the graph at the points where it crosses the x-axis.

Exercise 6E, page 93.

2. Find the derivative of the function at x=p. (Use the difference of two squares formula as often as you can.)

11. Show that the curves
have exactly one point in common, and use differentiation to find the gradient of each curve at this point.

Excercises 7C, page 110

16. A circular cylinder is to fit inside a sphere of radius 10 cm. Calculate the maximum possible volume of the cylinder. (It is probably best to take as your independent variable the height, or half the height, of the cylinder.)

Miscellaneous exercises 8, page 126 No. 8 & 9

8. An arithmetic progression has first term a and common difference -1. The sum of the first n terms is equal to the sum of the first 3n terms. Express a in terms of n.

9. Find the sum of the arithmetic progression 1, 4, 7, 10, 13, 16,...1000.
Every third term of the above progression is removed, i.e. 7, 16, etc. Find the sum of the remaining terms.

Miscellaneous exercises 9, page 136 No. 17 and No. 28 page 137.
17. Find and simplify the term independent of x in the expansion of

28. Prove that

Miscellaneous exercises 11, page 173.
Given that the function ; where x is a real number different from zero, find:


Miscellaneous exercises 9, page 137
Find an expression, in terms of n, for the coefficient of x in the expansion

Miscellaneous exercises 10, No 5 page 153:
Solve the equation 3cos2x=2, giving all the solutions in the interval 0<x<180.

Miscellaneous exercises 10, No 19 page 155:
The road to an island close to the shore is sometimes covered by the tide. When the water rises to the level of the road, the road is closed. (...) See pdf doc for a solution to this problem.

Excercise 13D, # 16 page 207:
The roof of a house has a rectangular base of side 4 metres by 8 metres. The ridge line of the roof is 6 metres long, and centred 1 metre above the base of the roof. Calculate the acute angle between two opposite slanting edges of the roof.

Answers to selected questions pdf doc here.

Online resources:
MathCentre, UK.
A level Mathematics: Resources for teachers.

Casio Scientific Calculator FX-115ES PLUS
Algebra, Review. Pdf online, 12 pages.
Properties of quadrilaterals, pdf, 2 pages.
Geometry, a study guide, pdf doc, 104 pages.
Proofs in Coordinate Geometry.
History of Mathematics in 50 Minutes by Mathematics Professor John Dersch (video lecture).

Rabbits in Mathematics
The Fibonacci sequence:

Let's begin with a pair of rabbits. Assume that that rabbits begin to reproduce two months after their own birth. This is, after reaching the age of two months each pair produces one male and one female, and another pair each month thereafter. Also assume that no rabbit dies. This is the sequence of numbers of rabbits generated by the original pair of rabbits:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on...

The Fibonacci Sequence!

Exponential Growth:

In 1859, an Australian farmer, Thomas Austin, imported two dozen wild English rabbits and set them free on his land. Within six years Austin's 24 rabbits had multiplied to 22 million. At a rate of 2.29, the following equation explains the rabbits population growth during that period of time:


Flies in Mathematics

1) Descartes & the coordinate system.
2) Von Neumann and the fly problem, By Idiom Zero (blog).

Notes, Practices & Assignments:

Pure Mathematics 1

Notice: Oct 20, 2014. I taught Pure 1 & 2 two years ago, 2012-2013. I kept the webpage, however. Reviewing the webpage stats I've learned that tens of people visit this page every day looking for resources. I am using La Tex (textmaker editor) to create the pdf file. It takes time the typesetting of math. Solutions of some problems are downloable for free.
I'm working on the rest of the practices. Practices with answers appear in red, see below.

The content of the practices --exercises, page number etc of Pure Mathematics 1, by Hugh Neill and Douglas Quadling, Cambridge University Press, 2002, are indicated below. The answers consists of step by step solutions. In some instances, like practice 4, the answers take 6 pages; in practices 5 & 6, five pages each.

Selected Questions: Suggested solutions by C. Sotuyo, here.

Typesetting math with LaTex: in order to get this, type \left(\frac{1}{2x}+x^3\right)^8.
LaTex Editor, texmaker. See here LaTex tutorial by Michelle Krummel.
A quick guide to LaTex here, by Dickinson College. Also, LaTex common expressions, by Aurora soft.
LaTeX article template (document header).

