The number of solution$(s)$ satisfying the equation $3x^2 - 2x^3 = \log_2 (x^2 + 1) - \log_2 x$ is

The ratio of the height of a cone of maximum volume inscribed in a sphere to its radius is

If $x=-2$ and $x=4$ are the extreme points of $y=x^3-\alpha x^2-\beta x+5$,then

Let $f(x) = \frac{\sin \pi x}{x^2}, x > 0$. Let $x_1 < x_2 < x_3 < \ldots < x_n < \ldots$ be all the points of local maximum of $f(x)$ and $y_1 < y_2 < y_3 < \ldots < y_n < \ldots$ be all the points of local minimum of $f(x)$. Which of the following statements are true?
$(1)$ $|x_n - y_n| > 1$ for every $n$
$(2)$ $x_1 < y_1$
$(3)$ $x_n \in (2n, 2n + \frac{1}{2})$ for every $n$
$(4)$ $x_{n+1} - x_n > 2$ for every $n$

The number of points of local maxima of the function $f(x) = x + \sin x$ is-

$A$ manufacturer sells $x$ items at a price of rupees $\left(6-\frac{x}{40}\right)$ each. The cost price of $x$ items is Rs. $\left(\frac{x}{5}+193\right)$. The maximum profit in Rs. is