HSC 2016 Maths Marathon (archive) (1 Viewer)

leehuan · Aug 14, 2016

Re: HSC 2016 2U Marathon

$$I think at a 2U level InteGrand would've been happy if you left your answer like this without bringing in the factorial notation.$\\ f^{(n)}(x)=n(n-1)(n-2)\dots(3)(2)(1)$

LC14199 · Aug 14, 2016

Re: HSC 2016 2U Marathon

$2.00 \times 10^4$

packpig123 · Aug 14, 2016

Re: HSC 2016 2U Marathon

LC14199 said:
$2.00 \times 10^4$

thanks!

InteGrand · Sep 17, 2016

Re: HSC 2016 2U Marathon

$\noindent \textbf{NEW QUESTION}$

$\noindent Show that for any positive numbers $a,b,c,d$, the number $\frac{a+c}{b+d}$ is strictly between $\frac{a}{b}$ and $\frac{c}{d}$.$

$\noindent \textbf{Bonus.} Provide a geometric interpretation of this result.$

Paradoxica · Sep 17, 2016

Re: HSC 2016 2U Marathon

InteGrand said:
$\noindent \textbf{NEW QUESTION}$

$\noindent Show that for any positive numbers $a,b,c,d$, the number $\frac{a+c}{b+d}$ is strictly between $\frac{a}{b}$ and $\frac{c}{d}$.$

$\noindent \textbf{Bonus.} Provide a geometric interpretation of this result.$

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.

InteGrand · Sep 17, 2016

Re: HSC 2016 2U Marathon

Paradoxica said:
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.

Correct! I'll leave the algebraic exercise as an exercise for 2U students.

InteGrand said:
$\noindent Show that for any positive numbers $a,b,c,d$, the number $\frac{a+c}{b+d}$ is strictly between $\frac{a}{b}$ and $\frac{c}{d}$.$

trecex1 · Sep 17, 2016

Re: HSC 2016 2U Marathon

InteGrand said:
Correct! I'll leave the algebraic exercise as an exercise for 2U students.

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)

$\text{New question:}\\ \text{Find the values of k such that } \frac{\log_{e}x}{x^2}=kx \text{ has two solutions}$

leehuan · Sep 17, 2016

Re: HSC 2016 2U Marathon

trecex1 said:
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)

$\text{New question:}\\ \text{Find the values of k such that } \frac{\log_{e}x}{x^2}=kx \text{ has two solutions}$

Nicely done, however the assumption that a/b < c/d isn't necessarily true. It may be possible that c/d < a/b.

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 · Sep 17, 2016

Re: HSC 2016 2U Marathon

trecex1 said:
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)

$\text{New question:}\\ \text{Find the values of k such that } \frac{\log_{e}x}{x^2}=kx \text{ has two solutions}$

Consider the equivalent equation (logx)/x³ = k

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 · Sep 28, 2016

Re: HSC 2016 2U Marathon

mini8658 said:
Find when the particle is at rest of the motion x=e^2t-7.

It's an exponential which has no rest points

KingOfActing · Sep 28, 2016

Re: HSC 2016 2U Marathon

mini8658 said:
Find when the particle is at rest of the motion x=e^2t-7.

$v = 2e^{2t} \neq 0\quad \forall t \in \mathbb R\\\therefore $The particle is never at rest, much like me$$

AfroNerd · Oct 4, 2016

Re: HSC 2016 2U Marathon

Differentiate y=x^2 + bx + c and hence find values of b and c if the line 3x+y-5=0 is a normal to the curve at the point X(3,-1).

davidgoes4wce · Oct 4, 2016

Re: HSC 2016 2U Marathon

AfroNerd said:
Differentiate y=x^2 + bx + c and hence find values of b and c if the line 3x+y-5=0 is a normal to the curve at the point X(3,-1).

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?

$\frac{dy}{dx}=2x+b$

The reason I ask is because if you substitute x=3

$\frac{dy}{dx}=6+b$

If the wording of the question is right, the gradient of the normal is -3, and the gradient of the tangent is $\frac{1}{3}$

In which case,

$\frac{1}{3}=6+b \implies b=\frac{-17}{3}$

AfroNerd · Oct 4, 2016

Re: HSC 2016 2U Marathon

Yep Im pretty sure 3x+y-5=0 is the normal.

Ty and also would that mean c=7

davidgoes4wce · Oct 4, 2016

Re: HSC 2016 2U Marathon

Yep I got c=7 from my working out.

duxxx · Oct 14, 2016

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.

Screen Shot 2016-10-14 at 2.13.05 PM.png

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 · Oct 14, 2016

Re: HSC 2016 2U Marathon

duxxx said:
The tidal heights over the next five days for the Yangtze River mouth in Shanghai are shown in screenshot below.
View attachment 33502
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.

Are you sure this is even 2 unit

duxxx · Oct 14, 2016

Re: HSC 2016 2U Marathon

Yeah it actually is, soooo hard.

duxxx · Oct 14, 2016

Re: HSC 2016 2U Marathon

Help me, I know you can you do extension 2!

InteGrand · Oct 15, 2016

Re: HSC 2016 2U Marathon

duxxx said:
The tidal heights over the next five days for the Yangtze River mouth in Shanghai are shown in screenshot below.
View attachment 33502
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.

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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