Author: gjooheng

H2 maths tutor

Challenging Normal Distribution problem 2

normal

See Square root of x squared

Juggling: shortcut to long division

Juggling is a shortcut to long division, especially in the case where the degree of the numerator is the same as the denominator. In H2 math, juggling is a useful technique in finding the asymptote of graphs and integration.

juggling

Triangular permutations

A group of 9 people sits at a triangular table, which is equilateral, with 3 persons at each side of the table. Find the number of different possible arrangements.

Solution

The 3 arrangements below are considered the same

triangle

Therefore, number of different possible arrangements = 9! / 3

Finding suitable replacement to solve new inequalities

ineq1

ineq2

ineq3

Graph transformation of oblique asymptote

Many students have difficulty with the graph transformation of oblique asymptote. Consider the oblique asymptote y = x-1 (red line)

oblique

i) y= 1/ f(x)

f(x) approaches infinity as x approaches infinity. 1 divided by infinity is 0.

Hence oblique asymptote y=x-1 becomes horizontal asymptote y= 0

For y= 1/ f(x), any oblique asymptote y=ax+b in f(x) will become horizontal asymptote y= 0 

ii) y= f ‘ (x)

The gradient of y= x-1 is 1. Hence oblique asymptote y=x-1 becomes horizontal asymptote y= 1

For y= f ‘ (x), any oblique asymptote y=ax+b in f(x) will become horizontal asymptote y= a

iii) y= f(2x+1)

Let original asymptote be f(x) = x-1

Therefore, f(2x+1) = (2x+1)-1 = 2x

Hence oblique asymptote y = x-1 is transformed to y=2x.

This method can be applied to any oblique asymptote.

Circular Permutations

Circular permutations often pose some difficulty to students. Let’s consider the following scenarios:

Scenario A: 10 people to be seated at a round table with 10 identical seats

Number of ways = (10-1)! = 9! = 362880

Scenario B: 5 people to be seated at a round table with 10 numbered seats

Number of ways = 10 P 5 = 30240

Scenario C: 6 people to be seated at a round table with 10 identical seats

circular

There will be 4 identical empty seats. Consider fixing 1 occupied seat, and permutate the other 9 seats around it. There are 4 identical seats among the 9 seats. So number of ways = 9!/4! = 15120