Each question contains a statement followed by Quantity $\mathrm{I}, \mathrm{II} \text{and} \mathrm{III}$. Read the information clearly and answer your questions accordingly.

The options represent the relations between these three quantities.

A) $>$

B) $<$

C) $=$

D) $\le$

E) $\ge$

For example:

Quantity$\mathrm{I} = 200$

Quantity $\mathrm{II}=300$

Quantity $\mathrm{III}=100$

Hence, Quantity $\mathrm{I} <$ Quantity $\mathrm{II} >$ Quantity $\mathrm{III}$

a) A, B

b) B, C

c) B, A

d) E, B

e) B, D

Answer is option: c)

Quantity $\mathrm{I}$: If the radius of circle to cylinder is $2: 1$ and the height of the cylinder is $7 \mathrm{cm}$and the volume of the cylinder is $1078 {\mathrm{cm}}^2$ , then what is the circumference of the circle?

Quantity$\mathrm{II}$: If the ratio of the length to breadth of the rectangle is $4: 3$and the side of the square is $2 \mathrm{cm}$ more than the length of the rectangle whose perimeter is $70 \mathrm{cm}$, then what is the perimeter of the square?

Quantity $\mathrm{III}$: A, B and C started the business with the investment in the ratio of $4: 5: 3$. After $4$ months A added $₹3000$ for his initial investment and the investment period of A, B and C is $8$ months, $12$ months and $10$ months respectively. If the profit of the business is $₹670$ and the profit share of C is$₹150$, then what is the $50\%$ of the profit share of A?

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.