## Is it possible to design a rectangular mango grove whose length is twice its breadth,

[question] Question. Is it possible to design a rectangular mango grove whose length is twice its breadth, and area is $800 \mathrm{~m}^{2}$ ? If so, find its length and breadth. [/question] [solution] Solution: Let x be the breadth and 2x be the length of the rectangle. $x \times 2 x=800$ $\Rightarrow 2 x^{2}=800$ $\Rightarrow x^{2}=400=(20)^{2}$ $\Rightarrow x=20$ Hence, the rectangle is possible and it has breadth $=20 \mathrm{~m}$ and length $=40 \mathrm{~m}$. [/solution]...

## A train travels 360 km at a uniform speed.

[question] Question. A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken I hour less for the same journey. Find the speed of the train. [/question] [solution] Solution: Let the speed of the train be x km/hr. We are given that 360 km distance is to be travelled at uniform speed of x km/hr. Time taken to cover the distance of $360 \mathrm{~km}=\frac{\mathbf{3 6 0}}{\mathbf{x}}$ hours. In case, the speed is increased by $5 \mathrm{~km} / \mathrm{hr}$, ...

## The difference of squares of two numbers is 180.

[question] Question. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers [/question] [solution] Solution: Let the larger and smaller number be x and y respectively. According to the given question, $x^{2}-y^{2}=180$ and $y^{2}=8 x$ $\Rightarrow x^{2}-8 x=180$ $\Rightarrow x^{2}-8 x-180=0$ $\Rightarrow x^{2}-18 x+10 x-180=0$ $\Rightarrow x(x-18)+10(x-18)=0$ $\Rightarrow(x-18)(x+10)=0$ $\Rightarrow x=18,-10$ However, ...

## Represent the following situations in the form of quadratic equations :

[question] Question. Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528 \mathrm{~m}^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. (ii) The product of two consecutive positive integers is 306. We need to find the integers. (iii)Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to...