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  1. Introduction to the course. Pretest. This diagnosis test is based on the Algebra II text book in use in Florida, USA.
  2. Geometry and coordinate geometry. Students should check online resources: Properties of quadrilaterals, pdf, 2 pages; also Proofs in Coordinate Geometry.
  3. Chapter 1: Coordinates, points and lines. Practice 01 page 15 exercises 1 to 12. Answers to practice 01.
  4. Aug 30th: review chapter 01. Next week, Tues 9/4, quiz chapter 1.
  5. Chapter 2. Surds and indices., Practice 02 here.
  6. Practice 03, ex 10 to 20 page 30 of the textbook.
  7. Quiz 02, surds and indices, based on practice 2 week 9/10-9/14.
  8. Week 09/17-9/21: Chapter 3. Functions and graphs & ex 3 to 19 page 49: Practice 04.
  9. Assignment chapter 01 & Assignment chapter 02 due 09/28.
    Assignment 01 is page 15 --textbook, except question 4.
    Assignment 02 is page 29 excercises 4 to 8, and miscellaneous from 1 tru 5, same page.
  10. Assignment 03, questions. Functions, domain, graphs. Assignment 03, answers.
  11. Online review: graphing polynomials, Paul's online math notes.
  12. Week 09/24-9/28: Chapter 4: Quadratics Practice 05, Miscellaneous ex 4, page 62.
  13. Week 10/08-10/12. Quiz 3. Quadratics.
  14. Chapter 5, Inequalities. Practice 06, page 72 (ex 1 tru 11)
  15. Assignment 04. Question 4 page 72, from a) to l). Due 10/12.
  16. Chapter 6, Differentiation. Practice 07, 6D page 86-87, #1 tru 13. Solutions here.
  17. Rules for differentiation, proofs. Practice 08 page 93, exercises 6E and miscellaneous ex 6, page 94, 1 tru 6.
  18. Assignment 05, page 86, chapter 6, exercise 6D # 5, 6, 7, 8, 9 and 14.
  19. Practice 09, differentiation, miscellaneous exercise 6 page 94, questions 7 tru 15. Questions 1 tru 6 included as part of Pratice08.
  20. Applications of differentiation. Practice 10a, Exercise 7C, from #6 to 16 page 110. Practice 10a, answers.
  21. Assignment 06, due Fri 11/2nd. Item c) for each question on exercise 7B.
  22. Derivatives as rates of change. Practice 10b, miscellaneous exercise 7, from question 1 to 17, pages 110 to 113.
    Practical Tips for Modeling Optimization Problems.
  23. Sequences, chapter 8. Practice 11, section 8C (from 8 to 11, p125) and miscellaneous exc 8, p125-126, form 1 to 15. Sequences answers here.
  24. Assignment 07, Sequences, problems 10 & 11 page 125.
  25. Chapter 9, The binomial Theorem. Practice 12 Ex 9B and Miscellaneous 9 from number 1 to 33, pages 135 to 137. Solutions here. Exercises 9B #6 to 13 (page 135) & Miscellaneous excercise 9, #1 to 11.
  26. Chapter 10, Trigonometry practice 13: excercises 10D, page 152 & miscellaneous 10, page 153 from 1 to 4.
  27. Trigonometry Practice 14, miscellaneous 10, page 153 to 155, # 5 tru 20. Selected solns #5 to 14, here.
  28. Assignment 10, transformation of graphs of trig functions.
  29. Chapter 11, combining and inverting functions, Practice 15, miscellaneous exercise 11 page 172-173 #1 thru #20. Assignment 11, p172: 1 - 4. Selected solutions: questions #5, #16, #17, #19 & 20 here.
  30. Chapter 12, extending differentiation. Related rates problems ! Practice 16: p182-183. All of them, 1 tru 9. Practice 17, differentiation, related rates: miscellaneous exercise 12, pages 184-186.
  31. Vectors, Practice 18 Miscellaneous exercise 13 p207-209. see also Michael Corral, vector Calculus, first 18 pages. Basic theory on vectors, recommended ! click here.
  32. Assignment 13, Vectors: exercise 13D: 12-16 page 207 questions & answers here.
  33. Geometric Sequences: Jan 7 & 9. Practice 19, exercises 14A p213-214; Practice 20, 14B p.217 & Practice 21 exercises 14C p 221. Quiz 7 Fri 1/11
    Assignment 14, Geometric seq: page 213, 14A. # 3, b) c) #4 c) #5 b) #6 c) and from 14B page 217 #2 a). Practice 22, geometric sequences miscellaneous exercise 14 pages 222-224.
  34. Second derivatives: Practice 23, 15A p228 & 229, also 15B p232.
  35. Second derivatives: Practice 24, miscellaneous exercise 15 pages 234-235.
  36. Integration: Practice 25, 16B p245 & p251, 16C. Assignment 15: p240 exer 10e, 14, 15e, & p244, 4e and #9. Integration, Practice 26, miscellaneous exercise 16 pages 255-257.
  37. Volumen of revolutions (Integration). Practice 27, ex 17 from 4 to 9 and miscellaneous exercise 17 pages 262-263. Assignment 16 p262 #2 d), #3 f), #6 & 7.
  38. Radians. Practice 28, miscellaneous exercise 18 p275-276. Assignment 17: Page 275 exercise 18D, question 2 a) b) c)

