Some very standard maths questions that I will be posting

The following is from RJC Prelims Year 2000 Maths C

Find all angles between 0° and 180° which satisfy the equation

cos x - sin 2x + cos 3x - sin 4x = 0

(Ans: 18°, 90° or 162°)

The diagram shows the frame of a writing desk. AB, DC, PQ, SR are perpendicular to the horizontal base BQRC. The rectangle ADSP represents the sloping plane of the desk. ABQP and DCRS are trapezia with SR = PQ = 30 cm, DC = AB = 6 cm, CR = BQ = 42 cm, and BC, AD, QR, PS are each 56 cm.

Giving each answer correct to the nearest 0.1°, find

(i) the angle between PC and plane PQRS

(ii) the angle between the lines RB and SA

(iii) the angle between the planes ADSP and PSXY where X, Y are points on CR and BQ respectively with CX = BY = 12 cm

Ans: (i) 33.5° (ii) 18.9° (iii) 15.3°

RJ Year 2000 Maths S paper question on induction

Show by induction that, for every positive integer n, both

cos (nx) and sin (nx) / sin x

can be expressed as polynomials in cox x with integer coefficients.

Answers to the first 2 questions has been posted on ExamWorld

Answer to the 3rd question will be posted tomorrow. Hint: must prove both together at the same time; i.e. P1 and G1 are true...

Originally posted by eagle:

RJ Year 2000 Maths S paper question on induction

Show by induction that, for every positive integer n, both

cos (nx) and sin (nx) / sin x

can be expressed as polynomials in cox x with integer coefficients.

For this question, it tests very much on the actual concepts of induction.

Solution also out on ExamWorld under Mathematical Induction

I will post it out here as well:

Let P_n be the statement: cos (nx) can be expressed polynomials in cos x with integer coefficients
Let G_n be the statement: sin (nx) / sin x can be expressed polynomials in cos x with integer coefficients

For cos (nx),
n=1 => cos x
n=2 => 2 cos² x - 1

P₁ and P₂ are true

For sin (nx) / sin x,
n=1 => 1
n=2 => 2 cos x

G₁ and G₂ are true

Suppose P_k and G_k are true.

P_k+1:
cos ([k+1]x) = cos kx cos x - sin kx sin x
= cos kx cos x - (sin kx / sin x) (sin² x)
= cos kx cos x - (sin kx / sin x) (1 - cos² x)

Since cos kx can be expressed as a polynomial in cos x with integer coefficients, cos kx cos x can also be expressed as a polynomial in cos x with integer coefficients.

Since sin kx / sin x can be expressed as a polynomial in cos x with integer coefficients, (sin kx / sin x) (1 - cos² x) can also be expressed as a polynomial in cos x with integer coefficients.

Therefore, suppose P_k and G_k are true, P_k+1 is true.

G_k+1:

sin ([k+1]x) / sin x = (sin kx cos x + cos kx sin x) / sin x
= (sin kx / sin x) cos x + cos kx

cos kx can be expressed as a polynomial in cos x with integer coefficients

Since sin kx / sin x can be expressed as a polynomial in cos x with integer coefficients, (sin kx / sin x) (cos x) can also be expressed as a polynomial in cos x with integer coefficients.

Therefore, suppose P_k and G_k are true, G_k+1 is true.

Combining, P_k and G_k are true implies that P_k+1 and G_k+1 is true

But P₁, P₂, G₁ and G₂ are true, i.e. P₁ and G₁ true implies that P₂ and G₂ are true. P₂ and G₂ true implies that P₃ and G₃ are true. And so on...

Hence, by mathematical induction, P_n and G_n are true for every positive integer n.
i.e. Both cos (nx) and sin (nx) / sin x can be expressed as polynomials in cos x with integer coefficients.

put all the questions in examworld also.

lol.

Hi,

  Visit my website for learning resources on H2 mathematics:

  www.freewebs.com/weews

  Btw, 3D trigo and solving of trigo equations are not examined in H2 mathematics.

  Thanks!

WS

Hi,

  Students preparing for H2 mathematics A-level exams may find practicing JJC, IJC, SRJC, TPJC prelim 2008 exams worthwhile. Try doing Cambridge 2007 and Specimen papers too. Good luck! Thank you.

WS

I just checked the SEAB website

# Last year of exam in 2009.

Wonder what they going to change now...

Originally posted by eagle:

I just checked the SEAB website

 

# Last year of exam in 2009.

 

Wonder what they going to change now...

Referring to H2 maths right? Coz 2008 is 1st year of new O level maths syllabuses? And these students will take H2 maths in 2010?

Originally posted by Mikethm:

Referring to H2 maths right? Coz 2008 is 1st year of new O level maths syllabuses? And these students will take H2 maths in 2010?

Probably... Anyway, no matter how they change, as long as we can do, then can liao.

Hi,

  The only change I have heard so far is the removal of geometry from H3 maths. Thanks!

WS

Hi,

  Students preparing for H2 maths can access Cambridge June '08 exam questions and answers on my website: www.freewebs.com/weews

  Good luck!

