ABCD is a rectangle. M and N are points on AB and DC respectively. MC and BN meet at X. M is the midpoint of AB.
(a) Prove that and are similar.
(b) Given that area of : area of =9:4, find the ratio of,
(i) DN: NC
(ii) area of rectangle ABCD: area of . (Challenging)
[Answer Key] (b) (i) 1:3
(vert. opp. angles)
Therefore, and are similar (AAA).
We now have a shorter solution, thanks to a visitor to our site! (see comments below)
From part (a), since and are similar, we have
This means that
Thus (the two triangles share a common height)
Now, note that
Hence area of
We conclude that area of rectangle ABCD: area of
Here is a longer solution, for those who are interested:
Let area of
Let area of
Let area of
We have since and have the same base BC and their heights have ratio 3:2.
Cross-multiplying, we get
since and have the same base BC and their heights have ratio 3:4.
Thus, area of
area of rectangle ABCD: area of =40:6=20:3
Given , find the value of .
Working with logarithm is tricky, we try to transform the question to an exponential question.
Then, we have
Here comes the critical observation:
Observe that .
Divide throughout by , we get .
Solving using quadratic formula (and reject the negative value since and has to be positive for their logarithm to exist),
We get .
If you have any questions, please feel free to ask me by posting a comment, or emailing me.
(I will usually explain in much more detail if I teach in person, than when I type the solution)
Given that a parabola intersects the x-axis at x=-4 and x=2, and intersects the y-axis at y=-16, find the equation of the parabola.
Sketch of graph:
Now, there is a fast and slow method to this question. The slower method is to let , and solve 3 simultaneous equations.
The faster method is to let .
Why? We know that x=4 is a root of the polynomial, so it has a factor of (x-4). Similarly, the polynomial has a factor of (x-2). The constant k (to be determined) is added to scale the graph, so that the graph will satisfy y=-16 when x=0.
So, we just substitute in y=-16, x=0 into our new equation.
In conclusion, the equation of the parabola is .
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This is a continuation from H2 Maths Tuition: Foot of Perpendicular (from point to line) (Part I).
Foot of Perpendicular (from point to plane)
From point (B) to Plane ( )
Where does F lie?
F lies on the plane .
Substitute Equation (II) into Equation (I) and solve for k.
[VJC 2010 P1Q8i]
The planes and have equations and respectively. The point has position vector .
(i) Find the position vector of the foot of perpendicular from to .
Let the foot of perpendicular be F.
Subst. (II) into (I)
Solve for k, .
H2 Maths Tuition
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Foot of Perpendicular is a hot topic for H2 Prelims and A Levels. It comes out almost every year.
There are two versions of Foot of Perpendicular, from point to line, and from point to plane. However, the two are highly similar, and the following article will teach how to understand and remember them.
H2: Vectors (Foot of perpendicular)
From point (B) to Line ( )
Where does F lie? F lies on the line .
Substitute Equation (I) into Equation (II) and solve for .
[CJC 2010 P1Q7iii]
Relative to the origin , the points , and have position vectors , and Find the shortest distance from to . Hence or otherwise, find the area of triangle .
[Note: There is a 2nd method to this question. (cross product method)]
Let the foot of perpendicular from C to AB be F.
For the next part, please read our article on Foot of Perpendicular (from point to plane).
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This is a Maths Tuition Flyer I created using the Maths software , for distribution in the following areas:
- Lorong Chuan
- Ang Mo Kio
- Toa Payoh
- Serangoon Gardens
PDF: Maths Tuition Flyer
This is a nice worksheet on Expansion and Factorisation by Hwa Chong Institution (HCI).
There are no solutions, but if you have any questions you are welcome to ask me, by leaving a comment, or by email.
Hope you enjoy practising Expansion and Factorisation.
The worksheet may be downloaded here:
Expansion and Factorisation by Hwa Chong Institution (HCI)
If your child needs help in Maths, please feel free to contact us for Maths Tuition. 🙂
In O Level, students are taught that
So naturally, students may think that (a is a constant)
Well, actually that is good pattern spotting, but unfortunately it is incorrect. Do not be too disheartened if you make this mistake, it is a common mistake.
