The perimeter of a bigger square is $12\,m$ more than the perimeter of a smaller square. Three times the area of the smaller square is $11\,m^2$ more than the area of the bigger square. Find the length of the side of the bigger square.

Out of the following triplets,$\ldots \ldots \ldots \ldots$ is not a Pythagorean triplet.

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.

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.

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

Write whether the following statements are true or false. Justify your answers.
$(i)$ If the coefficient of $x^{2}$ and the constant term have the same sign and if the coefficient of $x$ term is zero,then the quadratic equation has no real roots.
$(ii)$ Every quadratic equation has at least two roots.