HSC 2016 MX1 Marathon (archive) (1 Viewer)

KingOfActing · Nov 7, 2015

Re: HSC 2016 3U Marathon

leehuan said:
$\\ $leehuan guessed all 10 questions in his MX1 multiple choice. What is the probability that he got at least 9/10 right anyway?$$

$\begin{align*} P(\text{At least 9}) = P(9) + P(10) &= \binom{10}{9}\left (\frac{1}{4}\right )^9\left (\frac{3}{4}\right )^1 + \binom{10}{10}\left (\frac{1}{4}\right )^{10}\left (\frac{3}{4}\right )^0 \\&= 10\left(\frac{1}{4}\right)^{9}\frac{3}{4} + \left ( \frac{1}{4} \right )^{10} \\&= \frac{31}{1048576} \end{align*}$

Hope I did that right, I'm not good at probability

davidgoes4wce · Nov 11, 2015

Re: HSC 2016 3U Marathon

When asked to determine the x^3 term in the Binomial expansion. This was the answer from Fitzpatrick answers in his main text:

Would it be ok to state the other terms mC0 and nC0 in the answer?

leehuan · Nov 11, 2015

Re: HSC 2016 3U Marathon

Why is there a nC0 on the LHS?

But on the RHS I'd argue you actually should and Fitzpatrick is inaccurate (although not incorrect, obviously).

davidgoes4wce · Nov 11, 2015

Re: HSC 2016 3U Marathon

leehuan said:
Why is there a nC0 on the LHS?

But on the RHS I'd argue you actually should and Fitzpatrick is inaccurate (although not incorrect, obviously).

My bad I typed up nC0 on LHS when it shouldn't have been there.

davidgoes4wce · Nov 12, 2015

Re: HSC 2016 3U Marathon

I didn't quite understand the last time

I also assume in this question we have to rote memorise the binomial expansion to do any of these questions more efficiently?

leehuan · Nov 12, 2015

Re: HSC 2016 3U Marathon

If we are trying to extract the coefficient of x^r in:
1+(1+x)+(1+x)^2+...+(1+x)^n

We firstly note that the term in (1+x)^n with x^r is nCr x^r
We then note that the term before (1+x)^n is (1+x)^(n-1), so the term with x^r is (n-1)Cr
Then we can extract the next term, (n-2)Cr from (1+x)^(n-2)

Now, this question will naturally assume the conditions 1 <= r <= n

We note that if we spread out the thing further, it would look something like
1 + (1+x) + (1+x)^2 + ... + (1+x)^(r-1) + (1+x)^r + (1+x)^(r+1) + ... + (1+x)^n

We note that the 1 + (1+x) + (1+x)^2 + ... + (1+x)^(r-1) component will not even have a term including x^r, so we disregard it when trying to extract the coefficient

Hence, the last term that is included in the coefficient of x^r comes from (1+x)^r, which naturally must just be rCr x^r

davidgoes4wce · Nov 12, 2015

Re: HSC 2016 3U Marathon

Would I be right about saying this about the (1+x)^(n+1) expansion?

Drongoski · Nov 12, 2015

Re: HSC 2016 3U Marathon

davidgoes4wce said:
Would I be right about saying this about the (1+x)^(n+1) expansion?

Yes.

Did you do the IB at high school?

davidgoes4wce · Nov 12, 2015

Re: HSC 2016 3U Marathon

Drongoski said:
Yes.

Did you do the IB at high school?

No I didn't.

InteGrand · Nov 12, 2015

Re: HSC 2016 3U Marathon

davidgoes4wce said:
I also assume in this question we have to rote memorise the binomial expansion to do any of these questions more efficiently?

$Well yes, the binomial theorem $(a+b)^n = \sum _{k=0}^n \binom{n}{k}a^{n-k}b^k$ is expected to be known by students. From this, if we let $a=1$ and $b=x$, we have $a^{n-k}=1^{n-k}=1$ for all $k$, and so $(1+x)^n = \sum _{k=0}^n \binom{n}{k} x^k$ (but you should know this by heart as well for efficiency purposes; so $\binom{n}{k}$ is the coefficient of $x^k$ in the expansion of $(1+x)^n$, for $k=0,1,2,\ldots,n$). I think the new formula sheet to be used in HSC 3U from 2016 onwards has the binomial theorem formula on it though, so if you sit the HSC exam next year or later, you may technically not need to memorise the formula.$

leehuan · Nov 14, 2015

Re: HSC 2016 3U Marathon

NEXT QUESTION

$\\ \int { \frac { dx }{ \sqrt { 2x-{ x }^{ 2 } } } } $ using the substitution $u=x-1$$

Note: If you haven't been taught the resulting integral after the substitution, consult your formula sheet.

davidgoes4wce · Nov 14, 2015

Re: HSC 2016 3U Marathon

leehuan said:
NEXT QUESTION

$\\ \int { \frac { dx }{ \sqrt { 2x-{ x }^{ 2 } } } } $ using the substitution $u=x-1$$

