開這個Blog的目的有幾個:
1) 解釋一啲CE MC的題目，起每個Topic講一啲心得，幫同學溫習返Concepts，花15-20分鐘重溫返一個Topic，多啲時間做其他野/溫其他科。
2) 提供一個交流平台，同學可以起IG Direct/ Whatsapp send題目比我，我做完可以post起呢個Blog同大家分享一啲有深度既題目，互相學習。
3) 想睇下同學有咩需要，如果需求大會考慮開Youtube拍片講解，比文字講解更為清晰，但要睇同學需求程度。
本人以前未出過去補習，亦無學過所謂「神技」，答案目的主要係提供詳盡解答，令大家可以仔細思考每個步驟，從中學會答題方向及技巧等等。
有任何疑問/更快更好的答案，歡迎到IG找我，IG上亦有聯絡方法，謝謝。
Statistics
Recall the following terms.
Mean = all data sum up/ number of terms
Median = the data separating the upper half and lower half
Case 1 (Odd number of terms) : 1,2,3,4,5
5/2 = 2.5 which is not an integer. So, another way to go.
Left-hand side of 3 : 1,2 (2 terms)
Right-hand side of 3 : 4,5 (2 terms)
Thus, median is 3
Case 2 (Even number of terms) : 1,2,3,4,5,6
6/2 = 3
Thus, the upper half and lower half should each involve 3 terms.
Thus, median = (3rd term + 4th term) / 2 = (3 + 4) / 2 = 3.5
Range = largest term - smallest term
Lower quartile = median of the lower half
Upper quartile = median of the upper half
Inter-quartile = upper quartile - lower quartile
Standard deviation increases when data become dispersed.
(E.g. mean data is removed)
Standard deviation decreases when data become concentrated.
(E.g. mean data is added)
Variance = (Standard deviation)2
Mean = all data sum up/ number of terms
Median = the data separating the upper half and lower half
Case 1 (Odd number of terms) : 1,2,3,4,5
5/2 = 2.5 which is not an integer. So, another way to go.
Left-hand side of 3 : 1,2 (2 terms)
Right-hand side of 3 : 4,5 (2 terms)
Thus, median is 3
Case 2 (Even number of terms) : 1,2,3,4,5,6
6/2 = 3
Thus, the upper half and lower half should each involve 3 terms.
Thus, median = (3rd term + 4th term) / 2 = (3 + 4) / 2 = 3.5
Range = largest term - smallest term
Lower quartile = median of the lower half
Upper quartile = median of the upper half
Inter-quartile = upper quartile - lower quartile
Standard deviation increases when data become dispersed.
(E.g. mean data is removed)
Standard deviation decreases when data become concentrated.
(E.g. mean data is added)
Variance = (Standard deviation)2
Each
data plus x
|
Each
data times x
|
|
Mean
|
Plus x
|
Times x
|
Median
|
Plus x
|
Times x
|
Range
|
Unchanged
|
Times x
|
Upper
quartile
|
Plus x
|
Times x
|
Lower
quartile
|
Plus x
|
Times x
|
Inter-quartile
|
Unchanged
|
Times x
|
Standard
deviation
|
Unchanged
|
Times x
|
Variance
|
Unchanged
|
Times x2
|
Probability
Here are few types of questions.
1) A,B,C,D,E are sitting in a row. Find the probability that A and B are sitting together.
Answer : 2/5
Similar method for more people sitting together
Answer : E
Case 1: The ball is put into a non-empty bag. (4/7)
Pick a ball from an empty bag. (3/7)
Case 2: The ball is put into a empty bag. (3/7)
Pick a ball from an empty bag. (2/7)
Note that two 2s and two 5s are distinct. Sometimes, counting the combinations is a good choice.
Remarks:
1) Different people solve the same question by different methods. Some may use nCr and nPr. Some may just count the combinations. Some may multiply the probability of different events. Try different methods for the same question and you will learn more for sure.
1) A,B,C,D,E are sitting in a row. Find the probability that A and B are sitting together.
Answer : 2/5
Similar method for more people sitting together
Answer : E
Pick a ball from an empty bag. (3/7)
Case 2: The ball is put into a empty bag. (3/7)
Pick a ball from an empty bag. (2/7)
Answer : A
Note that two 2s and two 5s are distinct. Sometimes, counting the combinations is a good choice.
Remarks:
1) Different people solve the same question by different methods. Some may use nCr and nPr. Some may just count the combinations. Some may multiply the probability of different events. Try different methods for the same question and you will learn more for sure.
Sequence
Here are the formulae. Photo from Google.
I suggest you download the photos and look through each step carefully and see the remarks made by red and blue pens.
Sequence often appears in the last question of DSE.
Here is 2013 Q19.
For students aiming at 5*/5**, you should understand all parts.
For students aiming at 5, you should understand a and b parts.
For students aiming at 4, you should understand a part and better try b part.
I suggest you download the photos and look through each step carefully and see the remarks made by red and blue pens.
Polar Coordinates
Cosine formula is always used in this topic. The following question shows how to use the cosine formula.
Answer : C
Circles
Nothing special. Just some reminders.
Equation of circle: (x–h)2 + (y–k)2 = r2
Center = (h, k)
Radius = r
Another form of equation of circle: x2 + y2 + Dx + Ey + F =0
Center = (-D/2, -E/2)
Radius = √(D/2)2 + (E/2)2–F
Remember the coefficients of x2 and y2 must be 1 when you are
calculating the center and radius.
