The equation of the tangent to the curve $y=\mathrm{be}^{-x / \mathrm{a}}$ at the point where it crosses the Y axis 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 ₹ $\left(\frac{x}{5}+193\right)$. The maximum profit in ₹ __________ is

The equation of the tangent to the curve $\left(1+x^2\right) y=2-x$, where it crosses the X -axis, is

If $y=\alpha \log x+\beta x^3-x$ has extreme values at $x=-1$ and $x=1$, then $\alpha$ and $\beta$ are respectively