HSC 2013 Maths Marathon (archive) (1 Viewer)

[Shazer2]

Re: HSC 2013 2U Marathon

302 digits

Did you look it up? I want to see your working, interesting problem.

 

[Frie]

Re: HSC 2013 2U Marathon

Need a little last minute help:

Can someone explain why the discriminant has to be less than 0, because I thought that meant that there are no real roots.
And what the heck does the Note mean.

(Question at bottom and answer is at the top, it's from 2010 HSC Q8d)

Thanks, appreciated.

 

[andybandy]

Re: HSC 2013 2U Marathon

Did you look it up? I want to see your working, interesting problem.

no ill wrote it out now

 

[andybandy]

Re: HSC 2013 2U Marathon

<a href="http://www.codecogs.com/eqnedit.php?latex=\log_{10}2^{1000}&space;=&space;1000&space;\times&space;\log_{10}2&space;=&space;301.030" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\log_{10}2^{1000}&space;=&space;1000&space;\times&space;\log_{10}2&space;=&space;301.030" title="\log_{10}2^{1000} = 1000 \times \log_{10}2 = 301.030" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=2^{10}&space;=&space;1024&space;\textup{&space;hence&space;2&space;digits}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?2^{10}&space;=&space;1024&space;\textup{&space;hence&space;2&space;digits}" title="2^{10} = 1024 \textup{ hence 2 digits}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=10&space;\times&space;\log_{10}2&space;=&space;3.01&space;\therefore&space;\textup{&space;as&space;its&space;}&space;>3.&space;\textup{&space;4&space;digits}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?10&space;\times&space;\log_{10}2&space;=&space;3.01&space;\therefore&space;\textup{&space;as&space;its&space;}&space;>3.&space;\textup{&space;4&space;digits}" title="10 \times \log_{10}2 = 3.01 \therefore \textup{ as its } >3. \textup{ 4 digits}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=1000&space;\times&space;\log_{10}2&space;=&space;301.02&space;\textup{&space;and&space;therefore&space;302&space;digits.&space;}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?1000&space;\times&space;\log_{10}2&space;=&space;301.02&space;\textup{&space;and&space;therefore&space;302&space;digits.&space;}" title="1000 \times \log_{10}2 = 301.02 \textup{ and therefore 302 digits. }" /></a>

 

[Frie]

Re: HSC 2013 2U Marathon

<a href="http://www.codecogs.com/eqnedit.php?latex=\log_{10}2^{1000}&space;=&space;1000&space;\times&space;\log_{10}2&space;=&space;301.030" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\log_{10}2^{1000}&space;=&space;1000&space;\times&space;\log_{10}2&space;=&space;301.030" title="\log_{10}2^{1000} = 1000 \times \log_{10}2 = 301.030" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=2^{10}&space;=&space;1024&space;\textup{&space;hence&space;2&space;digits}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?2^{10}&space;=&space;1024&space;\textup{&space;hence&space;2&space;digits}" title="2^{10} = 1024 \textup{ hence 2 digits}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=10&space;\times&space;\log_{10}2&space;=&space;3.01&space;\therefore&space;\textup{&space;as&space;its&space;}&space;>3.&space;\textup{&space;4&space;digits}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?10&space;\times&space;\log_{10}2&space;=&space;3.01&space;\therefore&space;\textup{&space;as&space;its&space;}&space;>3.&space;\textup{&space;4&space;digits}" title="10 \times \log_{10}2 = 3.01 \therefore \textup{ as its } >3. \textup{ 4 digits}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=1000&space;\times&space;\log_{10}2&space;=&space;301.02&space;\textup{&space;and&space;therefore&space;302&space;digits.&space;}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?1000&space;\times&space;\log_{10}2&space;=&space;301.02&space;\textup{&space;and&space;therefore&space;302&space;digits.&space;}" title="1000 \times \log_{10}2 = 301.02 \textup{ and therefore 302 digits. }" /></a>

You wrote 1000 x log2 = 301.030 and then changed it to equal 301.02?

I don't understand the working , how is 3.01 = 4 digits?

 

[andybandy]

Re: HSC 2013 2U Marathon

You wrote 1000 x log2 = 301.030 and then changed it to equal 301.02?

I don't understand the working , how is 3.01 = 4 digits?

im sort of a begginer with latex, so its messy and i cant concentrate as properly haha..
but basically 10^ (of the answer of 10 times log 2) = 1024
apply the same process

 

[andybandy]

Re: HSC 2013 2U Marathon

Need a little last minute help:

Can someone explain why the discriminant has to be less than 0, because I thought that meant that there are no real roots.
And what the heck does the Note mean.

(Question at bottom and answer is at the top, it's from 2010 HSC Q8d)

Thanks, appreciated.

yeah i would love if someone helped with that, but what im thinking is, since its a cubic, only one side will continously increase, and so that root might be above the x axis?

