HSC 2014 MX2 Marathon (archive) (3 Viewers)

Trebla · Mar 19, 2014

Post any questions within the scope and level of Mathematics Extension 2. Once a question is posted, it needs to be answered before the next question is raised.

This thread is mainly targeting Q1-15 difficulty in the HSC. Any questions of Q16 difficulty or harder should be posted in this thread:
http://community.boredofstudies.org...4872/hsc-2014-4u-marathon-advanced-level.html

I'll get the ball rolling:

$\noindent$For what values of $k$ does $\dfrac{x^2}{4-k}+\dfrac{y^2}{k-6}=1$ describe a hyperbola?$$

Sy123 · Mar 19, 2014

Re: HSC 2014 4U Marathon

$\\ $Sketch the lines$ \ \ y=4-k, \ \ y=k-6 \ \ $the part of the plane where 1 function is positive and the other negative, the curve is a hyperbola, rejecting function values where both functions are positive or negative, hence$ \\ \\ k<4, \ \ k>6$

Parvee · Mar 20, 2014

Re: HSC 2014 4U Marathon

$\int \frac{dx}{a-\sqrt{x}}$

braintic · Mar 20, 2014

Re: HSC 2014 4U Marathon

Didn't ANYONE read OP's instruction?

"Once a question is posted, it needs to be answered before the next question is raised."

Sy123 · Mar 20, 2014

Re: HSC 2014 4U Marathon

fixed

Resource · Mar 20, 2014

Re: HSC 2014 4U Marathon

Can I just ask what program you use to post maths questions/answers Sy123?

Trebla · Mar 20, 2014

Re: HSC 2014 4U Marathon

Sy123 said:
$\\ $Sketch the lines$ \ \ y=4-k, \ \ y=k-6 \ \ $the part of the plane where 1 function is positive and the other negative, the curve is a hyperbola, rejecting function values where both functions are positive or negative, hence$ \\ \\ k<4, \ \ k>6$

Are you sure about k > 6?

Sy123 · Mar 20, 2014

Re: HSC 2014 4U Marathon

Resource said:
Can I just ask what program you use to post maths questions/answers Sy123?

Its built into the site, its called LaTeX. To use it you use tags like [.tex] and [./tex] (without the dots). So

[.tex] \frac{5}{3} [./tex]

Becomes

$\frac{5}{3}$

To get a good grasp of the latex language, you can use the buttons here
And then copy paste the generated code into tex tags here.

Trebla said:
Are you sure about k > 6?

I don't see anything wrong, and geogebra seems to say its correct.

Trebla · Mar 20, 2014

Re: HSC 2014 4U Marathon

Sy123 said:
I don't see anything wrong, and geogebra seems to say its correct.

I'm assuming you're taking the general equation of the hyperbola as

$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1$

If k > 6 then we cannot get this equation

Sy123 · Mar 20, 2014

Re: HSC 2014 4U Marathon

Trebla said:
I'm assuming you're taking the general equation of the hyperbola as

$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1$

If k > 6 then we cannot get this equation

Rather

$\\ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \\ \\ $or$ \\ \\ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$

Which are both hyperbola

Trebla · Mar 20, 2014

Re: HSC 2014 4U Marathon

Sy123 said:
Rather

$\\ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \\ \\ $or$ \\ \\ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$

Which are both hyperbola

Right. Forgot about that.

Sy123 · Mar 20, 2014

Re: HSC 2014 4U Marathon

Parvee said:
$\int \frac{dx}{a-\sqrt{x}}$

Bump

hit patel · Mar 21, 2014

Re: HSC 2014 4U Marathon

Parvee said:
$\int \frac{dx}{a-\sqrt{x}}$

Let x= a^2sin^4 (theta) you get ln (sec theta)-(sin ^2(theta))/2. then use the right angle triangle to find x values and sub them in for theta. Dont know how to use latex sorry.

dunjaaa · Mar 21, 2014

Re: HSC 2014 4U Marathon

Screen shot 2014-03-21 at 10.26.18 PM.png

Sy123 · Mar 22, 2014

Re: HSC 2014 4U Marathon

$\\ $The polynomial$ \ x^3+ax^2+bx+c \ $is such that$ \ c < 0 \\ \\ $Explain why only 1 or all 3 of the roots are positive$$

dunjaaa · Mar 23, 2014

Re: HSC 2014 4U Marathon

Product of roots > 0 as c<0
Case #1: all 3 roots are real and positive (trivial case)
Case #2: 2 of the roots are a complex conjugate pair and the other being real and positive (assuming a,b,c are integer coefficients) Note* when you multiply a complex conjugate pair, it will always be positive "process of real-ising", hence the real root must be positive in order for this condition to be satisfied

dunjaaa · Mar 23, 2014

Re: HSC 2014 4U Marathon

Next Question:

Screen shot 2014-03-23 at 3.42.18 AM.png

Sy123 · Mar 23, 2014

Re: HSC 2014 4U Marathon

dunjaaa said:
Product of roots > 0 as c<0
Case #1: all 3 roots are real and positive (trivial case)
Case #2: 2 of the roots are a complex conjugate pair and the other being real and positive (assuming a,b,c are integer coefficients) Note* when you multiply a complex conjugate pair, it will always be positive "process of real-ising", hence the real root must be positive in order for this condition to be satisfied

But you can have 2 negative roots and 1 positive root resulting in the product of roots being positive, yet not all roots are positive.

dunjaaa · Mar 23, 2014

Re: HSC 2014 4U Marathon

woops forgot about that haha

Axio · Mar 23, 2014

Re: HSC 2014 4U Marathon

View attachment Q.pdf
The altitudes AP and BQ in an acute-angled triangle ABC meet at H. AP produced cuts the circle through A, B and C at K.
Prove that HP = PK.

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