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March 30, 2024, 12:51:10 am

Author Topic: 4U Maths Question Thread  (Read 660024 times)  Share 

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phunky

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Re: 4U Maths Question Thread
« Reply #1965 on: August 18, 2018, 11:24:32 pm »
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hey guys,
could someone please explain how to graph the last one? Nut quite sure how to do those types of questions!
lol figuring out how to paste an image in was such a struggle

RuiAce

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Re: 4U Maths Question Thread
« Reply #1966 on: August 19, 2018, 09:51:13 am »
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hey guys,
could someone please explain how to graph the last one? Nut quite sure how to do those types of questions!
lol figuring out how to paste an image in was such a struggle
(Image removed from quote.)
Firstly, similar to \(y = f(|x|) \) you block out the part of the curve to the left of the \(y\)-axis. This is because regardless of what \(x\) is, \(x^2\) will always be positive, so you're always taking \(f\) of a positive number.

In particular, it's somewhat similar to \( y = f(|x|) \) in that you need to reflect the right-portion of the curve onto the left. However the fact that it involves \(x^2\) means that the graph becomes somewhat distorted as well, because it's not an ordinary dilation like \( y = f(3x)\).

Furthermore, note that since \( (4, 3) \) lies on \( y = f(x)\), we should expect \( (2,3) \) AND \( (-2,3) \) to lie on \( y = f(x^2) \). Since \( \left( \frac74, 0 \right) \) lies on \(y = f(x)\), we should expect \( \left( \frac{\sqrt7}{2}, 0 \right) \) AND \( \left( -\frac{\sqrt7}{2}, 0 \right) \) to lie on \( y = f(x^2)\).

(Picture was way too late but edited in anyway for completeness)
« Last Edit: August 19, 2018, 06:56:04 pm by RuiAce »

phunky

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Re: 4U Maths Question Thread
« Reply #1967 on: August 19, 2018, 04:37:14 pm »
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ahhh thanks, got it!   ;D

yammy

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Re: 4U Maths Question Thread
« Reply #1968 on: August 19, 2018, 06:08:11 pm »
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Hey Rui, can you please help me with Q15a) i and iii? -I don't understand the solutions provided
For i) why is p a double root if the hyperbola touches the ellipse at P?
For iii) what do they mean by the parameter at the point Q? Why is it -p? And how do they know that O is the midpoint of PQ?
Thank you in advance(:

RuiAce

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Re: 4U Maths Question Thread
« Reply #1969 on: August 19, 2018, 07:41:37 pm »
+2
Hey Rui, can you please help me with Q15a) i and iii? -I don't understand the solutions provided
For i) why is p a double root if the hyperbola touches the ellipse at P?
For iii) what do they mean by the parameter at the point Q? Why is it -p? And how do they know that O is the midpoint of PQ?
Thank you in advance(:
Because \(P\) is the point \( \left(cp, \frac{c}{p} \right) \), the parameter at \(P\) is \(p\). That's literally it - the parameter at a point is just the parameter that represents the coordinates of that point. The reason why \(Q\) must therefore be \(-p\) is because clearly the ellipse and the hyperbola intersect at only two distinct points. So because \( \left( cp, \frac{c}{p} \right) \) corresponds to \(P\), this leaves us with \( \left( -cp, -\frac{c}{p} \right) \) corresponding to \(Q\). So clearly \(-p\) must be the parameter representing the point \(Q\).

(It's the same as the parabola \(x^2 = 4ay\). If a point is marked \( P(2ap, ap^2)\), then the parameter at \(P\) is just \(p\).)

It is then easy to show by the midpoint formula that the midpoint of \(P\) and \(Q\) is the origin.



For example, the quadratic equation \( x^2 - 2x + 1 = 0\) has only one unique solution. (In particular, for that one it will be \(x = 1\).) That solution is also a double root of the equation.



It's quite abstract to argue formally, but essentially the idea is to generalise the intersection between a conic and a tangent line at the point of contact, to the intersection between two "touching" conics at their point of contact instead.

yammy

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Re: 4U Maths Question Thread
« Reply #1970 on: August 19, 2018, 08:48:54 pm »
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Because \(P\) is the point \( \left(cp, \frac{c}{p} \right) \), the parameter at \(P\) is \(p\). That's literally it - the parameter at a point is just the parameter that represents the coordinates of that point. The reason why \(Q\) must therefore be \(-p\) is because clearly the ellipse and the hyperbola intersect at only two distinct points. So because \( \left( cp, \frac{c}{p} \right) \) corresponds to \(P\), this leaves us with \( \left( -cp, -\frac{c}{p} \right) \) corresponding to \(Q\). So clearly \(-p\) must be the parameter representing the point \(Q\).