    Pure Mathematics 2


  39. Polynomials. Practice 27: Pure 2, page 16, misc 1 on 02/13 & 15; Wed 2/20, quiz. Recommended: College ALGEBRA, Univ of Texas. Tutorials 36, 37, 38.
  40. The modulus function: chp 2, pure 2. Practice 28, problems on page 28. Assignment 18: from 2a p23 1 b) & j. from 2b p28 1 f) 2 g) and 3 c).
    Paul's Online Math Notes: Absolute Value Equations, Absolute Value Inequalities and Rational Inequalities.
  41. Exponential and logarithmic functions. Practice 29, 3B 1 & 2; 3C 1 tru 8; 3D, 1 (a,b,c,d,e); miscellaneous 3 p48 from 1 tru 11. Assignment 19: 3A p34 #8 a, b, c; from 3C p40 #1 g, h, i.
  42. Differentiating exponentials and logarithmic functions. Practice 30, 4A p54. Practice 31: 4b ( 1 thru 6), 4C ( 1 tru 6), 4D ( 1, 2 & 3). Misc p62 1 tru 4. Assignment 20: 4A, 6 a), 7 f) 8 e) 4B 1 c) 4 a) 4C 1 f)
  43. Trigonometry (identities, equations): Practice 32: p68 ex 5A 7 to 12. Ex 5C p75 from 1 tru 10. Assignment 21: 5C p75, 10b only
  44. Video: Fx-115 ES Plus (This Webinar features the fx-115ESPlus calculator)

  45. Differentiating trigonometric functions: chapter 6. Practice 33, 6A p88 #1 to 6, 6B p92#s 1 tru 3. Assignment 22: 6A, p88 #2 d, g. #3, g. #5, b. #6, c.
  46. Differentiating products: chapter 7. Practice 34, 7A pg 102 # 2 thru 6, 7B p105 #s 1 thru 5. Assignment 23: 7A, p102 #2 e, f; #5, b. 7B, p105, #3 a, #4 a.

  47. Solving equations numerically: Practice 35, 8A p112 and 8B, p114.
    Assignment 24: 8B p114 # 2 c, & 3.
    Related topics: solving cubics by factoring, rational zeros theorem & Descartes rule of sign and Abel's impossibility theorem. Recommended video: chaos, fractals, and dynamics by Robert L. Devaney ( He explains the process of iteration in the context of dynamical systems.)

    Algebra resources: using GeoGebra, and Twelve Basic Parent Functions.

    Tutoring: Tues & Thursdays, 2.30-3.30pm room 2201

  48. The trapezium rule. Practice 36, excercise 9 p 125.
    Assignment 25: Exercise 9 p125 #4 a, b, c.
  49. Parametric equations. Derivatives of parametric eq.
    Practice 37, exc 10A p 133 & 10B p 136.
    Assignment 26: 10B, #9 a) b)
  50. Curves defined implicitily. Derivative of implicit functions.
    Practice 38,
    11 A p 147 #1 to 6, and 11B p150.
    Assignment 27: 11B #5 & #6.
  51. Discussion of final exams.
  52. Quiz based on past examinations: week April 29- May 3rd
  53. Final test: May 7, Pure 1, 1h 45 mins - 11 questions, max 60 points
    Pure Math 2, May 13, 1h 15 mins - 8 questions, max 40 points.
  54. Integration by substitution. A glimpse to AP Calc.
  55. Intro. History of math. The Golden ratio, etc
  56. School year 2012-2013 ends !

 

Numberphile, Cambridge University: amazing youtube channel:

Infinity is bigger than you think - Numberphile: