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If the derivative of a function is identically zero, what can you say about the function ?

Correct answer is C. First of all, there are six coins so there are 2^6 different results of guess, because each coin has two possible outcomes: correct guess or wrong guess. And for each of these 2^6 results, the probability is all same, equal to 1/2^6. Let x be the entrance fee. For the...

Re: HSC 2014 4U Marathon oops, how can i forget similar triangles :( \triangle{ADO}|||\triangle{BOC} (\angle{DAO}=\angle{OBC}, \angle{AOD}=\angle{BCO}) the reason why these pairs of angles are equal is already in my first solution

lol. I only change limits but keep other parts unchanged because in my example I did NOT have the term \sqrt{x+1} , so I did not need to change the integrand. I totally agree with SilentWaters. -- He did NOT change the substitution at all

anyone wanna a bet? lol

Re: HSC 2014 4U Marathon hm thinking

Re: HSC 2014 4U Marathon Here is my solution using trigonometry. Let AD, DC and CB touch the semicircle at points E, F and G respectively. Let the radius of the circle be r. $ Let $ \angle{EOA}=\angle{GOB}=\alpha (\triangle{AOE}\equiv\triangle{BOG}) $ Let $ \angle{EOD}=\angle{DOF}=\beta...

are you serious? $ Lets randomly do $ \int_0^1xdx, $ which is easily evaluated to be $ \frac12 $ Now lets take substitution $ x=u^2-1, $ and first take $ u=\sqrt{x+1}. $ So $ dx=2udu, $ and $ x=0\rightarrow u=1, x=1\rightarrow u=\sqrt2, $ thus the integral...

There are usually two different ways to solve this type of question. $ The first way: dividing the equation by $ 2: \frac{\alpha+\gamma}{2}=\frac{\beta+\delta}{2} the LHS represents the midpont of AC, while the RHS represents the midpont of BD. This equation means AC and BD bisect each...

Re: HSC 2014 4U Marathon Is it allowed to use the standard table of integrals to evaluate \int\frac{du}{u^2+2} ?

You can take either one, but not both. Because the substitution needs to be one-to-one mapping.

Re: HSC 2014 4U Marathon - Advanced Level yep well done

Re: HSC 2014 4U Marathon - Advanced Level maybe it is good, one odd and the other even, always like that

Re: HSC 2014 4U Marathon - Advanced Level well done. nice question and nicer solution. lol

Re: HSC 2014 4U Marathon - Advanced Level This one should be a good one: $ Let $ P(x)=\frac{d^n}{dx^n}(x^2-1)^n. $ Evaluate $ \int_{-1}^1P^2(x)dx.

Re: HSC 2014 4U Marathon - Advanced Level but boats are different, aren't they? Anyway, my answer divided by 5! will be the same as yours

If you don't have to use integration, here is one way: the enclosed area = area of the quarter circle - area of the triangle. $ So $ A=\frac14\pi(2)^2-\frac12\times2\times2=\pi-2

Re: HSC 2014 4U Marathon - Advanced Level ok here are the details. My previous calculation is for the case X, Y and theire wives all in one boat. 5\times{8 \choose 2}^2\times{6 \choose 2}^2\times{4 \choose 2}^2=31752000 They can be in any of the five boates, which is why 5 times; 8 C 2...

Re: HSC 2014 4U Marathon - Advanced Level hold on first, coz i am not sure i understood the question correctly: does the question mean that X, his wife, Y and his wife all four people in the same boat? or X and his wife in one same boat and Y and his wife in the other same boat?

Re: HSC 2014 4U Marathon $ Without loss of generality, let the circle be $ |z|=1, $ and the diameter be on the real axis. $ $ Let the points $ P, A $ and $ B $ represent complex numbers $ \cos\theta+i\sin\theta, -1 $ and $ 1. $ Now $ \overrightarrow{AP}=\cos\theta+1+i\sin\theta...