Arun Sharma Solutions for Exercise 1: Level of Difficulty

Find the number of solutions of $a^3+2^{a+1}=a^4$, given that $n$ is a natural number less than $100$.

The number of positive integral values of $\mathrm{x}$ that satisfy ${\mathrm{x}}^3-32\mathrm{x}-5{\mathrm{x}}^2+64\le 0$ is/are:

Find the positive integer value of $x$ that satisfies the equation $x^3-32x-5x^2+64=0$.

If $a$, $b$, $c$ are positive  integers, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{28}{72}$, how many sets of $(a$, $b$, $c)$ exist?

For how many real value of $x$ is $3^{x^2-2x-1}=9$?

For how many real values of $k$, the absolute value of the difference between the roots of the equation $x^2+kx+15=0$ is $2$.

If $f\left(x\right)=x^2-8x+12$, then for how many integer values of $x$ is $f\left(x\right)\le 0$.

If $f\left(x\right)=x^2-8x+12$, then for what value of $x$, does $f\left(x\right)$ attains its minimum value.