The displacement $x$ of a particle at time $t$ is given by $x = t^4 - kt^3$. If the velocity of the particle is maximum at $t = 2$,then $k = $ ..........

The function $f(x) = x^{-x}, (x \in R)$ attains a maximum value at $x =$

If $x$ is real and $\alpha, \beta$ are maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$ respectively,then $\alpha+\beta=$

$A$ box open from the top is made from a rectangular sheet of dimensions $a \times b$ by cutting squares of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum,then $x$ is equal to:

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ $(a>0)$ attains its maximum and minimum at $p$ and $q$ respectively and $p^2=q$. Then,$a=$

The perimeter of a sector is a constant. If its area is to be maximum,then the sectorial angle is