The number of integers $n$ for which $3 x^3-25 x+n=0$ has three real roots is
$1$
$25$
$55$
infinite
The value of $x$ in the given equation ${4^x} - {3^{x\,\; - \;\frac{1}{2}}} = {3^{x + \frac{1}{2}}} - {2^{2x - 1}}$is
Suppose $a, b, c$ are positive integers such that $2^a+4^b+8^c=328$. Then, $\frac{a+2 b+3 c}{a b c}$ is equal to
If $x$ is real, then the value of ${x^2} - 6x + 13$ will not be less than
The sum of the cubes of all the roots of the equation $x^{4}-3 x^{3}-2 x^{2}+3 x+1=10$ is
The number of solutions for the equation ${x^2} - 5|x| + \,6 = 0$ is