$A$ metal wire $108 \ m$ long is bent to form a rectangle. If the area of the rectangle is maximum,then its dimensions are

If $\overline{PQ}$ and $\overline{PR}$ are two sides of $\Delta PQR$,for what value of the angle between $\overline{PQ}$ and $\overline{PR}$ is the area of the triangle maximum?

$A$ curve with equation of the form $y = ax^4 + bx^3 + cx + d$ has zero gradient at the point $(0, 1)$ and also touches the $x$-axis at the point $(-1, 0)$. Then the values of $x$ for which the curve has a negative gradient are:

$A$ wire of length $22 \; m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then,the length of the side of the equilateral triangle,so that the combined area of the square and the equilateral triangle is minimum,is

If $p$ and $q$ are respectively the global maximum and global minimum of the function $f(x) = x^2 e^{2x}$ on the interval $[-2, 2]$,then $p e^{-4} + q e^4 =$

Let a rectangle $ABCD$ of sides $2$ and $4$ be inscribed in another rectangle $PQRS$ such that the vertices of the rectangle $ABCD$ lie on the sides of the rectangle $PQRS$. Let $a$ and $b$ be the sides of the rectangle $PQRS$ when its area is maximum. Then $(a+b)^2$ is equal to :