Arun Sharma Solutions for Exercise 3: Level of Difficulty

If all the three roots of the equation ${\mathrm{x}}^3—12{\mathrm{x}}^2 + \mathrm{px}—42=0$ are unequal and prime. Then $\mathrm{p}=?$ 

If $\mathrm{f}\left(\mathrm{x}\right)=(\mathrm{b}-1) {\mathrm{x}}^2+(\mathrm{c}+\mathrm{d})\mathrm{x}+\mathrm{e}$ and $\mathrm{a}:\mathrm{b}=1:2, \mathrm{b}:\mathrm{c}=2:3, \mathrm{c}:\mathrm{d}=3:4, \mathrm{d}:\mathrm{e}=4:5$. Then find the square of the difference of the roots of the equation $\mathrm{f}\left(\mathrm{x}\right)=0$.

The value of $\mathrm{p}$ for which the sum of the square of the roots of the equation ${\mathrm{x}}^2-(\mathrm{p}-3)\mathrm{x}+(\mathrm{p}-4)=0$ is minimum is:

If $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2.\mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^3-2^{\mathrm{x}}$, where $\mathrm{x}$ is a positive integer then for how many values of $\mathrm{x}$ is $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right)$.

The curve, shown below can possibly represent which of the following equation? (Assume the curve does not exist at $\mathrm{x}=0$).

If $\mathrm{f}\left(\mathrm{x}\right)=\max ({\mathrm{x}}^2-4, 2\mathrm{x}+4)$, then find the number of solutions of the equation $\mathrm{f}\left(\mathrm{x}\right)=\frac{5}{2}$.

Find the number of real solutions of the equation $\left|\left|\mathrm{x}\right|-1\right|={\mathrm{e}}^{\mathrm{x}}$.

Find the number of real solutions of the equation $\left|\left|\left|\mathrm{x}\right|-1\right|-2\right|={\left(\frac{1}{2}\right)}^{\mathrm{x}}$.