Ncert Solutions Maths class 11th

If the two lines are perpendicular then a = 2

D = 0       $\Rightarrow$    c = -3

(D = abc + 2fgh - af² – bg² - ch²)

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

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

(1)² + (2)² gives

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

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

A quadrilateral is a square and it is not a rhombus

area of quad PACB = LR = 8

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

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

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

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

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

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

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

$x^2+(y-1{)}^2=y^2$

$\Rightarrow x^2-2y+1+y^2=y^2\Rightarrow x^2=2y-1\Rightarrow x^2=2\left(y-\frac{1}{2}\right)$

Foci (0,1)

α=0, β=1;α+β=1

$9*9*3=81*3=243$

By property

If first and Last term of Andhra Pradesh and HP are equal (of nn  terms) then $a_r{H}_S=$  first *  x last term

If $r+s=n+1$ r+s=n+1

Single digit factor of $12\to \mathrm{1,2},\mathrm{3,4},6$

  $(x,y,z)$  where , $xyz=12$

$(\mathrm{6,2},1)(\mathrm{4,3},1)(\mathrm{3,2},2)$

$\begin{array}{ccc}\downarrow & \downarrow & \downarrow \\ \text{6Ways}& \text{6 Ways}& \text{3Ways}\end{array}$

 Total 15 Ways