H3 Mathematics Resource Page

H3 Mathematics is the pinnacle of the Junior College Mathematics syllabus in Singapore. It contains a glimpse of actual Math that Mathematicians do, and it requires true mathematical understanding and technique to do well. (H1/H2 math requires a lot of practice, but not true understanding. It is quite common for students to “apply the method” and get the correct answer without having any idea of what they are actually doing.)

Topics in H3 Mathematics include Functions, Sequence and Series, Combinatorics, and even Number Theory. Certain schools also include topics like Linear Algebra and Differential Equations. Certainly, the H3 Math questions have a Math Olympiad style to them.

Here are some practice questions for H3 Math (more will be added in the future), with some hints. Questions are adapted from actual H3 prelim papers.

Functions

Q1) The function f is such that f(x+2)=af(x+1)-f(x), for all real x and some constant a.

(i) In the case that f is a linear function, find all possibilities for f and a.

(ii) In the case that f(0)=0 and |f(1)|=1, use mathematical induction to prove that f(n-1)f(n+1)+1=[f(n)]^2 for all positive integers n.

(iii) In the case that a=2, sketch one possibility for f which is not linear.

Hint:

(i) Write f(x)=mx+c and substitute it into the question. You should reach two cases, a=2 or m=0. For the case a=2, there is no other restriction. For the case m=0, either a=2 or c=0.

(iii) Try a step function.

Sequence and Series

Combinatorics

Q1) Mr H uses a software to generate distinct codes of the form \overline{x_1x_2x_3x_4x_5}, where x_i\in\{0,1,2,3,4,5\}. What is the least number of codes Mr H should generate such that there are at least two distinct codes that satisfy: \boxed{x_1+x_3=x_2+x_4}. [3 marks]

Hint: No, the answer is not 12. Draw a possibility diagram for x_1+x_3, essentially a table with 6×6=36 entries detailing what combinations are there for x_1+x_3. Then list cases (a popular technique for combinatorics), based on each value of x_1+x_3. For example, the case x_1+x_3=0 only has 1 option namely x_1=x_3=0. Then x_2+x_4 can have 35 other options such that x_2+x_4 is different from x_1+x_3. The last digit x_5 has no restrictions so there are 6 options. Hence the total number of ways for this case is 1x35x6=210. Finally, after adding up all the cases, use pigeonhole principle (add one) to conclude the answer, which is 6901.

Number Theory

Q1) Let m, n, M and N be positive integers. Given that gcd(M,N)=1 and mN^2=nM^2, use Mathematical Induction to show that M^2\mid m and N^2\mid n. [5 marks]

Hint: First prove for the trivial case M=1 or N=1. Then write M=p_1p_2\dots p_k and N=q_1q_2\dots q_l, where p_i, q_j are primes. Use Euclid’s Lemma twice to prove that p_1^2\mid m. By induction, show that M^2\mid m. I think the key point is that Generalized Euclid’s Lemma is probably not allowed otherwise it is very easy.

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