Vectors Question #3

Vectors Question #3

JC Mathematics

This is a question a student sent me a few days back, and I shared with my class.

Find the Cartesian equation of the plane that is equidistant of the xy plane and xz plane.

The following should aid students to visualise.

xy-, xz-, yz-planes

Sidenote: I think Vectors is a very important topic for 9758 as its applications are wide. Students should do their best to understand the topic. I will share a few more applied questions next week when I have time.

A little reminder to students doing Calculus now

A little reminder to students doing Calculus now

JC Mathematics

When \frac{dy}{dx} = 0 , it implies we have a stationary point.

To determine the nature of the stationary point, we can do either the first derivative test or the second derivative.

The first derivative test:

First Derivative Test

Students should write the actual values of \alpha^-, \alpha, \alpha^+ and \frac{dy}{dx} in the table.

We use this under these two situations:
1. \frac{d^2y}{dx^2} is difficult to solve for, that is, \frac{dy}{dx} is tough to be differentiated
2. \frac{d^2y}{dx^2} = 0

The second derivative test:

Second Derivative Test

Other things students should take note is concavity and drawing of the derivative graph.

APGP Question #3

APGP Question #3

JC Mathematics, Mathematics

This is a question from a recent ACJC JC2 test.

A philanthropist started a donation matching programme to encourage more people to donate regularly to a particular charity.

(a) For a person who donates $a in the first month, and for each subsequent month donates $ b^2 more than the previous month, the philanthropist will donate $a in the first month and for each subsequent month, b times the amount he donated the previous month.

(i) John is a regular donor of the charity. Find the values of a and b such that the philanthropist donates $20 in the third month, and ten times more than John in the seventh month.

(ii) Find the total amount of money donated by John and the philanthropist in one year, leaving your answer to the nearest dollar.

(b) In a revised donation matching programme, if a donor makes a monthly donation of $c, the philanthropist will donate according to the following plan:

First month : No donation (0\% donation rate)
Second month : 10 \% of the total amount donated by the donor in the first two months
Third month : 20 \% of the total amount donated by the donor in the first three months
Fourth month : 30 \% of the total amount donated by the donor in the first four months

and so on.

(i) Show that the total amount of money the philanthropist will donate at the end of the 4^{th} month is $2c.

(ii) By expressing the total amount of money, the philanthropist will donate at the end of the n^{th} month as a summation, and using the result \sum_{r=1}^n r^2 = \frac{n}{6}(n+1)(2n+1), show that the total amount of money the philanthropist will donate at the end of the n^{th} month is $ \frac{(n-1)n(n+1)}{30} c .

Vectors Question #2

Vectors Question #2

JC Mathematics

If c = |a| b + |b| a, where a , b and c are all non-zero vectors, show that c bisects the angle between a and b.

Post-Results 2016

Post-Results 2016

Chemistry, JC Chemistry, JC General Paper, JC Mathematics, JC Physics, Mathematics, Studying Tips, University Mathematics

Let’s face it. Some of us will not get the dream results we want. Don’t give up and let fear conquer you.

For students unsure of the available courses, they can check out the following post. It contains the grade profile for local universities.

Our Team will be here if you need help/ advice. Feel free to text us.

P.S. Today, I saw an image shared by Mr Wee, which said that “You’re the architect of your own life”. So let’s not let the grades define us.

Things to note for 9758 H2 Mathematics

Things to note for 9758 H2 Mathematics

JC Mathematics, Studying Tips

It has been awhile since the A’levels. We talked and met up with several of our students. Some students are working and some are preparing their Personal Statements for overseas University Applications.

A few of them also remarked that they wish they put more efforts into studying A’levels.  An advice to this year JC2 – Do not wait till it is too late.

Students should also have a clear understanding of their syllabus, especially their scheme of exam. I still have JC2s this year who get stunned by the applications questions I threw at them (P.S. Aside from spending time with family, I wrote many sets of applied questions.). You may read more about your syllabus here. Or see the following images.

Integration & Applications Source: SEAB
Scheme of Examination Source: SEAB

How much practice or emphasis your school put on this, is up to them. But it is clear that application takes up 25% of your marks. The entire syllabus can be found here.

PLEASE DON’T WALK INTO THE EXAM HALL BLUR BLUR…

Checklist for Vectors

Checklist for Vectors

JC Mathematics

Many schools have been doing vectors recently. Thought I’ll share a little summary/ checklist I have done for my students.

