Maths

$\frac{x}{4}=cost-sint...\left(1\right)$

$\frac{y}{5}=cost+sint$ . . . . (2)

(1)² + (2)² gives

${\left(\frac{x}{4}\right)}^2+{\left(\frac{y}{5}\right)}^2=2$

$xdy+ydx=\frac{xydx}{\sqrt[]{1-x^2}}$

$\frac{d\left(xy\right)}{xy}=\frac{dx}{\sqrt[]{1-x^2}}$

Integrate both sides we get

logexy = sin1 x + C

$y^2=x,a=\frac{1}{4}$

$x^2=-y,b=\frac{1}{4}$

$A=\frac{16\left|ab\right|}{3}=\frac{1}{3}$

$\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=0$

$\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 6\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill \alpha +1\hfill & \hfill \beta -1\hfill & \hfill 1\hfill \end{array}\right|=0$

$8\beta -24=0\Rightarrow \beta =3$

$\left|\overrightarrow{c}\right|=\sqrt[]{6}$

${\left(\alpha +1\right)}^2+{\left(\beta -1\right)}^2+1=6\Rightarrow {\left(\alpha +1\right)}^2=1$

a = 1 = 1, 1 $\Rightarrow \alpha =-2,0$ $$

+ = 1, 3

Variance of 2001, 2003, 2006, 2007, 2009, 2010 is same as variance of 1, 3, 6, 7, 9, 10

mean =  $\frac{1+3+6+7+9+10}{6}=6$

var (x) =  $\frac{1^2+3^2+6^2+7^2+9^2+10^2}{6}?36$

var (x) = 10

  $\sim \left(p\to q\right)=p\wedge \sim q$

A quadrilateral is a square and it is not a rhombus

$f\left(x\right)=[\begin{array}{l}-\frac{1}{x}x\in \left(-\infty ,-1\right)\\ a{x}^2+bx\in \left(-1,1\right)\\ \frac{1}{x}x\in \left[1,\infty \right)\end{array}$

cont at x = 1, a + b = 1

diff at x = 1, 2a = 1 $\Rightarrow a=-\frac{1}{2}\Rightarrow b=\frac{3}{2}$

area of quad PACB = LR = 8

$P=1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+...+\infty$

$\frac{P}{3}=\frac{1}{3}=\frac{2}{3^2}+\frac{6}{3^3}+\frac{10}{3^4}+...+\infty$

$\frac{2P}{3}=1+\frac{1}{3}+\left[\frac{4}{3^2}+\frac{4}{3^3}+\frac{4}{3^4}+...+\infty \right]$

$\frac{2P}{3}=\frac{\frac{4}{3}}{1-\frac{1}{3}}\Rightarrow P=3$

$f\left(x\right)=x^2+x+\left(\frac{\lambda +1}{\lambda }\right)$

$f\left(1\right)<0\Rightarrow 2+\left(\frac{\lambda +1}{\lambda }\right)<0$

$\frac{3\lambda +1}{\lambda }<0\Rightarrow \lambda \in \left(-\frac{1}{3},0\right)$