The product of all solutions of the equation $e ^{5\left(\log _{ e } x \right)^2+3}= x ^8, x >0$, is :

Number of natural solutions of the equation $xyz = 2^5 \times 3^2 \times  5^2$ is equal to

Suppose the quadratic polynomial $p(x)=a x^2+b x+c$ has positive coefficient $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$ then what could be the possible value of $\alpha+\beta+\alpha \beta$ if $0 \leq \alpha+\beta+\alpha \beta \leq 8$

If $x$ be real, then the maximum value of $5 + 4x - 4{x^2}$ will be equal to

If $x$ is real and satisfies $x + 2 > \sqrt {x + 4} ,$ then

The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$, is