 

[Frie]

Re: HSC 2013 2U Marathon

im sort of a begginer with latex, so its messy and i cant concentrate as properly haha..
but basically 10^ (of the answer of 10 times log 2) = 1024
apply the same process

Oh right, I understand now, you round all the decimals up, so 3.01 becomes 4, 301.03 becomes 302. I see.

 

[nerdasdasd]

Re: HSC 2013 2U Marathon

The trick question of 2012 2U (Q16, b)

 

[vilst3r]

Re: HSC 2013 2U Marathon

Oh my god, I just did that question a few moments ago, I totally got destroyed and had to resort to answers , its the first part that's tricky and the rest should be easy

I didn't understand the solution to the first part, I'm hoping someone here knows how..

 

[andybandy]

Re: HSC 2013 2U Marathon

Oh my god, I just did that question a few moments ago, I totally got destroyed and had to resort to answers , its the first part that's tricky and the rest should be easy

I didn't understand the solution to the first part, I'm hoping someone here knows how..

i felt so happy getting all the questions right the first time! i was so suprised

 

[Drongoski]

Re: HSC 2013 2U Marathon

Oh right, I understand now, you round all the decimals up, so 3.01 becomes 4, 301.03 becomes 302. I see.

In the days before calculators were invented, logarithm tables (& slide rules) were used to find (approximately) the product or quotient of 2 numbers (otherwise one would have to do it by hand). So when looking up the log (to base 10) of 2,.002, 200, one would look up the log of 2.0, which is 0.3010 ...: for 2, the log is 0.3010 ..., for 20000, it is 4.3010 ..,(one was taught, if there are 5 whole number digits, the characteristic is 4, i.e. 1 less than 5), for 0.002, the log is -2.69897 ...(= -3 + 0.3010...). By the way, the 0.3010 ... is known as the mantissa. For -3.3010 ... is meant to be -3 + 0.3010 ... written with the '-' above the '3' rather than just before it.

So: the number of whole number digits = characteristic + 1


Note that:

 

[andybandy]

Re: HSC 2013 2U Marathon

The trick question of 2012 2U (Q16, b)

<a href="http://www.codecogs.com/eqnedit.php?latex=Mot&space;=&space;\frac{\sin\theta&space;}{\cos\theta}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?Mot&space;=&space;\frac{\sin\theta&space;}{\cos\theta}" title="Mot = \frac{\sin\theta }{\cos\theta}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=Mpt&space;=&space;-\frac{\cos\theta&space;}{\sin\theta}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?Mpt&space;=&space;-\frac{\cos\theta&space;}{\sin\theta}" title="Mpt = -\frac{\cos\theta }{\sin\theta}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y&space;-&space;y_{1}&space;=&space;m(x&space;-&space;x^{1})&space;\textup{&space;at&space;}&space;(\cos\theta,&space;\sin\theta)&space;\textup{&space;gradient&space;}&space;=&space;-\frac{\cos\theta}{\sin\theta}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y&space;-&space;y_{1}&space;=&space;m(x&space;-&space;x^{1})&space;\textup{&space;at&space;}&space;(\cos\theta,&space;\sin\theta)&space;\textup{&space;gradient&space;}&space;=&space;-\frac{\cos\theta}{\sin\theta}" title="y - y_{1} = m(x - x^{1}) \textup{ at } (\cos\theta, \sin\theta) \textup{ gradient } = -\frac{\cos\theta}{\sin\theta}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y&space;-&space;\sin\theta&space;=&space;-\frac{\cos\theta}{\sin\theta}(x&space;-&space;\cos\theta)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y&space;-&space;\sin\theta&space;=&space;-\frac{\cos\theta}{\sin\theta}(x&space;-&space;\cos\theta)" title="y - \sin\theta = -\frac{\cos\theta}{\sin\theta}(x - \cos\theta)" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y\sin\theta&space;-&space;\sin^{2}\theta&space;=&space;-x\cos\theta&space;&plus;&space;\cos^{2}\theta" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y\sin\theta&space;-&space;\sin^{2}\theta&space;=&space;-x\cos\theta&space;&plus;&space;\cos^{2}\theta" title="y\sin\theta - \sin^{2}\theta = -x\cos\theta + \cos^{2}\theta" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y\sin\theta&space;&plus;x\cos\theta&space;=&space;\sin^{2}\theta&space;&plus;&space;\cos^{2}\theta" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y\sin\theta&space;&plus;x\cos\theta&space;=&space;\sin^{2}\theta&space;&plus;&space;\cos^{2}\theta" title="y\sin\theta +x\cos\theta = \sin^{2}\theta + \cos^{2}\theta" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y\sin\theta&space;&plus;x\cos\theta&space;=&space;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y\sin\theta&space;&plus;x\cos\theta&space;=&space;1" title="y\sin\theta +x\cos\theta = 1" /></a>

