HSC 2016 Maths Marathon (archive) (1 Viewer)

davidgoes4wce · Feb 19, 2016

Re: HSC 2016 2U Marathon

The previous exercises were about 'investments' and 'superannuation investments', so they did it something along the lines of 1000 x 1.07

But Im just gathering that if doing a 'repayment, we should express it as 1000 x 0.07. But for some reason the text they go back to the GP formula by converting r from 0.07 to 1.07 (which I found a bit confusing). If this is the way to do repayments I'll make a note of that.

Paradoxica · Mar 10, 2016

This question is 2U level difficulty, right?

$$\noindent The line $y = bx + c$ intersects the parabola $y =ax^2$ in two distinct points $P$ and $Q.\\$a) Find the points of intersection in terms of $a,b,c. \\ $b) Find the point on the parabola $R$ whose $x$ value is the average of the points of intersection.$\\ $c) By using the trapezoidal rule with a single strip on the points $P,Q$, and then two strips on the points $P,Q,R$, determine the area of $\Delta PQR. \\ $d) Find the area of the parabolic segment bound by the chord $PQ$ and the parabola.$\\$e) Hence prove that the area of the parabolic segment is $\frac{4}{3}$ times the area of the inscribed triangle $PQR.$

leehuan · Mar 10, 2016

Re: This question is 2U level difficulty, right?

Paradoxica said:
$\\ $d) Find the area of the parabolic segment bound by the chord $PQ$ and the parabola.$\\$e) Hence prove that the area of the parabolic segment is $\frac{4}{3}$ times the area of the inscribed triangle $PQR.$

Since when were parabolic arcs and segments anywhere in the HSC courses

EDIT: Nvm if you have the correct intercepts it can probably be done by aid of trapezoidal rule. But they would probably miss that easily.

Nailgun · Mar 10, 2016

Re: HSC 2016 2U Marathon

Before I go on is the answer to c (feels wrong lel)

$A_{ \triangle PQR }=\frac { (2a-{ b }^{ 2 }+ac)\sqrt { b^{ 2 }-4ac } }{ { 2 }a^{ 2 } } ?$

MilkyCat_ · Mar 14, 2016

Re: HSC 2016 2U Marathon

Does anyone know how to do this question ?

The line y=3x+4-p cuts the parabola y=x^2-3x in two places

find p

davidgoes4wce · Mar 14, 2016

Re: HSC 2016 2U Marathon

MilkyCat_ said:
Does anyone know how to do this question ?

The line y=3x+4-p cuts the parabola y=x^2-3x in two places

find p

davidgoes4wce · Mar 14, 2016

Re: HSC 2016 2U Marathon

$As a visualisation, if p $ < $ 13 , as shown in the graph below, the red line would shift to the left . If p=13, there is exactly 1 solution$

leehuan · Mar 14, 2016

Re: HSC 2016 2U Marathon

davidgoes4wce said:
Quadratic formula answer

You could've just used the quadratic discriminant b^2-4ac instead of write out the entire formula

MilkyCat_ · Mar 14, 2016

Re: HSC 2016 2U Marathon

Okay sweet I got that I just didn't know if they had to be certain values but that diagram helped

InteGrand · Mar 15, 2016

Re: HSC 2016 2U Marathon

$\noindent \textbf{NEW QUESTION}$

$\noindent Suppose that $f$ is a linear function of $x$. Let $a,b\in \mathbb{R}$. Show that $\int _{a}^{a+b}f(x) \text{ d}x = \frac{b}{2}\left(f(a)+f(a+b)\right)$.$

Nailgun · Mar 16, 2016

Re: HSC 2016 2U Marathon

InteGrand said:
$\noindent \textbf{NEW QUESTION}$

$\noindent Suppose that $f$ is a linear function of $x$. Let $a,b\in \mathbb{R}$. Show that $\int _{a}^{a+b}f(x) \text{ d}x = \frac{b}{2}\left(f(a)+f(a+b)\right)$.$

$f\left( x \right) =mx+b\\ \int _{ a }^{ a+b } f(x){ d }x=\frac { b }{ 2 } \left( f(a)+f(a+b) \right) \\ L.H.S=\int _{ a }^{ a+b } f(x){ d }x\\ =\left[ \frac { mx^{ 2 } }{ 2 } +bx \right] _{ a }^{ a+b }\\ =\frac { m(a+b)^{ 2 } }{ 2 } +b(a+b)-\frac { m{ a }^{ 2 } }{ 2 } -ba\\ =\frac { m{ b }^{ 2 } }{ 2 } +{ b }^{ 2 }+mab\\ R.H.S=\frac { b }{ 2 } \left( f(a)+f(a+b) \right) \\ =\frac { b }{ 2 } (ma+b+m(a+b)+b)\\ =\frac { b }{ 2 } (2ma+2b+mb)\\ =\frac { m{ b }^{ 2 } }{ 2 } +{ b }^{ 2 }+mab\\ \therefore \quad L.H.S=R.H.S\\ \\ $ Q.E.D $\\ \\$P.S. did I use that right lel?$$

leehuan · Mar 16, 2016

Re: HSC 2016 2U Marathon

Good enough

Lol. Love the Q.E.D.