Basic Concepts

  • Operations on Vectors
    • Addition & Subtraction
    • Scalar multiplication
    • Dot Product (Scalar)
      1. a • a = |a|2
      2. If a ⊥ b, then a • b = 0
      3. a • b = b • a
    • Cross Product (Vector)
      1. a × a = 0
      2. a × b = − b × a
  • Unit Vectors
  • Parallel Vectors ( a = k)
  • Collinear Vectors ( Parallel with a common point )
  • Ratio Theorem ( Found in MF26)
    • Midpoint Theorem
  • Directional Cosines
  • Angles between two Vectors
  • Length of Projection
  • Perpendicular Distance

Lines

  • Equations
    • Vector Form ( : r = a + λb, λ∈ ℜ )
    • Parametric Form
    • Cartesian Form
  • Line & Line
    • Parallel ( Directions are parallel to each other. )
    • Same ( Same Equations )
    • Intersecting ( There is a unique solution for λ and μ. )
    • Skewed ( Not parallel AND not Intersecting. )
  • Angle between two lines ( Angle between their directions )
  • Point & Line
    • Foot of Perpendicular
    • Perpendicular (Shortest) distance
    • Point on Line

Planes

  • Equations
    • Parametric Form ( π r = a + λb + μc, λ, μ ∈ ℜ )
    • Scalar Product Form ( r • n = a • n  = d )
    • Cartesian Form
  • Point & Plane
    • Foot of Perpendicular
    • Perpendicular (Shortest) distance
    • Distance from O to Plane
    • Point on Plane
    • Reflection of Point
  • Line & Plane
    • Relationships
      1. Parallel
        • Line intersects Plane entirely ( Infinite Solutions )
        • Do not intersect ( No Solution )
      2. Not Parallel
        • Intersects at a point ( One Solution )
    • Intersection Point
    • Angle between Line & Plane
    • Reflection of Line
  • Plane & Plane
    • Relationships
      1. Parallel
        • Same ( Infinite Solutions )
        • Do not intersect ( No Solution )
      2. Not Parallel
        • Intersects at a line ( Infinite Solutions )
    • Intersection Line ( Use of GC )
    • Angle between two Planes ( Angle between their normals )
A little history of e

A little history of e

JC Mathematics, University Mathematics

Some students remarked on why I actually recognise e, that is, e=2.718281828.... Well, e is a rather unique constants. Firstly, for all JC students, we see it our daily algebra & complex numbers. Students exposed to university statistics will see e appearing in the formula for normal distribution, that is, f(x | \mu , \sigma^2) = \frac{1}{\sqrt{2 \sigma^2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}.

Secondly, the story of how it came about is pretty cool as you will observed in the video below.

The Story of e

Hopefully it provides you with another perspective towards this constants! And now you should be more cautious when signing up savings plans that give interest per annum or per month.

P.S. I once confused a banker when I asked her about this. 🙂

How to maintain your self discipline to study?

How to maintain your self discipline to study?

JC General Paper, Studying Tips

Dear all, this is a general post on how to keep your study momentum up during this december holidays. As you already know, this december holidays is not only a time for you to rejuvenate yourself from the hectic JC life, it is also an important revision time for you! It is a time when you consolidate all that you have learnt in this year, so that you would have a good foundation to start JC2. (For those who didn’t do well in the promos, you should be doing catch up. For those who did well, you could possibly revise your learning and embark on a head-start program.)

So what are the 3 tips for you to have self-discipline?

One of the reason why students do not have self-discipline is because they do not believe in themselves and their abilities to attain academic success. They attach negative talk to themselves such as they are stupid, lazy and not cut out for straight As. Hence, their inner beliefs shape their actions and their revision process. One way to change it would be to change your identity. Start thinking to yourself that you deserve Straight As everyday and you will soon internalize it and manifest these behaviours. 

Secondly, you should attach pain to the notion of you not achieving your goals. Each time when you think about how you would not achieve your goals, it will automatically propel you to take action! It could possibly be a lack of self esteem, respect from your parents and society. Regret for not achieving your aims etc.

Finally, you could think about how to reward yourself whenever you achieve your goals and to get a trusted person to be accountable for you. One way to do this is to tell your friends or parents what you desire to achieve for the A levels, and to get them to monitor your revision schedule. I know this sounds unappealing to you, but this could perhaps be the best way to make sure you stay discipline to your goals!!

Have a good break everyone! Of course have a fruitful revision time too!