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HSC 2016 Maths Marathon (archive) (1 Viewer)
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packpig123
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Paradoxica
-insert title here-
Re: HSC 2016 2U Marathon
Vector addition results in a parallelogram, whereby the resultant vector must lie strictly within the angular region bounded by the rays formed by the two original vectors.
Re: HSC 2016 2U Marathon
Correct! I'll leave the algebraic exercise as an exercise for 2U students.
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trecex1
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Re: HSC 2016 2U Marathon
ad < bc
ad + ab < ab + bc
a(b+d ) < b(a+c)
a/b < (a+c)/(b+d)
similarly bc +cd > cd + ad
c(b+c) > d(a+c)
c/d > (a+c)/(b+d)
a/b < c/d so
ad < bc
ad + ab < ab + bc
a(b+d ) < b(a+c)
a/b < (a+c)/(b+d)
similarly bc +cd > cd + ad
c(b+c) > d(a+c)
c/d > (a+c)/(b+d)
Last edited:
leehuan
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Re: HSC 2016 2U Marathon
This is a type of assumption that's made because in essence, the other case is proved in the exact same manner. Should state that "without loss of generality, assume that a/b < c/d"
Nicely done, however the assumption that a/b < c/d isn't necessarily true. It may be possible that c/d < a/b.a/b < c/d so
ad < bc
ad + ab < ab + bc
a(b+d ) < b(a+c)
a/b < (b+d)/(a+c)
similarly bc +cd > cd + ad
c(b+c) > d(a+c)
c/d > (a+c)/(b+d)
This is a type of assumption that's made because in essence, the other case is proved in the exact same manner. Should state that "without loss of generality, assume that a/b < c/d"
Paradoxica
-insert title here-
Re: HSC 2016 2U Marathon
The curve attains a global maximum with stationary point at x = ∛e, so the value of k must be less than 1/(3e)
However, the curve also never drops below 0 for all sufficiently large values of x, so k must be greater than 0.
Thus:
0 < k < 1/(3e)
∎
Consider the equivalent equation (logx)/x³ = ka/b < c/d so
ad < bc
ad + ab < ab + bc
a(b+d ) < b(a+c)
a/b < (a+c)/(b+d)
similarly bc +cd > cd + ad
c(b+c) > d(a+c)
c/d > (a+c)/(b+d)
The curve attains a global maximum with stationary point at x = ∛e, so the value of k must be less than 1/(3e)
However, the curve also never drops below 0 for all sufficiently large values of x, so k must be greater than 0.
Thus:
0 < k < 1/(3e)
∎
pikachu975
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KingOfActing
lukewarm mess
davidgoes4wce
Well-Known Member
Re: HSC 2016 2U Marathon
Just want to check that you mean the equation 3x+y-5=0 is the normal equation?
The reason I ask is because if you substitute x=3
If the wording of the question is right, the gradient of the normal is -3, and the gradient of the tangent is
In which case,
Have you got an answer for b and c?
Just want to check that you mean the equation 3x+y-5=0 is the normal equation?
The reason I ask is because if you substitute x=3
If the wording of the question is right, the gradient of the normal is -3, and the gradient of the tangent is
In which case,
davidgoes4wce
Well-Known Member
Re: HSC 2016 2U Marathon
Yep I got c=7 from my working out.
Yep I got c=7 from my working out.
Re: HSC 2016 2U Marathon
The tidal heights over the next five days for the Yangtze River mouth in Shanghai are shown in screenshot below.
The heights in the tidal prediction chart show the height above the chart datum which is 7m. For example, a low tide of 0.5m is actually a depth of 7.5m.
a)Use the tidal data to synthesise a model for the depth of water at the Yangtze River mouth for all the data you have captured.
A horizontal translation is required in your model. Use the following information to assist you
The general trigonometric function
f(x) = asinb)x+c) + d
has an amplitude of a, a period of 2pi/b, a horizontal translation of c units to the left and a vertical translation of d units up.
b) Produce a single graph containing the actual data given in the tidal prediction chart and your model. Comment on the strength and limitations of the model.
The tidal heights over the next five days for the Yangtze River mouth in Shanghai are shown in screenshot below.
The heights in the tidal prediction chart show the height above the chart datum which is 7m. For example, a low tide of 0.5m is actually a depth of 7.5m.
a)Use the tidal data to synthesise a model for the depth of water at the Yangtze River mouth for all the data you have captured.
A horizontal translation is required in your model. Use the following information to assist you
The general trigonometric function
f(x) = asinb)x+c) + d
has an amplitude of a, a period of 2pi/b, a horizontal translation of c units to the left and a vertical translation of d units up.
b) Produce a single graph containing the actual data given in the tidal prediction chart and your model. Comment on the strength and limitations of the model.
pikachu975
Premium Member
Re: HSC 2016 2U Marathon
Since you have the actual data, you can get a computer package to fit the data to that model, as well as produce the graphs.
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