InteGrand · Mar 16, 2016

Re: HSC 2016 2U Marathon

Nailgun said:
$f\left( x \right) =mx+b\\ \int _{ a }^{ a+b } f(x){ d }x=\frac { b }{ 2 } \left( f(a)+f(a+b) \right) \\ L.H.S=\int _{ a }^{ a+b } f(x){ d }x\\ =\left[ \frac { mx^{ 2 } }{ 2 } +bx \right] _{ a }^{ a+b }\\ =\frac { m(a+b)^{ 2 } }{ 2 } +b(a+b)-\frac { m{ a }^{ 2 } }{ 2 } -ba\\ =\frac { m{ b }^{ 2 } }{ 2 } +{ b }^{ 2 }+mab\\ R.H.S=\frac { b }{ 2 } \left( f(a)+f(a+b) \right) \\ =\frac { b }{ 2 } (ma+b+m(a+b)+b)\\ =\frac { b }{ 2 } (2ma+2b+mb)\\ =\frac { m{ b }^{ 2 } }{ 2 } +{ b }^{ 2 }+mab\\ \therefore \quad L.H.S=R.H.S\\ \\ $ Q.E.D $\\ \\$P.S. did I use that right lel?$$

$\noindent Well done!$

$\noindent \textbf{NEW QUESTION}$

$\noindent Let $a$ be a positive constant. Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(a^{a^x}\right)$.$

leehuan · Mar 16, 2016

Re: HSC 2016 2U Marathon

Nek min logarithmic differentiation

jathu123 · Mar 17, 2016

Re: HSC 2016 2U Marathon

InteGrand said:
$\noindent Well done!$

$\noindent \textbf{NEW QUESTION}$

$\noindent Let $a$ be a positive constant. Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(a^{a^x}\right)$.$

$$ let $ y=a^{a^{x}}, u=a^{x} \\ y=a^{u} \\ log(y) =ulog(a)\\ \frac{1}{y}\frac{dy}{du} = log(a) \\ \frac{dy}{du} = a^{a^{x}}log(a) \\ \\ log(u)=xlog(a)\\ \frac{1}{u}\frac{du}{dx}= log(a)\\ \frac{du}{dx}=a^{x}log(a) \\ \\ \therefore \frac{dy}{dx} = a^{a^{x}}log(a)\cdot a^{x}log(a) \\ = a^{a^{x}+x}\cdot log^{2}(a)$

lol I think I did a really long method

Paradoxica · Mar 17, 2016

Re: This question is 2U level difficulty, right?

Paradoxica said:
$$\noindent The line $y = bx + c$ intersects the parabola $y =ax^2$ in two distinct points $P$ and $Q.\\$a) Find the points of intersection in terms of $a,b,c. \\ $b) Find the point on the parabola $R$ whose $x$ value is the average of the points of intersection.$\\ $c) By using the trapezoidal rule with a single strip on the points $P,Q$, and then two strips on the points $P,Q,R$, determine the area of $\Delta PQR. \\ $d) Find the area of the parabolic segment bound by the chord $PQ$ and the parabola.$\\$e) Hence prove that the area of the parabolic segment is $\frac{4}{3}$ times the area of the inscribed triangle $PQR.$

Bump

Nailgun · Mar 17, 2016

Re: This question is 2U level difficulty, right?

Paradoxica said:
Bump

$Okay, I've given it another attempt. I think I've gotten a,b,c but I'm struggling with d and e$
$Let\quad b^{ 2 }+4ac=z\\ a)\quad P\left( \frac { b-\sqrt { z } }{ 2a } ,\frac { { b }^{ 2 }-b\sqrt { z } }{ 2a } +c \right) \quad and\quad Q\left( \frac { b+\sqrt { z } }{ 2a } ,\frac { { b }^{ 2 }+b\sqrt { z } }{ 2a } +c \right) \\ b)\quad R\left( \frac { b }{ 2a } ,\frac { { b }^{ 2 } }{ 4a } \right) \\ c)\quad { A }_{ \Delta PQR }=\frac { { b }^{ 2 }\sqrt { z } }{ 8a^{ 2 } }$

$Is\quad d)\quad \int _{ P }^{ Q }{ bx+c-a{ x }^{ 2 } } dx?$

Paradoxica · Mar 17, 2016

Re: This question is 2U level difficulty, right?

Nailgun said:
$Okay, I've given it another attempt. I think I've gotten a,b,c but I'm struggling with d and e$
$Let\quad b^{ 2 }+4ac=z\\ a)\quad P\left( \frac { b-\sqrt { z } }{ 2a } ,\frac { { b }^{ 2 }-b\sqrt { z } }{ 2a } +c \right) \quad and\quad Q\left( \frac { b+\sqrt { z } }{ 2a } ,\frac { { b }^{ 2 }+b\sqrt { z } }{ 2a } +c \right) \\ b)\quad R\left( \frac { b }{ 2a } ,\frac { { b }^{ 2 } }{ 4a } \right) \\ c)\quad { A }_{ \Delta PQR }=\frac { { b }^{ 2 }\sqrt { z } }{ 8a^{ 2 } }$

$Is\quad d)\quad \int _{ P }^{ Q }{ bx+c-a{ x }^{ 2 } } dx?$

Indeed

leehuan · Mar 18, 2016

Re: HSC 2016 2U Marathon

$\\$Use your knowledge of the basis of Simpson's Rule and integration to prove the formula$ \\ \int_{a}^{b}{f\left(x\right)\,dx}\approx\frac{b-a}{6}\left[f\left(a\right)+4f \left(\frac{a+b}{2}\right) + f\left(b\right)\right]$

The proof is soundly difficult, so hints will be provided if left unanswered.

Drsoccerball · Mar 18, 2016

Re: HSC 2016 2U Marathon

leehuan said:
$\\$Use your knowledge of the basis of Simpson's Rule and integration to prove the formula$ \\ \int_{a}^{b}{f\left(x\right)\,dx}=\frac{b-a}{6}\left[f\left(a\right)+4f \left(\frac{a+b}{2}\right) + f\left(b\right)\right]$

The proof is soundly difficult, so hints will be provided if left unanswered.

Slight correction the '=' sign is supposed to be an approximation sign as the RHS is just another way of writing Simpson's rule.

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