HSC 2015 MX1 Marathon (archive) (1 Viewer)

[kawaiipotato]

Re: HSC 2015 3U Marathon

dy/dx = x/2a ==> dy/dx (at x = 2ap) = p
Hence, tangent at P : y - ap^2 = p (x-2ap)
y = px - ap^2
Similarly, tangent at Q: y = qx - aq^2
Subtracting, px - qx - ap^2 + aq^2 = 0
(p-q)x = a(p-q)(p+q)
x = a(p+q)
Substituting x = a(p+q) into one of the equations:
y = apq
Intersection (a(p+q),apq)

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

My bad, it should be 'vertices'.

Im concentrating on just one circle. A quadrilateral within one circle.

 

[Ambility]

Re: HSC 2015 3U Marathon

dy/dx = x/2a ==> dy/dx (at x = 2ap) = p
Hence, tangent at P : y - ap^2 = p (x-2ap)
y = px - ap^2
Similarly, tangent at Q: y = qx - aq^2
Subtracting, px - qx - ap^2 + aq^2 = 0
(p-q)x = a(p-q)(p+q)
x = a(p+q)
Substituting x = a(p+q) into one of the equations:
y = apq
Intersection (a(p+q),apq)

But what is the equation of the locus?

 

[InteGrand]

Re: HSC 2015 3U Marathon

Use the fact that pq = -1 for a focal chord.

 

[kawaiipotato]

Re: HSC 2015 3U Marathon

But what is the equation of the locus?

Oops didn't see it asked for locus
If the line connecting P and Q is a focal chord (passes through the focus S(0,a)) then gradient PS = SQ
When you equate this, you will find a relationship between p,q which is pq = -1 (what InteGrand said)
so intersection = (a(p+q),-a)
Locus : y = -a
Dunno what to say about the x ordinate. It should still satisfy when sketching the locus and the parabola

Alternatively you could find equation PQ and sub in the focus S(0,a) giving the same relationship: pq=-1

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

Consider a rectangle inscribed in a semi circle of radius r, that is, two of the vertices of the rectangle are on the circular arc and one side of the rectangle lies on part of the diameter.

a) What are the dimensions of the rectangle of largest area which can be inscribed in a semi circle as above?

b) What are the dimensions of the rectangle with largest perimeter which can be inscribed in a semicircle as above?

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon



Im guessing with this formula its very easy to get the f and f' mixed up? I admit it , I just put it in the wrong order. Won't be doing that again.

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

Can someone confirm if there has been a mistake for this question? (This is from the Fitzpatrick, Aus, Curran textbook) Use of the Half-Angle Formulae. Am I correct in saying that they have done a typo in their answer by putting in sin ^2 45 , when it should be just sin 45?

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

I enjoy deriving the half-angle formulae by drawing out the diagrams. But should we memorize it in order to do the question faster? Is this what teachers tell their students? (Remember time is an important facet of an exam)

 

[InteGrand]

Re: HSC 2015 3U Marathon

Can someone confirm if there has been a mistake for this question? (This is from the Fitzpatrick, Aus, Curran textbook) Use of the Half-Angle Formulae. Am I correct in saying that they have done a typo in their answer by putting in sin ^2 45 , when it should be just sin 45?

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

Yes my bad I did get the formula the wrong way around. But according to my calculator, I still think it should be 1/sin 45 which in this case is sqrt(2) and not 2 as listed in the answer.

 

[leehuan]

Re: HSC 2015 3U Marathon

'sin^2' is wrong.

http://www.wolframalpha.com/input/?i=(1+tan^2(22.5))/(2tan22.5+degrees)

http://www.wolframalpha.com/input/?i=sqrt2

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

'sin^2' is wrong.

http://www.wolframalpha.com/input/?i=(1+tan^2(22.5))/(2tan22.5+degrees)

http://www.wolframalpha.com/input/?i=sqrt2

OK thanks. Looks like Grove text is not that only one that is free from error.

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon



Curious to know when integrating , ive been always taught to use the absolute value when integrating something that involves the natural logarithm. Would the value with the absolute value signs also be a sufficient enough of an answer? (The listed answer was the 2nd line of working)

 

[InteGrand]

Re: HSC 2015 3U Marathon

Curious to know when integrating , ive been always taught to use the absolute value when integrating something that involves the natural logarithm. Would the value with the absolute value signs also be a sufficient enough of an answer? (The listed answer was the 2nd line of working)

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

There has been another error in the Heather Brown, David Coffey, Michelle Hill and Peter Muddle Worked Out Solutions for the text book Fitzpatrick, Bob Aus and Curran book.





It should be :



#justsaying

 

[Drsoccerball]

Re: HSC 2015 3U Marathon

This isnt twitter you dont have to post everything

 

[sharoooooo]

Re: HSC 2015 3U Marathon

help me pls with this question pls pls

Q. Consider the digits 9,8,7,6,5,4,3,2,1 and 0. Find how many five-digit numbers are possible if the digits are to be in:
a) i) ascending order
a) ii) descending order
b) Why do these 2 questions involve unordered selections?

Answer:
a) i) 126
a) ii) 252


pls halp i dont understand

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

Another error in Fitzpatrick, Aus & Curran.






Answer should be:

 

[davidgoes4wce]

Re: HSC 2015 3U Marathon

help me pls with this question pls pls

Q. Consider the digits 9,8,7,6,5,4,3,2,1 and 0. Find how many five-digit numbers are possible if the digits are to be in:
a) i) ascending order
a) ii) descending order
b) Why do these 2 questions involve unordered selections?

Answer:
a) i) 126
a) ii) 252


pls halp i dont understand

a)
Pick any 5 numbers 'Once' from 1 to 9 (can't choose 0) . 9C5=126

b)
Pick any 5 Numbers 'Once' from 1 to 10 (not ordering. 10C5=252