Equation of circle: (x–h)2 + (y–k)2 = r2
Center = (h, k)
Radius = r
Another form of equation of circle: x2 + y2 + Dx + Ey + F =0
Center = (-D/2, -E/2)
Radius = √(D/2)2 + (E/2)2–F
Remember the coefficients of x2 and y2 must be 1 when you are
calculating the center and radius.
Straight lines
Questions are straightforward. The main problem should be about the concepts only. So, let's recall the concepts one by one below.
Slope can be analyzed from two perspectives, the sign and the value (absolute value).
Slope can be analyzed from two perspectives, the sign and the value (absolute value).
Also, lines can be cut into parts and coordinates of points can be found according to the ratios. 口訣: 乘對面
Some reminders
Equation of straight line: y = mx + c
slope = m
y-intercept = c
Another form of equation of straight line: ax + by + c = 0
slope = -a/b
x-intercept = -c/a (not recommand to recite, just sub y=0)
y-intercept = -c/b (not recommand to recite, just sub x=0)
You can change the given equation to either form if you do not want to recite too many formulas.
Trigonometric Graph
Photo from Google.
Remarks:
Periodicity of sine curve = 360°
Periodicity of cosine curve = 360°
Periodicity of tangent curve = 180°
Usually combine with the topic "Transformation of graph" for questions in DSE.
Trigonometric Equations
-x
|
90°-x
|
90°+x
|
180°-x
|
180°+x
|
270°-x
|
270°+x
|
360°-x
|
360°+x
|
|
sin
|
-sin x
|
cos x
|
cos x
|
sin x
|
-sin x
|
-cos x
|
-cos x
|
-sin x
|
sin x
|
cos
|
cos x
|
sin x
|
-sin x
|
-cos x
|
-cos x
|
-sin x
|
sin x
|
cos x
|
cos x
|
Tan
|
-tan x
|
1/tan x
|
-1/tan x
|
-tan x
|
tan x
|
1/tan x
|
-1/tan x
|
-tan x
|
tan x
|
Remarks:
When theta is 90° ± x or 270° ± x, sinàcos, cosàsin, tanà 1/tan. The sign is determined by the original sin/cos/tan in the given quadrant, assuming 0° ≤ x ≤ 90°.
For example: sin (90°+ x), assuming 0° ≤ x ≤ 90°
90°+ x should be in quadrant II and sin is positive in quadrant II.
So, sin (90°+ x) = +cos x
Actually, the answer can be easily checked just by randomly assigning a value for x and see whether sin (90°+ x) = +cos x
The following are some tricky questions.
Answer : A
Try to find out the pattern
Answer : B
To make a fraction smaller, we want the numerator(分子) smaller and the denominator(分母) greater.
Also, recall sin x ranges from -1 to 1
Centers in a triangle
Centroid is the intersection of 3 medians.
Incenter is the intersection of 3 angle bisectors.
Orthocenter is the intersection of 3 altitudes.
Circumcenter is the intersection of 3 perpendicular bisectors.
Remarks:
1) Orthocenter and circumcenter lie outside an obtuse-angled triangle.
Similar and Congruent Triangles
Reasons for Congruent Triangles
SSS, SAS, AAS, ASA, RHS (NO ASS! NO DIRTY STUFF!)
Reasons for Similar Triangles
AAA
3 sides proportional
ratio of 2 sides, included angle
Plane geometry -- Circles
My solutions will not involve the reasons. I hope you can remember all of them and know how to use them. Here, I would like to focus on the steps to do questions about circle.
Step 1: Underline all the clues from the questions
Answer : C
Clues: tangent, arc BC = arc AC
Answer : B
Origin should be center.
This is a difficult question. Try to remember the method used. Also, recall the different methods of calculating the area of a triangle.
Answer : D
This kind of questions has appeared in DSE. The method is simple and please try your best to remember it.
Clue 1: tangents
Clue 2: 90 degree
Thus, you should be aware that theorem about tangent and 90 degree must be used.
Answer : E
Origin should be center.
Learn how to add the lines and help you finish the question. It is a common question.
pi/6 = 30 degree
Answer : E
We only know how to find area of sectors and triangle. So we try to divide the figure into sectors and triangle only by drawing OC.
A similar question appears in DSE. Just recall the area you have learnt to compute.
Step 1: Underline all the clues from the questions
Step 2: Label the required angle as x
Step 3: Use the clues one by one and you will get the answer.
If you have some clues unused, you better check again to see any mistakes are made.
Answer : C
Clues: tangent, arc BC = arc AC
Answer : B
Origin should be center.
This is a difficult question. Try to remember the method used. Also, recall the different methods of calculating the area of a triangle.
Answer : D
This kind of questions has appeared in DSE. The method is simple and please try your best to remember it.
Clue 1: tangents
Clue 2: 90 degree
Thus, you should be aware that theorem about tangent and 90 degree must be used.
Answer : E
Origin should be center.
Learn how to add the lines and help you finish the question. It is a common question.
pi/6 = 30 degree
Answer : E
We only know how to find area of sectors and triangle. So we try to divide the figure into sectors and triangle only by drawing OC.
A similar question appears in DSE. Just recall the area you have learnt to compute.
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