HSC 2013 Maths Marathon (archive) (1 Viewer)

Sy123 · Oct 18, 2013

Re: HSC 2013 2U Marathon

$$Consider the graph of$ \ \ y= \frac{1}{x}$

$$i) Using the trapezium rule for$ \ \ n+1 \ \ $function values$ \ \ x=n \ \ $to$ \ \ x=2n$

$Show that$

$\frac{1}{n} + \frac{1}{n+1} + \dots + \frac{1}{2n} > \ln 2 + \frac{3}{2n} \ \ \ \fbox{3}$

$ii) State a way to underestimate the area$

$$ underneath the curve from$ \ \ x=n \ \ $to$ \ \ x=2n \ \ \ \fbox{1}$

$iii) Use this way to Show that$

$\ln 2 > \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} \ \ \ \fbox{1}$

$iv) Hence by taking the limit as$ \ n \ $ goes to infinity$

$of both inequalities, show that$

$\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n+k} = \ln 2 \ \ \ \ \fbox{2}$

---

RealiseNothing · Oct 18, 2013

Re: HSC 2013 2U Marathon

RealiseNothing said:
Prove that:

$\log_{a+b}(ab) < 2$

For $a>b>1$

Consider the quadratic:

$x^2-ax+b$ which has positive real roots such that $a>b>1$

Show that:

$\ln(1) < \int_{a}^{b} \frac{dx}{x\ln(x)} < \ln(2)$

girlworld_club · Oct 18, 2013

Re: HSC 2013 2U Marathon

Having problems with this question, so easy, but can't get the answer.

The sum of 50 terms of an arithmetic series is 249 and the sum of 49 terms of the series is 233. Find the 50th term of the series.

nerdasdasd · Oct 18, 2013

girlworld_club said:
Having problems with this question, so easy, but can't get the answer.

The sum of 50 terms of an arithmetic series is 249 and the sum of 49 terms of the series is 233. Find the 50th term of the series.

Maths in focus ...,,
Hint: you can find the difference and then figure out everything from there.

girlworld_club · Oct 18, 2013

Re: HSC 2013 2U Marathon

i know how to do it but i think my algebra is stuffing me up. First find d, sub to find a in Sn formula. Then use d and a to find the 50th term using Tn formula.

nerdasdasd · Oct 18, 2013

girlworld_club said:
i know how to do it but i think my algebra is stuffing me up. First find d, sub to find a in Sn formula. Then use d and a to find the 50th term using Tn formula.

I'll post up a solution later if you can't get it .. On the train ATM.

HeroicPandas · Oct 18, 2013

Re: HSC 2013 2U Marathon

girlworld_club said:
i know how to do it but i think my algebra is stuffing me up. First find d, sub to find a in Sn formula. Then use d and a to find the 50th term using Tn formula.

Whats the value of 'd'?

integral95 · Oct 18, 2013

Re: HSC 2013 2U Marathon

RealiseNothing said:
Consider the quadratic:

$x^2-ax+b$ which has positive real roots such that $a>b>1$

Show that:

$\ln(1) < \int_{a}^{b} \frac{dx}{x\ln(x)} < \ln(2)$

Correct me if I'm wrong but I think the limits of dat integral is meant to be from b to a and not a to b

SpiralFlex · Oct 18, 2013

Re: HSC 2013 2U Marathon

anthony_wheels said:
I know it's a very 'shonky' proof/deduction lol... How could I improve it? View attachment 28805

Perhaps you can improve it by explaining your last line. (Although probably sufficient with most teachers)

HeroicPandas · Oct 18, 2013

Re: HSC 2013 2U Marathon

codislife said:
ok,

a = -151/25
d = 551/1225

50th term = 16

Remember that:

$T_n -T_{n-1} = d$ (this can be view differently)

n=50

d = 249-233 = 16

HeroicPandas · Oct 18, 2013

Re: HSC 2013 2U Marathon

my brain exploded sorry and yeh 'a' shouldn't be too nice

girlworld_club · Oct 18, 2013

Re: HSC 2013 2U Marathon

can someone explain how you got a and 50th term, i solved the two Sn questions simultanously, and got 0.45 but when i sub into the two different equations i end up with 2 diffrent values for a which is wrong.

RealiseNothing · Oct 18, 2013

Re: HSC 2013 2U Marathon

girlworld_club said:
Having problems with this question, so easy, but can't get the answer.

The sum of 50 terms of an arithmetic series is 249 and the sum of 49 terms of the series is 233. Find the 50th term of the series.

If 50 terms = 249 and 49 terms = 233 then isn't the 50th term just 16 lol?

RealiseNothing · Oct 18, 2013

Re: HSC 2013 2U Marathon

MethewYan said:
Correct me if I'm wrong but I think the limits of dat integral is meant to be from b to a and not a to b

Yer I think the condition that $a>b>1$ is wrong.

girlworld_club · Oct 18, 2013

Re: HSC 2013 2U Marathon

codislife said:
how many marks is that question worth?

3, it's from an independent paper which ripped it off maths in focus.

girlworld_club · Oct 18, 2013

Re: HSC 2013 2U Marathon

can someone pleaseeee show working for finding a with substitution as well as 50th term.

Menomaths · Oct 18, 2013

Re: HSC 2013 2U Marathon

girlworld_club said:
can someone pleaseeee show working for finding a with substitution as well as 50th term.

isn't 50th term what realize said?

girlworld_club · Oct 18, 2013

Re: HSC 2013 2U Marathon

Menomaths said:
isn't 50th term what realize said?

yes it's 16. but i'm getting frustrates cause i used substitution and then subbed 0.45 (b) into two different equations to get a, and got 2 different values. So if anyone did it the long way i would appreciate if they could show me how they did it.

ColdLipstick · Oct 18, 2013

Re: HSC 2013 2U Marathon

Carrotsticks said:
2011 HSC 2U Q9 (d) =)

View attachment 28803

You seem to enjoy writing questions. Very few students do that, good to see. The forum has about 2-3 other regulars (HSC Students) who also write questions.

Yes, I have written full papers (2U,3U,4U) that I keep as a way to explore my interest for mathematics

nerdasdasd · Oct 18, 2013

Re: HSC 2013 2U Marathon

I will post one up soon!

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