HSC 2013-14 MX1 Marathon (archive) (1 Viewer)

dunjaaa · Aug 12, 2013

Re: HSC 2013 3U Marathon Thread

so i was right hah

JJ345 · Aug 12, 2013

Re: HSC 2013 3U Marathon Thread

Sy123 said:
$$Prove for some integers$ \ m, n$

$1+ \frac{m}{1!} + \frac{m(m+1)}{2!} + \dots + \frac{m(m+1) \dots ( m+n-1)}{n!} = \frac{(m+n)!}{m! n!}$

What I did here was use sum of the following geometric series: (1+x)^m-1 +(1+x)^m+.........+(1+x)^m+n-1 =[ (1+x)^m+n -(1+x)^m-1/x]

Then I equated coefficients of x^m-1 and rewrote them in the form (m+n)!/m!n!....Does it work??

Sy123 · Aug 12, 2013

Re: HSC 2013 3U Marathon Thread

JJ345 said:
What I did here was use sum of the following geometric series: (1+x)^m-1 +(1+x)^m+.........+(1+x)^m+n-1 =[ (1+x)^m+n -(1+x)^m-1/x]

Then I equated coefficients of x^m-1 and rewrote them in the form (m+n)!/m!n!....Does it work??

Yep that worked for me

Well done

========

$$Prove for integers$ \ n \ $and$ \ k \ \ n>2k$

$\binom{n}{k} 2^k = \binom{n}{n} \binom{n}{k} + \binom{n}{n-1} \binom{n-1}{k-1} + \binom{n}{n-2} \binom{n-2}{k-2} + \dots + \binom{n}{n-k} \binom{n-k}{0}$

JJ345 · Aug 17, 2013

Re: HSC 2013 3U Marathon Thread

This is a pretty neat question (I think it appeared in some ancient HSC paper)--> Nothing tricky, just slightly more conceptual =)

A woman travelling along a straight flat road passes three points at intervals of 200m. From these points she observes the angle of elevation of the top of the hill to the left of the road to be respectively 30°, 45°, and again 45°. Find the height of the hill.

asianese · Aug 17, 2013

Re: HSC 2013 3U Marathon Thread

This was the last question of a 4U HSC paper in the 90s. (or 80s..)

JJ345 · Aug 18, 2013

Re: HSC 2013 3U Marathon Thread

asianese said:
This was the last question of a 4U HSC paper in the 90s. (or 80s..)

Probably, can't remember...but looking at my method I think its alot simpler than a typical 4U Q8. from the 1980s, I found it more 3U style.

Sy123 · Aug 18, 2013

Re: HSC 2013 3U Marathon Thread

Sy123 said:
Yep that worked for me

Well done

========

$$Prove for integers$ \ n \ $and$ \ k \ \ n>2k$

$\binom{n}{k} 2^k = \binom{n}{n} \binom{n}{k} + \binom{n}{n-1} \binom{n-1}{k-1} + \binom{n}{n-2} \binom{n-2}{k-2} + \dots + \binom{n}{n-k} \binom{n-k}{0}$

Take the identity:

$(1+2x)^n = (x + (1+x))^n$

$\sum_{k=0}^n \binom{n}{k} 2^k x^k = \sum_{k=0}^n \binom{n}{k} x^{n-k} (1+x)^k$

Then equate the co-efficients of x^k on both sides.

===========================

$$Prove by mathematical induction for integers$ \ n \geq 1$

$2! \cdot 4! \cdot 6! \cdot 8! \cdot \dots \cdot (2n)! \geq ((n+1)!)^n$

asianese · Aug 24, 2013

Re: HSC 2013 3U Marathon Thread

$$Use calculus to sketch the curve $ y = \frac{\ln x}{x} $ and find the maximum function value. Use this to deduce that if $ e\le a<b $ then $ b^a<a^b $ so that in particular, $\\ \pi^e<e^\pi.$

dunjaaa · Aug 25, 2013

Re: HSC 2013 3U Marathon Thread

Sketch the curve and develop and inequality, with x=pi and you should obtain that result

Sy123 · Aug 25, 2013

Re: HSC 2013 3U Marathon Thread

$$i) Verify that$ \ \ a^n + b^n = (a+b)(a^{n-1} + b^{n-1}) - ab(a^{n-2}+ b^{n-2})$