(It's the same as the parabola \(x^2 = 4ay\). If a point is marked \( P(2ap, ap^2)\), then the parameter at \(P\) is just \(p\).)

It is then easy to show by the midpoint formula that the midpoint of \(P\) and \(Q\) is the origin.



For example, the quadratic equation \( x^2 - 2x + 1 = 0\) has only one unique solution. (In particular, for that one it will be \(x = 1\).) That solution is also a double root of the equation.



It's quite abstract to argue formally, but essentially the idea is to generalise the intersection between a conic and a tangent line at the point of contact, to the intersection between two "touching" conics at their point of contact instead.

ahh i understand, thank youu!!
While you're at it, can you please help me with this question as well?
bi, ii and iv)

For i) i found f'(x) and let f'(x)=0 to find the stat point, but I cant really get an answer

RuiAce

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Re: 4U Maths Question Thread
« Reply #1971 on: August 19, 2018, 09:12:28 pm »
+2
ahh i understand, thank youu!!
While you're at it, can you please help me with this question as well?
bi, ii and iv)

For i) i found f'(x) and let f'(x)=0 to find the stat point, but I cant really get an answer
Hint for iv): You want \(n!\) to appear under the power. But you know that \( n! = 1\times 2\times \dots \times n\)....................

If you've correctly found the stationary point \( x = \frac{c}{n} \), you should then proceed to prove that it is a local minimum and argue that it is a global minimum. You're essentially doing the same things I did in my trial survival lecture regarding the global minimum.


yammy

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Re: 4U Maths Question Thread
« Reply #1972 on: August 20, 2018, 12:09:56 am »
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Hint for iv): You want \(n!\) to appear under the power. But you know that \( n! = 1\times 2\times \dots \times n\)....................

If you've correctly found the stationary point \( x = \frac{c}{n} \), you should then proceed to prove that it is a local minimum and argue that it is a global minimum. You're essentially doing the same things I did in my trial survival lecture regarding the global minimum.



ohhh thanks rui i get it  :D
Can you also help me with the 2nd part of di)?

RuiAce

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Re: 4U Maths Question Thread
« Reply #1973 on: August 20, 2018, 08:51:10 am »
+3
ohhh thanks rui i get it  :D
Can you also help me with the 2nd part of di)?




3.14159265359

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Re: 4U Maths Question Thread
« Reply #1974 on: August 23, 2018, 12:19:32 pm »
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hey can someone please help me with q10
thank you


RuiAce

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Re: 4U Maths Question Thread
« Reply #1975 on: August 23, 2018, 05:44:18 pm »
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hey can someone please help me with q10
thank you


I’ve attempted the question but because of its very vague wording there was very little that I could salvage from it. Please provide the final answer for me to confirm with before I post any solution.

3.14159265359

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Re: 4U Maths Question Thread
« Reply #1976 on: August 23, 2018, 09:09:04 pm »
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I’ve attempted the question but because of its very vague wording there was very little that I could salvage from it. Please provide the final answer for me to confirm with before I post any solution.


there is no solutions :(

RuiAce

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Re: 4U Maths Question Thread
« Reply #1977 on: August 23, 2018, 11:43:07 pm »
+2

there is no solutions :(
That's cool, but like was looking for just the final **answer**. Anyway here is a sketch solution.


yammy

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Re: 4U Maths Question Thread
« Reply #1978 on: August 23, 2018, 11:50:51 pm »
+1
hello, can someone please help me with d) - mathematical induction question and 16a)? i cant seem to get a solution and there are no solutions provided

RuiAce

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Re: 4U Maths Question Thread
« Reply #1979 on: August 24, 2018, 08:47:10 am »
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hello, can someone please help me with d) - mathematical induction question and 16a)? i cant seem to get a solution and there are no solutions provided
The induction question is already addressed in the compilation as it is a part of the 2016 paper.

That is a very long question. What have you attempted thus far?