A nice coordinate geometry problem (suitable for Form 7 pure)

Let be a point on the curve

For my IMO Students (Basic and Advanced)

Here is good problem for my IMO students.

Given a real number . Find the greatest area of triangle ABC with circum-radius

For my Form 7 Pure students: A proof of the Rational Root Theorem

Let Suppose , where is a rational root.

To my Form 7 students: I am impressed!

Just finished my CB class tonight and I am impressed by two students. They both expressed interests in the following problem:

Suppose is a polynomial of degree . Suppose further that

(i) for all and

(ii)

Find the leading coefficient of and the degree of

Selected solutions for my F7 Pure (Polynomials)

Book 2, pg 14. (00IQ12)(c)
(i) Let y = x^2, then we have ay^2 – by + a.
Considering the discriminant yields: b^2 – 4a^2 > 0. Therefore, it has two distinct roots: If they are both positive, then we are done. Otherwise, they must be both negative since (Note since .)
Suppose Then we have

Challenging problem in Linear Algebra

Let

Solve the following matrix equation where

Mental exercises for my Pure Complex Class (1)

Dear Students,

Please avoid as much calculations as possible. In ideal situation, you should not do any calculation at all.

Problem 1
Let be a complex number.
(a) Find
(b) Find
(c) Find
(d) Find
(e) Find
(f) What is
(g) Suppose , write in terms of

Problem 2
(a) Write in the form
(b) Find the two square roots of and represent the two square roots on the Gaussian Plane.

Selected Solutions to My Pure Complex Class (1)

Lesson 1:
Page 26
where the last line follows from part a(ii).

Concept Check for my Linear Algebra Students

Let (E) be the following system:

Solution to “Possible HKAL Pure Problems”

1(a) Expanding along the last row, we have:

Therefore, is a cubic in with leading coefficient