For the given pair of quadratic equations, solve the equation and find the relation between the variable $x$ and $y$.

$$\begin{gathered} \mathrm{I}. x^2-23x+132=0 \\ \mathrm{II}. y^2-25y+156=0 \end{gathered}$$

I.$(7{)}^{450}>(9{)}^{300}$

II .${\left(2\right)}^{900}>{\left(7\right)}^{450}$

Consider the following inequalities:

$$\begin{gathered} 1. \frac{a^2-b^2}{a^2+b^2}>\frac{a-b}{a+b}, \text{where} a>b>0 \\ 2. \frac{a^3+b^3}{a^2+b^2}>\frac{a^2+b^2}{a+b}, \text{where} a>b>0 \end{gathered}$$

Which of the above is/ are correct?

Quantity $1$: A pipe alone can fill a cistern in $60$ minutes. But due to leakage pipe filled only $80\%$ of the cistern in $60$ minutes. $"X"$ is the capacity of the cistern in litres and due to leakage $60$ litres were leaked out in $1$ hour.

Quantity $2$: $250$ litres.

Two equations are given, you have to solve the both the equations:

$\left(i\right) X^2=196$

$\left(ii\right) Y^2+2y-48=0$

The speed of a boat in still water and the speed of the current is in ratio $5:1$. If the difference between distances covered by the boat in going $2 \mathrm{hours}$ upstream and $2 \mathrm{hours}$  downstream is $8 \mathrm{km}$.

I. Quantity I - Speed of boat in still water

II. Quantity II -A cyclist goes $30 \mathrm{km}$ in $3 \mathrm{hours}$ 

Find the relation between these two quantities.

Consider the following statements:
(i) When $0<\mathrm{x}<1$, then $\frac{1}{1+\mathrm{x}}<1-\mathrm{x}+{\mathrm{x}}^2$.

(ii) When $0<\mathrm{x}<1$, then $\frac{1}{1+\mathrm{x}}>1-\mathrm{x}+{\mathrm{x}}^2$.

(iii) When $-1<\mathrm{x}<0$, then $\frac{1}{1+\mathrm{x}}<1-\mathrm{x}+{\mathrm{x}}^2$.

(iv) When $-1<\mathrm{x}<0$, then $\frac{1}{1+\mathrm{x}}>1-\mathrm{x}+{\mathrm{x}}^2$.

Then the correct statements are

I. Simple interest for $3 \mathrm{years}$  at $11\%$ rate on $\mathrm{x}$.

II. Compound interest for $3 \mathrm{years}$  at $10\%$ rate on $\mathrm{x}$.

Find the relation between these two quantities.

$\mathrm{Quantity} \mathrm{I}=18 {\mathrm{x}}^3{\mathrm{y}}^3, \mathrm{Quantity} \mathrm{II}=12 {\mathrm{x}}^4{\mathrm{y}}^4, \mathrm{if} \mathrm{x}>0 \& \mathrm{y}<0$

Find the relation between these two quantities.