The line $x + y = 0$ touches the curve $y^2 = ax^3 + b$ at $(1, -1)$ then values of $a$ and $b$ respectively are ...........

The function $f(x) = x(x + 3)e^{-\left(\frac{1}{2}\right)x}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$, then $c =$

A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\ \text{cm}^3/\text{min}$. When the thickness of ice is $5$ cm, the rate at which the thickness of ice decreases is...

If the line $ax + by + 5 = 0$ is a normal to the curve $xy = 1$ then .....