Q: There are five buildings. There are 4 different colours of paint. Each building must be painted with only 1 colour. Find the number of ways to paint all the 5 buildings if all 4 colours must be used.
Solution: 5 Choose 2 to have the same colour, then multiply by 4! for the colour permutation. Ans = 240
Principle used: Grouping principle.
Questions on factorial can appear in H2 math binomial expansion, partial fractions, method of difference, mathematical induction and probability. Knowing how to simplify and manipulate factorials are crucial to doing well.
Two commonly used techniques
1. Replace n! with n(n-1)!
2. Multiply by k/k or k! / k!
Example 1
Synthetic division can be used to divide a polynomial by a linear divisor, which is commonly required in H2 math and Amath.
It is faster than the traditional long division.
Partial fraction is commonly used in binomial expansion, method of difference and integration.
When the divisor consists only of linear factors, cover up rule can be used to find the constants in the partial fraction quickly.
Example
Express 
in partial fraction.
By cover up rule,
Therefore
0 = A+C
Therefore,
Some questions on functions require students to understand the difference between ff-1 and f-1f. Though both = x, they are two different functions because their domains might be different.
while 
Therefore,
while 
Example f(x) = x+2, x> 0
H2 math syllabus 9740 did not state that periodic functions are included in the syllabus. However, questions on periodic functions have appeared in A level 2009 and 2013.
So it is good to learn periodic, even and odd functions.
Periodic functions
A periodic function has a graph with a basic pattern that repeats at regular intervals.
For example, sin x is periodic and its period is 2 pi.
sin(x) = sin(x+ 2pi)
For a periodic function with period a
f(x) = f(x+a) = f(x+2a) = ..
Simiarly
f(x) = f(x-a) = f(x-2a) = …
i.e. f(x) = f(x+ka) where k is an integer
Even functions
A function is said to be even if f(x) = f(-x) for all values of x. The graphs of all even functions are symmetrical about the vertical axis.
Odd functions
A function is odd if f(x) = -f(-x) for all values of x. The graphs of all odd functions are symmetrical about the origin. One section of the graph can be rotated about the origin through 180 degrees to give the other section.
The concept of increasing function, point of inflexion and concavity can be understood with an illustration.
a. Strictly increasing. f(x) is strictly increasing when f'(x) > 0. ( x<1.423 or x> 2.577) . These points also coincide with the maximum and minimum point.
b. Strictly decreasing. f(x) is strictly decreasing when f'(x) < 0. (1.423 < x < 2.577)
c. Point of inflexion is the point where the curve changes curvature or concavity. f'(x) does not change sign, and f” (x) = 0 and changes sign. In the above graph, the curve changes from convex to concave at x = 2.
d. Concave upwards. f(x) concave upwards when f”(x) > 0 or f'(x) is strictly increasing. f(x) concave upwards at x> 2, after f'(x) reaches a minimum, at the point of inflexion.
e. Concave downwards or convex. f(x) concave downwards when f”(x) < 0 or f'(x) is strictly decreasing. f(x) concave downwards for x < 2.
In the transformation of graphs, knowing the order of transformation is important. Knowing whether to scale or translate first is crucial to getting the correct transformation.
Let’s look at this example to illustrate the difference:
Example 1
Original point on y=f(x) is x=8
For f(2x+4), we do translation first, then scaling. ie. Move left by 4 units, then scale parallel to the X axis by a factor of 1/2. Hence, the original point becomes x= (8-4)/2 = 2
If we want to do scaling first, we need to factorise into f 2(x+2). So scale parallel to the X axis by a factor of 1/2, then move left by 2 units. Hence, the original point becomes x= (8/2)-2 = 2
Example 2
Describe the transformation of 3f(2x-4) + 5.
Translate 4 units in the positive X direction
Scale by a factor of 1/2 parallel to the X axis
Scale by a factor of 3 parallel to the Y axis
Translate 5 units in the positive Y direction
In summary,
cf(bx+a)+d = Translate by a units in the negative X direction, then scale by a factor of 1/b parallel to the X-axis, then scale by a factor of c parallel to the Y-axis, then translate by d units in the positive Y direction.
c[f a(x+b)]+d = Scale by a factor of 1/a parallel to the X-axis, then translate by b units in the negative X direction, then scale by a factor of c parallel to the Y axis, then translate by d units in the positive Y direction.
Binomial expansion can be done quickly using the GC.
Consider binomial expansion of 
, where a is a real number, and q is a fraction or negative integer.
The recurrence formula for the coef works out to be
Example 1
Expand 
in ascending powers up to and including the term in 
Using TI 84 Plus (OS 2.55)
Therefore, the expansion is
The recurrence formula can also be used to expand in descending powers of x.
Example 2
Expand 
in descending powers of x up to the 7th term.
Using TI 84 Plus (OS 2.55)
Therefore, the expansion is
JC A Level H2 Math Tuition Singapore 
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