 

[vilst3r]

Re: HSC 2013 2U Marathon

<a href="http://www.codecogs.com/eqnedit.php?latex=Mot&space;=&space;\frac{\sin\theta&space;}{\cos\theta}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?Mot&space;=&space;\frac{\sin\theta&space;}{\cos\theta}" title="Mot = \frac{\sin\theta }{\cos\theta}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=Mpt&space;=&space;-\frac{\cos\theta&space;}{\sin\theta}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?Mpt&space;=&space;-\frac{\cos\theta&space;}{\sin\theta}" title="Mpt = -\frac{\cos\theta }{\sin\theta}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y&space;-&space;y_{1}&space;=&space;m(x&space;-&space;x^{1})&space;\textup{&space;at&space;}&space;(\cos\theta,&space;\sin\theta)&space;\textup{&space;gradient&space;}&space;=&space;-\frac{\cos\theta}{\sin\theta}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y&space;-&space;y_{1}&space;=&space;m(x&space;-&space;x^{1})&space;\textup{&space;at&space;}&space;(\cos\theta,&space;\sin\theta)&space;\textup{&space;gradient&space;}&space;=&space;-\frac{\cos\theta}{\sin\theta}" title="y - y_{1} = m(x - x^{1}) \textup{ at } (\cos\theta, \sin\theta) \textup{ gradient } = -\frac{\cos\theta}{\sin\theta}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y&space;-&space;\sin\theta&space;=&space;-\frac{\cos\theta}{\sin\theta}(x&space;-&space;\cos\theta)" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y&space;-&space;\sin\theta&space;=&space;-\frac{\cos\theta}{\sin\theta}(x&space;-&space;\cos\theta)" title="y - \sin\theta = -\frac{\cos\theta}{\sin\theta}(x - \cos\theta)" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y\sin\theta&space;-&space;\sin^{2}\theta&space;=&space;-x\cos\theta&space;+&space;\cos^{2}\theta" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y\sin\theta&space;-&space;\sin^{2}\theta&space;=&space;-x\cos\theta&space;+&space;\cos^{2}\theta" title="y\sin\theta - \sin^{2}\theta = -x\cos\theta + \cos^{2}\theta" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y\sin\theta&space;+x\cos\theta&space;=&space;\sin^{2}\theta&space;+&space;\cos^{2}\theta" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y\sin\theta&space;+x\cos\theta&space;=&space;\sin^{2}\theta&space;+&space;\cos^{2}\theta" title="y\sin\theta +x\cos\theta = \sin^{2}\theta + \cos^{2}\theta" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=y\sin\theta&space;+x\cos\theta&space;=&space;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y\sin\theta&space;+x\cos\theta&space;=&space;1" title="y\sin\theta +x\cos\theta = 1" /></a>

I don't understand the part [Mot = \frac{\sin\theta }{\cos\theta}], like I can't comprehend any method of getting the gradient other than the rise/run method. Can you clarify, I may be lacking basic knowledge here lol

 

[andybandy]

Re: HSC 2013 2U Marathon

I don't understand the part [Mot = \frac{\sin\theta }{\cos\theta}], like I can't comprehend any method of getting the gradient other than the rise/run method. Can you clarify, I may be lacking basic knowledge here lol

<a href="http://www.codecogs.com/eqnedit.php?latex=\cos\theta&space;=&space;\frac{x_{t}}{OT}&space;\textup{&space;where&space;OT&space;=&space;1&space;}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\cos\theta&space;=&space;\frac{x_{t}}{OT}&space;\textup{&space;where&space;OT&space;=&space;1&space;}" title="\cos\theta = \frac{x_{t}}{OT} \textup{ where OT = 1 }" /></a>

since its a unit circle, and a right angled triangle, so it gives you the point of t, and sintheta will give you the y value, and you find the gradient from their

 

[vilst3r]

Re: HSC 2013 2U Marathon

Isn't Cos0 = OT/OP, whats the Xt you referring to?

 

[andybandy]

Re: HSC 2013 2U Marathon

Isn't Cos0 = OT/OP, whats the Xt you referring to?

you draw a line down from t to the x axis

 

[vilst3r]

Re: HSC 2013 2U Marathon

Ohhhhh I think I got it, so you draw a perpendicular line from T to the x-axis and thats where you get cosO=xt/OT and sinO=yt/OT, right?

 

[vilst3r]

Re: HSC 2013 2U Marathon

you draw a line down from t to the x axis

God, they should mention that in the solutions, I got so confused LOL

 

[andybandy]

Re: HSC 2013 2U Marathon

Ohhhhh I think I got it, so you draw a perpendicular line from T to the x-axis and thats where you get cosO=xt/OT and sinO=yt/OT, right?

yes thats it!