Cheers,
WS

welcome back from reservist

Hi,

  Thanks, eagle :)

  Students preparing for H2 maths may refer to Jan '08 AQA (i.e. an exam board in UK) exam questions and mark schemes from this site:

  http://www.aqa.org.uk/qual/gceasa/mathematics_assess.php

  I have included the link on my website too :)

  Please look at relevant questions (by now, you should be able to identify them) from Pure Core 2 - 4 and Further Pure 1 - 4 papers.

  Thanks! Feel free to share these questions with your friends so that everyone benefits!

Cheers,
Wen Shih
www.freewebs.com/weews

Dear students,

There are many key words you often encounter in mathematics exam questions.

Some examples of process words (or key words) and their implied meanings are:

1. 'Write down', 'State', 'Give', 'Express/Express briefly', 'List', 'Specify' mean write down without justification, i.e. no working need be shown (although you may include appropriate working if it helps you).

2. 'Find', 'Calculate', 'Determine', 'Simplify', 'Derive', 'Solve', 'Evaluate' 'Transform', 'Expand', 'Factorise', 'Differentiate', 'Integrate' mean work out and show your working, using standard results and techniques.

3. 'Prove', 'Show', 'Deduce', 'Explain', 'Indicate', 'Justify', 'Demonstrate', 'Determine', 'Decide (if, whether ...)', 'Verify', 'Confirm', 'Test', 'Predict', 'Illustrate', 'Identify' mean you must justify each step and provide a convincing argument.

4. 'Assume', 'Consider', 'Suppose', 'Apply', 'Use', 'Define (in terms of ...)', 'Sketch', 'Draw', 'Plot', 'Graph', 'Compile (a table ...)', 'Make (a table, list, ...)' indicate you must answer the question in a particular way indicated by whatever words immediately follow each of these cueing terms.

5. 'Explore', 'Investigate', 'Devise', 'Design', 'Obtain', 'Find', 'Adapt', 'Construct', 'Produce (an algorithm, argument, diagram, ...)', 'Translate (from, into, ...)' indicate doing a mathematical activity and then reporting on the process and result.

Source: Success with Mathematics by Heather Cooke, pp. 49

Please familiarise yourself with these words, and share them with your friends. Thanks!

Cheers,
Wen Shih
www.freewebs.com/weews

Hi,

  I have posted answers to the 2008 H2 Maths exam on my website:

  http://www.freewebs.com/weews/2008examanswers.htm

  We could start discussing any doubts you may have on the questions :) The fully worked solutions will be available in the form of a new solution book edition from Feb '09 onwards. Do look out for it.

Cheers,
Wen Shih

hi wenshih

the solutions are already out on www.exampaper.com.sg

Hi,

  I know :) 

  The solutions that I write include student-friendly explanations and GC screenshots, which is not present in www.exampaper.com.sg :P Anyhow, there is an error in the Argand diagram in that solution. The required region is the line segment within the circle including the 2 end-points on the circle :P

Cheers,
Wen Shih

Clap clap clap :D

Originally posted by wee_ws:

Hi,

  I have posted answers to the 2008 H2 Maths exam on my website:

  http://www.freewebs.com/weews/2008examanswers.htm

  We could start discussing any doubts you may have on the questions :) The fully worked solutions will be available in the form of a new solution book edition from Feb '09 onwards. Do look out for it.

Cheers,
Wen Shih

Hi Mr Wee,

Perhaps you can have a page devoted to the analysis of 2007 and 2008 questions and classified them into the different topics in a tabular format in your coming 2009 solution book to be published by Dyna Publisher (MOE authorised TYS publisher).

Thank you for your kind attention.

Regards,

ahm97sic

Hi,

  Thanks for your suggestion :) I have included a detailed analysis of the 2007 exam questions in the previous version of the book (please refer to the last 2 pages of the document below):

  http://www.freewebs.com/weews/Worked%20Solutions%20to%202007%20H2%20Mathematics%20Papers%201%20&%202.pdf

  I am happy to continue with this feature in the 2009 version.

Cheers,
Wen Shih

Hi Mr Wee,

Oh, I bought the earlier 2007 edition. So, it did not have the detailed analysis of questions then.

I have visited your website too and you intend to write a guide for H2 Maths. There are a number of guides (by yellowreef, cosmic, epb etc) and textbooks (by epb, pan pacific, elbs etc) for H2 Maths now. However, there is no formula booklet and the essential steps and concepts for each topic for H2 maths to help the students for quick reference and revision.

Perhaps you will be interested to write one.

Thank you for your kind attention.

Regards,

ahm97sic

Hi ahm97sic,

  I have changed my mind about writing the guide, as I'm aware of those books you have mentioned. I guess the website that I have developed is sufficient to meet the needs of students facing specific learning difficulties :P

  Perhaps I should do a print version of the website. What do you think?

  Yes, the formula booklet is an idea I am exploring. The essential steps and concepts are already indicated in the solutions manual :) Typically, students buy the solutions manual for quick revision before their A-level.

  Thanks again!

Cheers,
Wen Shih