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Re: 2012 HSC MX2 Marathon \hspace{-16}\bf{\int%20\frac{(4x^5+5x^4)}{(x^5+x+1)^2}dx}

Re: 2012 HSC MX2 Marathon...

Thanks Fus Ro Dah for answer I have got (1) one It is diverge at x = \frac{3}{\sqrt{10}}\in \left(0,\frac{3}{2}\right)

\hspace{-16}$If%20both%20the%20equation%20$\bf{\begin{Bmatrix}%20\bf{x^2-ax+b=0}%20\\\\%20\bf{x^2-bx+a=0}%20\\\\%20\end{Bmatrix}}$\\\\%20have%20positive,Integral%20and%20Distinct%20Roots,%20Then%20$\bf{a}$%20and%20$\bf{b}$%20are

Re: 2012 HSC MX2 Marathon...

Sorry Moderator for typing wrongly in Latex editor for question (3) actually original question is (Typo) \hspace{-16}\bf{\mathbb{I}=\int_{2}^{3}\frac{1}{\ln(x)}dx}$\\\\ options (A) is less than 2 (B) is equal to 2 (C) lies in the interval...

\hspace{-16}\bf{(1)\;\;%20\int_{0}^{\frac{3}{2}}\frac{2x+\sqrt{9-x^2}}{x.\sqrt{81-9x^2}+x^2-9}dx}$\\\\\\%20$\bf{(2)\;\;%20}$%20If%20$\bf{a%3Cb}$,%20Then%20Max.%20value%20of%20$\bf{\mathbb{I}=\int%20\frac{3}{4-x-x^2}dx}$\\\\%20for%20all%20values%20of%20$\bf{a\;,b}$\\\\\\%20$\bf{(3)\;\;%20}$%20If%2...

Yes I didnot understand how you get a = b, Now I understand that things Thanks jyu.

Thanks for answer but i did not understand if circle and Rectangular hyperbola are symmetrical about y = x then how can i get a = b and (ii) doubt is how we can say that y(ax+by) = c represent Rectangular Hyperbola

Sorry Drongoski actual question is \displaystyle \int \frac{(1+x^{-2})^{\frac{1}{3}}}{x^2}dx

\displaystyle \int \frac{\left(1+x^{-2}\right)}{x^2}dx

I gave got the point that if hyperbola is tangent then there is only point of intersection but did not understand how can we get a = b Thanks

Thanks jyu but I did not understand the line .: d^2=2a/(a+b) and the hyperbola must be tangential to the circle which requires a=b .: d=1 Thanks

jyu how can you get the answer would you like to explain it to me

Thanks moderator got it

Let A_{1}(x_{1},y_{1})\;,A_{2}(x_{2},y_{2})\;,.......A_{n}(x_{n},y_{n})\; be the only point to satisfying the hyperbola y(ax+by)=c and x^2+y^2=d^2 If x_{r} = y_{r}\forall r\in \left\{1,2,3,.......,n\right\} and c=a. then d= options 1,2,4,none

Thanks moderator but I did not Understand the line w x u = v - w (ie: it lies on the same plane as v and w ) Thanks

\hspace{-16}$If%20$\bf{O}$%20is%20the%20orthocenter%20of%20a%20$\bf{\triangle%20ABC}$%20having%20sides%20$\bf{BC\;,CA\;,AB}$\\\\%20Where%20$\bf{BC%20=%20a\;\;,CA%20=%20b\;\;,AB%20=%20c}$.\;Then%20the%20ratio%20of%20radious%20of%20circle\\\\%20circumscribing%20the%20$\bf{\triangle%20BOC\;\;,\trian...

Yes Moderator you are saying Right

Thanks jyu got it