The product of the digits of a two-digit number is $15$. If the digits of this number are interchanged,the new number so obtained is $18$ more than the original number. Find the original number.

In $\Delta ABC$,$m\angle B = 90^{\circ}$,$AB = 2x + 3$,$BC = x + 2$,and $AC = 3x - 1$. Find the value of $x$.

State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has at most two roots.
$(ii)$ If the coefficient of $x^{2}$ and the constant term of a quadratic equation have opposite signs,then the quadratic equation has real roots.

In the centre of a rectangular lawn of dimensions $50\, m \times 40\, m$,a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be $1184\, m^{2}$ [see $Fig.$]. Find the length and breadth of the pond. (in $m$)

For a two-digit number,if the sum of its digits is equal to the product of its digits,then the number is .............

The sum of two numbers is $15$. If the sum of their reciprocals is $\frac{3}{10}$,then find the numbers.