The above is a conceptual error as only holds when n is a constant.
Fortunately, this question is rarely tested, though it is quite possible that it can come up in A Levels.
To fully understand the following steps, it would help read my other post (Why is e^(ln x)=x?) first.
First, we write .
After fully understanding the above steps, you may memorize the formula if you wish:
Memory Tip: If you let a=e, you should get
The above steps involve the chain rule, which I will cover in a subsequent post.
Many parents have feedback to me that their child often makes careless mistakes in Maths, at all levels, from Primary, Secondary, to JC Level. I truly empathize with them, as it often leads to marks being lost unnecessarily. Not to mention, it is discouraging for the child.
Also, making careless mistakes is most common in the subject of mathematics, it is rare to hear of students making careless mistakes in say, History or English.
Fortunately, it is possible to prevent careless mistakes for mathematics, or at least reduce the rates of careless mistakes.
From experience, the ways to prevent careless mistakes for mathematics can be classified into 3 categories, Common Sense, Psychological, and Math Tips.
- Firstly, write as neatly as possible. Often, students write their “5” like “6”. Mathematics in Singapore is highly computational in nature, such errors may lead to loss of marks. Also, for Algebra, it is recommended that students write l (for length) in a cursive manner, like to prevent confusion with 1. Also, in Complex Numbers in H2 Math, write z with a line in the middle, like Ƶ, to avoid confusion with 2.
- Leave some time for checking. This is easier said than done, as speed requires practice. But leaving some time, at least 5-10 minutes to check the entire paper is a good strategy. It can spot obvious errors, like leaving out an entire question.
- Look at the number of marks. If the question is 5 marks, and your solution is very short, something may be wrong. Also if the question is just 1 mark, and it took a long time to solve it, that may ring a bell.
- See if the final answer is a “nice number“. For questions that are about whole numbers, like number of apples, the answer should clearly be a whole number. At higher levels, especially with questions that require answers in 3 significant figures, the number may not be so nice though. However, from experience, some questions even in A Levels, like vectors where one is suppose to solve for a constant , it turns out that the constant is a “nice number”.
Mathematical Tips are harder to apply, unlike the above which are straightforward. Usually students will have to be taught and guided by a teacher or tutor.
- Substitute back the final answer into the equations. For example, when solving simultaneous equations like x+y=3, x+2y=4, after getting the solution x=2, y=1, you should substitute back into the original two equations to check it.
- Substitute in certain values. For example, after finding the partial fraction , you should substitute back a certain value for x, like x=2. Then check if both the left-hand-side and right-hand-side gives the same answer. (LHS=1/3, RHS=1/2-1/6=1/3) This usually gives a very high chance that you are correct.
Thanks for reading this long article! Hope it helps! 🙂
I will add more tips in the future.
This book is a New York Times Bestseller by actress Danica McKellar, who is also an internationally recognized mathematician and advocate for math education. It should be available in the library. Hope it can inspire all to like Maths!
Why is ?
This formula will be useful for some questions in O Level Additional Maths, or A Level H2 Maths.
There are two ways to show or prove this, first we can let
Taking natural logarithm (ln) on both sides, we get
So . Substitute the very first equation and we get . 🙂
Alternatively, we can view and as inverse functions of each other. So, we can let and . Then, by definition of inverse functions. This may be a better way to remember the result. 🙂
The above method of inverse functions can be used to remember too.
Question from http://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=29&t=8315&start=780
The mass of particles of a certain radioactive chemical element is halved every 10 months. During a chemical experiment, the initial mass of particles of the chemical element is 3mg.
(i) write down an expression, in terms of t, for the mass of particles after t years.
(ii) Hence, find the value of t, if the mass is reduced to 0.046875 mg after t years.
How many 10 months are there in ? (Ans: )
Hence, the mass of particles after years is mg.
We need to solve .
Dividing by 3, we have .
Ln both sides, we have .
If you liked our solution above, please consider signing up for Maths Tuition with us! 🙂