4u Mathematics Marathon v2.0 (1 Viewer)

[pLuvia]

Time to start this again, point of this thread is to test each other's maths skills, general revision etc.

General rule: A person posts up a question (not homework-help questions) and then another person answers it in spoilers and after they answer they put up another question for the next person to answer

How to do spoilers : [spoiler ]Like this[/spoiler ], but without the spaces in between the brackets so without it, it would be something like this

Spoiler

Like this

I'll start off to get the ball rolling, last question from the previous marathon

Question

How many possible four letter words can be selected from the word MATHEMATICS?

 

[kony]

isn't it a bit early to ask this stuff?

like the whole methods of counting thing - i think most schools won't have touched on this yet.

and why is this in ext2? isn't this .. 3unit?

and you copied this straight from a hsc paper lol (i dont remember how many units it was though)

 

[pLuvia]

isn't it a bit early to ask this stuff?

like the whole methods of counting thing - i think most schools won't have touched on this yet.

and why is this in ext2? isn't this .. 3unit?

and you copied this straight from a hsc paper lol (i dont remember how many units it was though)

I did? I've never encountered this I don't think, I made it up on the spot

 

[Mill]

How many possible four letter words can be selected from the word MATHEMATICS?

Spoiler

There are 8 unique letters in the word "Mathematics": A, C, E, H, I, M, S, T. We desire 4 letter words. There are 8P4 = 1680 such words.

 

[Mill]

My Question:

The sum of the ages of the four children of a family is equal to half the age of their father. The difference of the squares of the ages of the oldest and youngest children is equal to the age of the father, which is also equal to twice the sum of the differences of the squares of the ages of the two younger and two older children. In sixteen years time, the sum of the ages of the children will exceed the age of their father by his present age. Find the age of every member of this family.

 

[velox]

mill do you have a younger brother?

 

[Mill]

I do not. Why do you ask?

 

[pLuvia]

I was thinking along the lines of

Spoiler

Case 1: 8 unique letter - 8P4=1680
Case 2: 2 of the same letters plus two different letters - 3C1*7C2*4!/2!=756
Case 3: 2 pairs of the same letters - 3C2*4!/2!2!=18

Hence the number of words would be 1680+756+18=2454

 

[Mill]

I was thinking along the lines of

Spoiler

Case 1: 8 unique letter - 8P4=1680
Case 2: 2 of the same letters plus two different letters - 3C1*7C2*4!/2!=756
Case 3: 2 pairs of the same letters - 3C2*4!/2!2!=18

Hence the number of words would be 1680+756+18=2454

Oh yah you are clearly correct. For some reason when I wrote the answer I thought you had specified that letters could not be used more than once.

 

[Riviet]

which is also equal to twice the sum of the differences of the squares of the ages of the two younger and two older children.

Could you clarify what you mean by this?

 

[pLuvia]

I think that means:

Let the younger brothers be a and b like wise with the older children let them be x and y, from my interpretation it means

2[(a²-b²)+(x²-y²)]

 

[ravdawg]

Spoiler

a=6 b=5 c=3 d=2 f=32

 

[Mill]

I think that means:

Let the younger brothers be a and b like wise with the older children let them be x and y, from my interpretation it means

2[(a²-b²)+(x²-y²)]

Yes, if you're assuming that from eldest to youngest we have: a,b,x,y.

 

[Mill]

Spoiler

a=6 b=5 c=3 d=2 f=32

This is correct.

 

[Slidey]

ravdawg, post a question

Try and keep it within the scope of the 4u syllabus, of course.

 

[ravdawg]

Oh yeah, sorry, i kinda posted teh answer and completely forgot to put up a new question.

Ok here goes, not really a hsc type question but yeah...

Two cars are 1.5km away from each other. Both are travelling at a constant velocity to each other, one at 5m/s and the other 10m/s. There is a plane in between them, starting at one of the cars which travels at 20m/s. Every time it reaches one of the cars it turns around insantaneously and goes in the opposite direction till it meets the other car etc. When the two cars meet, how far would the plane have travelled in metres.

 

[LoneShadow]

although you mention that the question is not HSC type; but even a general maths student knows how to that question. Could probably bet a year 10 kid could do that question. All you need is a pencil that can last long enough to draw a few semi straight lines back and forth.

It doesn't belong in Ext2 section.


that being said here are a few question that shouldn't probably be posted here either :

(1) Show that for all positive x and y: Exp(x+y)/(xy)>=Exp(2).
(2) Fow what value of k does the equation Exp(2x)=kSqrt(x) has exactly one solution?
(3) For which positive number a is it true that a^x>=1+x for all x?

[questions randomly chosen from a general calculus book]

 

[ravdawg]

although you mention that the question is not HSC type; but even a general maths student knows how to that question. Could probably bet a year 10 kid could do that question. All you need is a pencil that can last long enough to draw a few semi straight lines back and forth.

It doesn't belong in Ext2 section.


that being said here are a few question that shouldn't probably be posted here either :

(1) Show that for all positive x and y: Exp(x+y)/(xy)>=Exp(2).
(2) Fow what value of k does the equation Exp(2x)=kSqrt(x) has exactly one solution?
(3) For which positive number a is it true that a^x>=1+x for all x?

[questions randomly chosen from a general calculus book]

If you're trying to figure out the answer to this question by drawing lines back and forward you are making it hard for yourself. The reason i gave this question is that you can do it in one line and about 5 seconds if you use your head a bit...

 

[vafa]

I have an interesting question:

Prove that Sigma (from k=0 to n) coskx={cos(nx/2)sin((x(n+1))/2)}/sin(x/2)

and sigma (from k=0 to n) sinkx={sin(nx/2)sin((x(n+1))/2)}/sin(x/2)

note: there are lots of ways to show those two identities.but the main ways are by using mathematical induction or by using the property of complex numbers (Euler's equation e^(ix)=cosx+isinx)). but mathematical induction needs more work so i suggest to use euler's equation

 

[kony]

Oh yeah, sorry, i kinda posted teh answer and completely forgot to put up a new question.

Ok here goes, not really a hsc type question but yeah...

Two cars are 1.5km away from each other. Both are travelling at a constant velocity to each other, one at 5m/s and the other 10m/s. There is a plane in between them, starting at one of the cars which travels at 20m/s. Every time it reaches one of the cars it turns around insantaneously and goes in the opposite direction till it meets the other car etc. When the two cars meet, how far would the plane have travelled in metres.

it takes the cars 1500/(5+10) seconds to meet. in that time, the plane hasnt stopped moving. therefore, the plane has travelled 20 * 1500/(5+10) meters.

20*1500/15 = 20*100 = 2000m = 2 km.