MC for finding LCM in mathematics

2004 NO.38
Find the LCM of (2-b) (4-b^2) (8-b^3).

There must be at least one that can factorized into another form. In this question we consider the middle one first.
(4-b^2)=(2^2-b^2) =(2+b)(2-b)
then the third one
(8-b^3) =(2^3-b^3) =(2-b)(2^2+2xb+b^2)
...
You can then work out the answer.

Take out variable instead of cancel out in solving equations

How many roots does the equation tan x =2sinx have? For 0<=x<=360 in degree.
sinx/cosx = 2sinx
sinx = 2sinx cosx

Now if you cancel out the term sinx, and the equation becomes 1=2cosx, you will miss 3 answers.

The correct steps are as follow:
0 = 2sinx cosx -sinx
0 = sinx(2cosx -1)
when sinx =0, x can be 0, 180, 360 in degree.
when cosx=0.5, x can be 60, 300 in degree.

There are 5 answers. This technique is very important in both math and A-math. Especially dueling with =0 or trigo problem.

Using Acidified silver nitrate solution to test for the presence of halides

Acidified silver nitrate can test for halides to give precipitate. Remember it do give positive result for halides, not only chloride ions.

(pale) Yellow precipitate forms when iodide (bromide) ions is used, white precipitate forms when there are chloride ions.

Here are the reasons why silver nitrate has to be acidified for testing purpose from Wikipedia:
Silver nitrate are typically acidifed with dilute nitric acid to remove interfering ions, e.g. carbonate ions and sulfide ions. This step avoids confusion of silver sulfide or silver carbonate precipitates with that of silver halides. Addition of AgNO3 to the resulting solutions produces a precipitate in the presence of halides. The color of precipitate varies with the halide.

Misconception on Alkene

The statement' It can decolorize acidified potassium dichormate solution ' has two mistakes:

1. Acidified potassium dichormate solution will change from orange to green when being reduced. NOT decolorize
2. Acidified potassium dichormate is not strong enough to oxidize alkene to alkanediol.

*HOWEVER, acidified potassium dichormate solution can oxidize methanol and ethanol to methanoic acid and ethanoic acid repectively. Further down the homologous series propan-1-ol and Butan-1-ol need a more stronger oxidizing agent that is acidified potassium permanganate.

Question on alpha paricle scattering experiment

Option (3) is not correct because the positive charge of the nucleus of the gold atom will repell the positively charged alpha particle. Instead, the alpha particle should bend downward.

Nucleons = neutrons + protons.

Electrostatics: Charge 不變定則before earthing

在一幅圖內，不論charge 如何induce，圖內的total charge 是不會改變的，在earthing之前而言。
(稍後會附圖解釋。)

當earthing 發生後，圖內的total charge 是 0 。

是就整體而言，非指個別物體。

Electrostatics: Earth 的獨特性

接下來要說明的物理學重點頗為抽像，用中文解釋可以幫助香港學生理解。

一樣沒有被earth的普通conductor 是不可以discharge 一個帶charge 的 insulator(當接觸時).只有像Earth 那般巨大的物體才可以做到share 所有的charge.

如要discharge 一個insulator，可以用火去燒。

adding line to the given graph to solve equation

y= x^2 -2x -3 is given, to solve 2x^2 -6x -3 =0, what straight line should be added in order to solve the equation?

1.make the equation contains =0 to be like the y=... equation. NOT VICE VERSA.
x^2 -3x-3/2 =0

2.duel with the y= equation, substitute the line equation into y so that it becomes the treated
equation in procedure 1
x^2 -2x -3 = x -3/2

The equation of the line is y= x -3/2.

arithmatic sequence

given sum of first n of the arithmatic sequence= n^2, find T(10).

Difference between two consecutive arithmatic sequence is the last term:T(n)=S(n)-S(n-1)
e.g. T10 =S10-S9 = 10^2 - 9^2 =19

This technicque is quite useful .
----------------------------------------------------------------------------------
reminder:
T(n)= a + (n-1)d to find T(10).
sum of arithmatic sequence is 頭加尾乘項數除2.

solving trigo equation

cos x = cos 2x
cos x = 2 (cos x)^2 -1
2(cos x)^2 - cos x - 1=0
(2cos x +1)(cos x - 1)=0
.
.

The skill is to make similar trigo terms, e.g. with cos x and cos 2x, you should convert cos 2x in terms of cos x.

Probability

Two dice are thrown, what is the probability of getting one '6'?

(1/6 x 5/6) x 2,
(1st=6 and 2st<>6) or (1st<>6 and 2st=6)

Not 1/6 + 1/6 !
Every time you due with the question concerning probabilty,
think of the combination first!

Getting at least one '6'?

(1/6 x 5/6) x 2 + (1/6 x 1/6)
(1st=6 and 2st<>6) or (1st<>6 and 2st=6) or (1st=6 and 2st=6)