Solutions to the Ext 1 HSC Paper of 2003 (1 Viewer)

Will Hunting

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Hey,

I'm sure a question like this has surfaced in the forums before, but I'm just after solutions (in reasonable detail) to the 2003 HSC Paper for Ext 1 Maths (or, at the very least, a solution to that 4 mark humdinger on projectile motion at the end). If anyone could lend a hand with this, I'd really appreciate it. Thanks!

P.S. Good luck to everyone for tomorrow! Just relax and have fun! All the best from Will!
 

Will Hunting

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Thanks heaps, man. I just did the question myself, though, while I was waiting for a reply! lol

WHOOOO! So stoked about it, too. I'm not sure how everyone else went with it, but I'm posting my solution up anyway!!! Maybe that'll bring me down from my high!

The Question is 7(b)iii. from the 2003 paper; check it out before reading my sol'n.

dmax occurs when α = π/4

This holds for all h ≤ H - S (and ∴ all d ≤ the value of d when α = π/4, h = H - S)

When α = π/4, h = H - S, and ∴

( v2sin2α )/2g = H - S

v2(1/√2)2 = 2g(H - S)

v2 = 4g(H - S)

For values of d ≤ the value of d when α = π/4, h = H - S,

v2 ≤ 4g(H - S)

(Since both h and d are decreasing, and α remains the same, v, and ∴ v2 must decrease)

And d = ( v2sin2α )/g = [v2sin(2.π/4)]/g = v2(1)/g = v2/g

But, for values of d ≥ the value of d when α = π/4, h = H - S,

α must decrease and v must increase to sustain both a constant h and an increase in d

(The fixing of h to H - S is based upon the fact that, ceteris paribus, the closer α is to π/4, the greater will be the range of the projectile - I didn't include this in my proof lol)

i.e. v2 ≥ 4g(H - S)

Bring the constant h into the equation for d (where v and α now vary):

d = ( v2sin2α )/g =

[2( v2sinαcosα )]/g =

( v2sin2α )/2g x 4cosα/sinα =

4(H - S)cotα =

4√[(H - S)2cot2α] =

4√[(H - S)2(cosec2α - 1)] (From 1 + cot2x = cosec2x)

But ( v2sin2α )/2g = H - S

i.e. v2/2g = (H - S)/sin2α = (H - S)cosec2α

So, d becomes:

d = 4√[(H - S)(H - S)(cosec2α - 1) - (H - S)2] =

4√[(H - S)(v2/2g) - (H - S)2]

Q. E. D.

Now, that is one f**king nasty 3 Unit Question! Enjoy!
 
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香港!

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hahaha i knew u wud be able to do it as i've seen ur posts long time ago and u seem quite high lvl:p

that 2003 paper.. i didn't like it, it had 23 marks worth of GEOMETRY+PROBABILITY all up...
those are my worst topics lol
hope dis yrs paper is not like dat...
 

CM_Tutor

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The syllabus changed only very slightly between 1983 and 2019, and a lot of that material is still relevant to the new syllabus.

Further, plenty of 2020 exams borrowed from older papers. For example:
  • 2020 Advanced paper from North Sydney Boys had a question from the 1986 2u HSC that has also been asked at other schools over the last 30 years.
  • 2020 Sydney Tech Advanced paper had a question that was used by Kings in 2005 and North Sydney Boys in 2011.
  • 2020 Sydney Boys Advanced paper had a question from the 1989 2u HSC that has also been asked by Fort Street (2015), Shore and St George Girls High (both in 2010), and in a modified form at Northern Beaches Secondary College (Manly Selective Campus) in 2011.
  • 2020 Killara High Advanced paper had a question from the Catholic paper in 2009 that was also used by Knox Grammar (2013) and Normanhurst Boys High (2017)
  • 2020 Baulkham Hills High Advanced paper has a question used by Newcastle Grammar in 2003 and North Sydney Girls in 2017
  • 2020 James Ruse Advanced paper had a question very similar to the 2017 Independent Schools Trial
  • 2020 Fort Street Advanced trial has a question from the 2007 Sydney Grammar paper
... and that list is incomplete and looks at only one particular type of question.
 

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