3U Maths Question Thread

[J.B]

Thank you.

[Sukakadonkadonk]

Hi,
How would you use binomial expansions and the binomial theorem to find the value of this?

Thanks

[J.B]

Hi,
In this question the answers used the proof "converse of angles on the same arc" and "equal arcs subtend equal chords". I'm just a bit confused with which theorem's they are stating, as I thought that the angles had to be subtended at the centre?
Thanks

[RuiAce]

Hi,
How would you use binomial expansions and the binomial theorem to find the value of this?

Thanks

[RuiAce]

Hi,
In this question the answers used the proof "converse of angles on the same arc" and "equal arcs subtend equal chords". I'm just a bit confused with which theorem's they are stating, as I thought that the angles had to be subtended at the centre?
Thanks

Edit: I realised what was going on.

The theorem you're familiar with is "equal arcs subtend equal angles to the centre". But combine this, along with the theorem "the angle subtended to the centre of the circle is twice the angle subtended to anywhere on the circumference", and you further get "equal arcs subtend equal angles to the circumference".

It's the converse of THIS which is being used.

[anotherworld2b]

I don't understand how probability density functions work. So I'm having trouble understanding what to do for these questions
Q: Each graph shows a probability density function, find k

Mod Edit for anyone reading: This isn't 3U content

[Shadowxo]

I don't understand how probability density functions work. So I'm having trouble understanding what to do for these questions
Q: Each graph shows a probability density function, find k

So the probability density function just shows what the probability of each number is and how the probability is distributed. The total probability is the area under the graph which should always be 1.
So for 5) It's saying the probability of getting any number between 1 and 2 is the same. eg We can see that the probability of getting above 1.5 is 50%, and below 1.5 is 50% by dividing the graph into two sections or thinking about it. The question is asking what k is if the probability of getting above k is 20% (0.2). The easiest way to do this is to divide the graph up into a part to the left with 80% of the area and a part to the right with 20% of the area. We can see that k would just be 1.8, as 20% of the numbers (and area) are above it and 80% below.
Other questions follow the same method just with an extra calculation
6. Pr(X > 3)=0.5 (half the area and numbers)
Pr(X > k | x > 3) = Pr(X > k n X > 3) /0.5 = 0.25
Pr(X > k) n Pr(X > 3) = 1/8
So we can see 1/8th of the area is above 4.5
k=4.5

[anotherworld2b]

Oh okay I think I get it
But I am a bit confused for the ones involving given.
Does that mean I should just focus on the fact from 3 and bigger has an area of 1? And do the same thing for this question?
Pr(X > k | x > 3) = Pr(X > k n X > 3) /0.5 = 0.25

So the probability density function just shows what the probability of each number is and how the probability is distributed. The total probability is the area under the graph which should always be 1.
So for 5) It's saying the probability of getting any number between 1 and 2 is the same. eg We can see that the probability of getting above 1.5 is 50%, and below 1.5 is 50% by dividing the graph into two sections or thinking about it. The question is asking what k is if the probability of getting above k is 20% (0.2). The easiest way to do this is to divide the graph up into a part to the left with 80% of the area and a part to the right with 20% of the area. We can see that k would just be 1.8, as 20% of the numbers (and area) are above it and 80% below.
Other questions follow the same method just with an extra calculation
6. Pr(X > 3)=0.5 (half the area and numbers)
Pr(X > k | x > 3) = Pr(X > k n X > 3) /0.5 = 0.25
Pr(X > k) n Pr(X > 3) = 1/8
So we can see 1/8th of the area is above 4.5
k=4.5

[Shoesta]

I know how simple these questions are, I just can never get the hang of them! Any help would be greatly appreciated

[Opengangs]

I know how simple these questions are, I just can never get the hang of them! Any help would be greatly appreciated

[Shlomo314]

Hi, I am current doing both extension one and two. I am finding circle geometry hard as usually can not see the what I have to prove. I recently did my Ext 2 trial and i skipped all the circle geoms which I would like to improve on. I have been practice them but it still doesn't click to me as the other topics. Any suggestions?

Cheers

[RuiAce]

Hi, I am current doing both extension one and two. I am finding circle geometry hard as usually can not see the what I have to prove. I recently did my Ext 2 trial and i skipped all the circle geoms which I would like to improve on. I have been practice them but it still doesn't click to me as the other topics. Any suggestions?

Cheers

From a while back

As for geometry in any circumstance, though notably in particular with circle geometry, it all comes down to being able to "SEE" it. This is what takes a lot of practice.

To "see" it better, it helps to know what all the theorems look like. For example, the angles standing on same arc (angles in same segment) really just remind me of two triangles mirrored together. The angle in a semicircle is an opposite one. If by some coincidence there's a cyclic quadrilateral is there that's a giveaway, and the alternate segment theorem looks like a triangle with a tangent attached with it.

Once you are able to visualise it more clearly, focus on details. In a question, if they give you a tangent, you'll probably need to use a tangent theorem. Otherwise, stick to everything else and only that. Also, when focusing, don't look at the entire diagram at once; look at a few lines at a time. If you know what your theorems "look like", you can try to see their shapes better.

[Shadowxo]

Oh okay I think I get it
But I am a bit confused for the ones involving given.
Does that mean I should just focus on the fact from 3 and bigger has an area of 1? And do the same thing for this question?
Pr(X > k | x > 3) = Pr(X > k n X > 3) /0.5 = 0.25

For given, first treat it as a usual given probability : Pr (A | B) = Pr (A n B)/Pr (B)
Using 6 as an example, we can see Pr (X>3) = 0.5 as half of the total area is above 3.
So we get
Pr(X > k n X > 3) is just Pr (X > k) as k is already greater than 3 when we solve, (and if it were less than 3, it would just be Pr(X > 3) which is greater than 1/8 )
So we end up with Pr(X > k) = 1/8, and we can see 1/8th of the area is above 4.5. k=4.5

Hope this answers your question

[zemilyx]

hey, does anyone know how to solve this? even the answers don't make sense to me
thanks

[jamonwindeyer]

hey, does anyone know how to solve this? even the answers don't make sense to me
thanks

Hey! Is there a bit to the question above this? The screenshot is missing things I think