Selected Solutions to My Pure Complex Class (1)

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

Solutions to selected problems for my Olympiad Class

Book 1

Page 1
Answer which can be looked up from the Pascal Triangle.

Page 2
Answer:

Page 2
Answer:

Page 4
Answer: because we have to travel 4 steps in total, and we need only to decide the 2 horizontal steps.

Alternatively, one can count and add along the way to see that the Pascal triangle appears again as the number of paths.

Concept Check for my Linear Algebra Students

Let (E) be the following system:

Review Problems for 324 Final

Chapter 14 Review, pg 946. Q37, 39, 47, 48.

Mid-Term 2 Solution

1) (a)

Math 324 Mid-Term 1 Solution

1) (a)

(b and c)

Tricks of Integration (1) – The Koop Subs

In this article, I will talk about some of my favorite substitutions for integration, which I will named them as the “Koop subs”. They are particularly useful in mathematics competitions such as the Putnam competition.

Example 1) Compute the integral:

Solution to “Possible HKAL Pure Problems”

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

Therefore, is a cubic in with leading coefficient

Possible HKAL Pure Problem (2)

Prove by contradiction or otherwise, show that is not a polynomial with real coefficients.

A possible HKAL Pure problem?

Let

(a) By expanding the determinant along the last row (or otherwise), show that det(A) is a cubic polynomial in and we call (1 point)

(b) Show that and hence conclude that for some in terms of and (5 points)

(c) Using the relationship between the roots and coefficients of a polynomial (or otherwise), show that (3 points)