Note: If you haven't been taught the resulting integral after the substitution, consult your formula sheet.

leehuan · Nov 14, 2015

Re: HSC 2016 3U Marathon

NEXT QUESTION

$\\ $Evaluate$ \\ $a)$ \lim _{ x\rightarrow 4 }{ \frac { { x }^{ 3 }-35x+76 }{ 2{ x }^{ 3 }-16{ x }^{ 2 }+33x-4 } } \\ $b)$\lim _{ x\rightarrow \infty }{ \frac { { x }^{ 3 }-35x+76 }{ 2{ x }^{ 3 }-16{ x }^{ 2 }+33x-4 } }$

davidgoes4wce · Nov 14, 2015

Re: HSC 2016 3U Marathon

leehuan said:
NEXT QUESTION

$\\ $Evaluate$ \\ $a)$ \lim _{ x\rightarrow 4 }{ \frac { { x }^{ 3 }-35x+76 }{ 2{ x }^{ 3 }-16{ x }^{ 2 }+33x-4 } } \\ $b)$\lim _{ x\rightarrow \infty }{ \frac { { x }^{ 3 }-35x+76 }{ 2{ x }^{ 3 }-16{ x }^{ 2 }+33x-4 } }$

Part (a) should be undefined.Left hand limits is not equal to the Right Hand Limit. (Try x=4.01 and Try x=3.99 as proof)

Part (b) limit as x approaches infinity, approaches 1/2 (by Dominating terms)

leehuan · Nov 14, 2015

Re: HSC 2016 3U Marathon

davidgoes4wce said:
Part (a) should be undefined.Left hand limits is not equal to the Right Hand Limit. (Try x=4.01 and Try x=3.99 as proof)

Part (b) limit as x approaches infinity, approaches 1/2 (by Dominating terms)

Your margin of error is too large. Try 4.0001 and 3.9999 instead. Tbh, the limit can actually be evaluated without testing both sides.

In terms of extracurricular maths (or maybe even 4u??), yep happy with the answer for b). Within the 3u course they want you to use lim x->infinity (1/x) = 0 though.

So you divide top and bottom by 1/x^3 to apply the limit. Because naturally it follows that lim x->infinity (1/x)^n = 0 for |n|>=1

davidgoes4wce · Nov 16, 2015

Re: HSC 2016 3U Marathon

This was an error in Margaret Grove Exercise 1.1 Q2f. The answer should be 1 (not negative 1)

leehuan · Nov 16, 2015

Re: HSC 2016 3U Marathon

There's hundreds of errors in MIF. Hate to sound rude but like I said, nobody really cares just to look here to pick one out. You're not informing anyone anything even though your intentions are clearly pure.
-------------------------------------------
NEXT QUESTION:
$\\ $Find the equation of the tangent to the curve ${y}^{2}=4ax$ at the point $(a{p}^{2},2ap)$

Carrotsticks · Nov 16, 2015

Re: HSC 2016 3U Marathon

leehuan said:
Your margin of error is too large. Try 4.0001 and 3.9999 instead. Tbh, the limit can actually be evaluated without testing both sides.

What is 'too large'? What makes your margin of error 'better' than another one?

And as you mentioned, these questions are ideally done without any sort of numerical analysis.

davidgoes4wce · Nov 16, 2015

Re: HSC 2016 3U Marathon

leehuan said:
There's hundreds of errors in MIF. Hate to sound rude but like I said, nobody really cares just to look here to pick one out. You're not informing anyone anything even though your intentions are clearly pure.
-------------------------------------------
NEXT QUESTION:
$\\ $Find the equation of the tangent to the curve ${y}^{2}=4ax$ at the point $(a{p}^{2},2ap)$

leehuan · Nov 16, 2015

Re: HSC 2016 3U Marathon

Carrotsticks said:
What is 'too large'? What makes your margin of error 'better' than another one?

And as you mentioned, these questions are ideally done without any sort of numerical analysis.

Nothing apart from the fact the magnitude of the values I chose are more closer to that of the given value (and equal in sign, since this isn't the case of a limit to 0).
-----------------------------------
I'll give some benefit of the doubt; it's not intuitive to realise that direct substitution of x=4 results in 0/0 because the calculator will say "math error", which can be interpreted as a/0 (a is any complex number except 0+0i). But when giving it to a 3U student arbitrarily I would have hoped that they'd then realise (x-4) is a factor of both the numerator and denominator, and thus perform polynomial long division (or any other correct approach) to find a limit that can be evaluated on the spot.

But that's what I would've done, which is like you said, avoidance of numerical analysis.
----------------------------
NEXT QUESTION:
$\\ $Since a 50c coin is a regular dodecagon, find the value of an exterior angle on a 50c coin.$$

(This is actually very simple 2U. But I'm leaving it here because apparently it was hard in the VCE and 2U has an unanswered question I think?)

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