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Re: HSC 2015 3U Marathon \int\frac{1-\sqrt{x}}{1-\sqrt[4]{x}} \, \mathrm{d}x

re: HSC Chemistry Marathon Archive Usually grams.

Re: HSC 2015 3U Marathon Prove that the product of 4 consecutive integers is always one less than a perfect square.

Thanks! Do you know what the harder types of these questions are like? Like instead of \alpha^2, \beta^2, \gamma^2, what else can there be?

Ah okay. So then just do the same for beta and gamma, add those, rearrange, sub in values etc.?

$The polynomial $x^3-3x+2=0$ has roots $\alpha, \beta, \gamma$. Find $\alpha^4+\beta^4+\gamma^4$. What's the easiest way to do this? Also, how do you do those questions where you are asked to find a polynomial with roots $\alpha^2, \beta^2, \gamma^2, given that another polynomial (e.g...

re: HSC Chemistry Marathon Archive For the first one: Using H=-mCΔT, we have H=-12540J=-12.54kJ. -12.54/-1367=0.00917 moles. Multiplying by the molar mass of ethanol you get 0.423g. For the second one: The reaction is: (COOH)2.2H2O + 2NaOH -> Na2(COO)2.2H2O + H2O. Using n=m/M, we have 0.100...

Yep, that's the only setup which follows the conditions. What's the answer?

He established that (x-1)S_1 = x^{2n}-1 by using the sum of a geometric series. To get S=\sum_{r=1}^n \binom{n}{r}x^{r+1}=x\left(\sum_{r=1}^n \binom{n}{r}x^{r}\right) he factored out x from the sum. Is that what you were unsure about?

There are some here.

f(x) is the expression they've stated, x2 - c. f'(x) would be 2x, as c is a constant.

re: HSC Chemistry Marathon Archive One Dobson Unit is: 1 m * 1 m * 0.01 mm = 1 m * 1 m * (0.01)x10-3 m = 1x10-5 m3. So we have: 2.7x1020 molecules / DU 4.48x10-4 moles / 1x10-5 m3 (Since there are 6.022x1023 molecules in a mole) 44.84 moles / m3 (Dividing through by 1x10-5) Multiplying through...

$Use $x_1=x_{0}-\frac{f(x_0)}{f'(x_0)}$, where $x_{0}=a$, to find the second approximation. The margin of error between this approximate root $x_1$ and the actual root $\sqrt{c}$ is given by the distance between them, i.e. $\left |x_1-\sqrt{c} \right |$. Simplify this to get the required...

2009 HSC Question 19.

Re: HSC 2015 3U Marathon Here are some more random ones:

Re: HSC 2015 3U Marathon Ah okay thanks for clearing that up.

Re: HSC 2015 3U Marathon It was k=1, my bad. Is it just me or does it not work for n=1?

If you sketch it you should be able to see that the maximum separation occurs at x=\frac{\pi}{2}.

Just so we can see what you wrote while Codecogs isn't working: Thanks for the help!

I can't get the LHS inequality. I can now see that with the RHS you show that approximating the area under the graph with rectangles gives an area that is lower than the exact integral, but with the LHS I'm still not sure. Could you explain what you mean by your hint?