Challengers*

Consider the following linear equation in two variables $\sqrt{mn}x=mny+5$. After finding the values of $a, b$ and $c$. Find the value of $a^2+b$.

If thee times the larger of the two numbers is divided by the smaller one, we get $4$ as quotient and $3$ as remainder. Also, if seven times the smaller number is divided by the larger one, we get $5$ as quotient and $1$ as remainder. Both the equations so obtained is

The area of a rectangle gets reduced by $9 \mathrm{sq} \mathrm{units}$, if its length is reduced by $5$ units and breadth is increased by $3$ units. Taking the length and breadth of the rectangle as $x$ and $y$ respectively, write the equation. If we increase the length by $3$ units and breadth by $2$ units, then the area is incraesed by $67 \mathrm{sq} \mathrm{units}$. Another equation so obtained.

A graph is shown below

If we rotate a line segment $AB$ in the anti-clockwise direction with $A$ being fixed, then the area covered by this line segment is

The ratio of girls and boys in a class is $1:3$. An equation between the students of class and boys is