$ii) A sequence$ \ u_1, u_2, u_3, \dots \ \ $is such that$

$u_n = u_{n-1} + u_{n-2} \ \ $and$ \ u_1 = 1 \ u_2 = 3$

$Using the result from part$ \ (i) \ $prove that$

$u_n = \left( \frac{1+\sqrt{5}}{2} \right )^n + \left(\frac{1-\sqrt{5}}{2} \right )^n$

$$For integer$ \ n$

===

$Using a similar method, find in terms of $ \ \ n \ \ S_n \ \ $Where$

$S_{n} = 11S_{n-1} - 31S_{n-2} + 30S_{n-3} \ \ \ $and$ \ S_1 = 11 \ S_2 = 38 \ S_3 = 160$

Sy123 · Sep 1, 2013

Re: HSC 2013 3U Marathon Thread

$A plane curve is shown below$

$The curve has parametric equations$

$x=\cos^3 \theta$

$y= \sin^3 \theta$

$$for$ \ 0 \leq \theta \leq 2\pi$

$i) Show that the tangent to the curve at$ \ P(\cos^3 \theta, \sin^3 \theta) \ \ $is given by$

$y \cos \theta + x \sin \theta = \sin \theta \cos \theta$

$$ii) The tangent cuts the$ \ y$-axis$ \ $at$ \ R \ $and the$ \ x$-axis at$ \ S$

$Prove that$ \ RS \ $is constant$

Sy123 · Sep 3, 2013

Re: HSC 2013 3U Marathon Thread

$Find$

$\sum_{k=2}^n \frac{1}{\log_k u}$

RealiseNothing · Sep 3, 2013

Re: HSC 2013 3U Marathon Thread

Sy123 said:
$Find$

$\sum_{k=2}^n \frac{1}{\log_k u}$

$\log_u (n!)$ ?

RealiseNothing · Sep 3, 2013

Re: HSC 2013 3U Marathon Thread

RealiseNothing said:
$\log_u (n!)$ ?

Using the identity:

$\log_a(b) = \frac{\ln(b)}{\ln(a)}$

$\sum_{k=2}^{n} \frac{1}{\log_k u} = \sum_{k=2}^{n} \frac{\ln(k)}{\ln(u)}$

$\frac{\ln(2) + \ln(3) + ... + \ln(n)}{\ln(u)}$

Using:

$\ln(A) + \ln(B) = \ln(AB)$

$\frac{\ln(n!)}{\ln(u)}$

$\log_u(n!)$

Sy123 · Sep 3, 2013

Re: HSC 2013 3U Marathon Thread

Yep that's pretty much it

$$Prove by mathematical induction for integer$ \ n\geq 2$

$\left( \frac{n+1}{2} \right)^n > n!$

Difficulty: Q14

Sy123 · Sep 5, 2013

Re: HSC 2013 3U Marathon Thread

I had a go at making a parabola question, its not as brute force as it looks.

$$The point$ \ P(2ap, ap^2) \ $lies on the parabola$ \ x^2=4ay$

$$The tangent to$ \ P \ $is drawn and intersects with the parabola$ \ x^2=-4ay \ $at the points$ \ \ R \ $and$ \ S$

$$Where the$ \ x$-co ordinate of$ \ S \ $is lower than$ \ R$

$$The point$ \ S \ $is then reflected about the origin to make point$ \ S'$

$$Show that the gradient of the line$ \ RS' \ $is$ \ \ \frac{3p}{2}$

Difficulty: Q13

Sy123 · Sep 8, 2013

Re: HSC 2013 3U Marathon Thread

$$i) For the polynomial$ \ \ P(x) = (x-a)^2 Q(x)$

$$Show that$ \ \ x=a \ \ $is a root of$ \ P'(a)$

$$ii) The polynomial$ \ \ S(y) = ay^3 + by + c \ \ $has a double zero at$ \ y= 1$

$$and has a remainder of$ \ 4 \ $when divided by$ \ \ y+1$

$$Find$ \ a,b,c$

Difficulty: Q13

obliviousninja · Sep 8, 2013

Re: HSC 2013 3U Marathon Thread

Sy, next to the questions you post, can you write its difficulty, ie question number it belongs to (section of the paper).

Sy123 · Sep 8, 2013

Re: HSC 2013 3U Marathon Thread

obliviousninja said:
Sy, next to the questions you post, can you write its difficulty, ie question number it belongs to (section of the paper).

I'll try

Most if not all questions I post will be at the higher end of difficulty in terms of HSC, for both the 2U and 4U marathons

Sy123 · Sep 9, 2013

Re: HSC 2013 3U Marathon Thread

$$A projectile is fired from the origin with initial velocity$ \ \ U$

$$to pass through the point$ \ (a,b)$

$Prove that there are 2 possible trajectories if$

$(U^2-gb)^2 > g^2(a^2+b^2) \ \ \ \ \fbox{5}$

Difficulty: Q14

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