Had Ajita scored $10$ more marks in her mathematics test out of $30$ marks,$9$ times these marks would have been the square of her actual marks. How many marks did she get in the test?

Solve the following quadratic equation using the quadratic formula,if the equation has a solution in $R$: $48m^{2} - 13m - 1 = 0$.

$A$ two-digit number is such that the product of its digits is $6$. When $9$ is subtracted from the number,the resulting number is the number obtained by interchanging the digits. Find the original number.

Comparing the quadratic equation $5x^{2}-4x-1=0$ with the standard form $ax^{2}+bx+c=0$,we get $b=\ldots$

Find the roots of the following quadratic equation using the quadratic formula,if they exist: $\frac{1}{x+1} + \frac{2}{x+2} = \frac{4}{x+4}$; $(x \neq -1, -2, -4)$

The lengths of the sides forming the right angle of a right-angled triangle are $x \, m$ and $(x+2) \, m$. If the area of the triangle is $84 \, m^2$,find the lengths of the sides which form the right angle.