Search results

Re: MX2 Integration Marathon \displaystyle \int\frac{1}{(x+\alpha)^{1+\frac{1}{n}}\cdot (x+\beta)^{1-\frac{1}{n}}}dx = \int \frac{1}{\left(\frac{x+\alpha}{x+\beta}\right)^{1+\frac{1}{n}}}\cdot \frac{1}{(x+\beta)^2}dx $Now \; let \; $\displaystyle \left(\frac{x+\alpha}{x+\beta}\right) =...

Re: MX2 Integration Marathon $How\ can\; we\; prove \; \displaystyle%20\int_{0}^{\frac{\pi}{4}}\tan^{-1}\left(\frac{\sqrt{2}\cos3%20\phi}{\left(2\cos%202%20\phi+%203\right)\sqrt{\cos%202%20\phi}}\right)d\phi%20=%200$

Re: HSC 2014 4U Marathon - Advanced Level For (1) part:: \\ $For$ \ p_4(x) = \sum_{k=0}^4 \frac{x^k}{k!} = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24} = \frac{1}{24}\left(x^4+4x^3+12x^2+24x+24\right) $Now\;\; \displaystyle \left(x^4+4x^3+12x^2+24x+24\right) =...

Re: HSC 2014 4U Marathon Solution :: $Here $ \ \ (x-2)(x-5)\cdot (x-3)(x-4) = 24\Rightarrow (x^2-7x+10)\cdot(x^2-7x+12) = 24 $Let$ \ \ (x^2-7x) = y\;, $Then \; equation \; is \;$ (y+10)\cdot(y+12) = 24 \Rightarrow y^2+22y+120 = 24\Rightarrow y^2+22y+96 = 0\Rightarrow y = -16\; y = -6...

$Let $z_{1}, z_{2}, z_{3}$ be complex numbers of unit modulus such that$ $\left|z_{1}-z_{2}\right|^2+\left|z_{1}-z_{3}\right|^2 = 4.$ $Then $\left|z_{1}+z_{3}\right|$ is equal to$

$If $'a'$ is a complex number such that $\mid a \mid = 1$ .Then the values of $a$ so$ $that the equation $az^2+z+1=0$ has one purely imaginary root$

$(1)Total number of integer ordered pair,s $(n,k)$ which satisfy $\displaystyle \binom{n}{k} = 120.$ \hspace{-20}$(2)Total number of integer ordered pair,s $(n,k)$ which satisfy $\displaystyle \binom{n}{k} = 2002.$ \hspace{-20}$(3)Total number of integer ordered pair,s $(n,k)$...

Re: MX2 Integration Marathon \displaystyle \bf{\int\frac{1}{(x^2-x+1)\sqrt{x^2+x+1}}dx} \displaystyle \bf{ \int\sqrt{\frac{1+\tan x}{\csc^2 x+\sqrt{\sec x}}}dx}

Re: MX2 Integration Marathon \displaystyle \bf{=\int\frac{\sin^2 x}{\cos^2 x \cdot \left(1+\sec^2 x\cdot \frac{1}{\tan^2 x}\right)}dx} \displaystyle \bf{ = \int\frac{\tan^{4}x}{\tan^2 x+1+\tan^2 x}dx = \int\frac{\tan^4 x}{2\tan^2 x+1}dx} \displaystyle \bf{Now\;\; Let\;\; \tan x = t...

Calculation of no. of real values of x in \{x\} = \{x^2\} = \{x^3\} where \{x\} = fractional part of x

To pdang i did not understand how can i calculate after that would you like to give me full solution. Thanks

Thanks jyu

\displaystyle \sum_{r=1}^{\infty}\frac{r^3+(r+1)^3}{(r^4+r^2+1).(r^2+r)} =

Value of \displaystyle i^{i^{i^{i................\infty}}} where i = \sqrt{-1} is a Complex no.

Question:: If z = \cos \theta + i.\sin \theta. Then prove that (1) \displaystyle \arg(z^2+\bar{z}) = \frac{\arg(z)}{2}\;\;, If \displaystyle 0 \leq \theta <\frac{\pi}{3}\;\;,\pi < \theta <\frac{5\pi}{3} (2) \displaystyle \arg(z^2+\bar{z}) = \pi+\frac{\arg(z)}{2}\;\;, If...

Thanks braintic

\displaystyle \sin \left(\frac{\pi}{7}\right).\sin \left(\frac{2\pi}{7}\right).\sin \left(\frac{3\pi}{7}\right) =

A parabola y^2=4x is given. Also family of lines 2ax+by+2c=0 is given where a,b,c are in Arithmetic Progression (1). The common point from which all the members of the given family of lines passes through is: a) (1,4) b) (1/2, -2) c) (1, -4) d) none 2. combined...

Thanks mathman got it

for the parabola x^2+2y=4x-3, the length of the double ordinate conjugate to the directrix is