$A$ tank with a rectangular base and rectangular sides, open at the top is made. The depth of the tank is $4 \text{ m}$ and its volume is $36 \text{ m}^3$. The cost of the base material is $\text{Rs. } 100 \text{ per m}^2$ and the cost of the side material is $\text{Rs. } 50 \text{ per m}^2$. Find the minimum cost of the tank.

Let $A$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is ..................

The minimum value of $2x^2+x-1$ is:

$A$ wire of length $28 \, m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ such that $\frac{d^2 f}{d x^2}(x) > 0$ for all $x \in (-1, 1)$. For $f \in S$,let $X_f$ be the number of points $x \in (-1, 1)$ for which $f(x) = x$. Then which of the following statements is(are) true?
$(A)$ There exists a function $f \in S$ such that $X_f = 0$
$(B)$ For every function $f \in S$,we have $X_f \leq 2$
$(C)$ There exists a function $f \in S$ such that $X_f = 2$
$(D)$ There does $NOT$ exist any function $f$ in $S$ such that $X_f = 1$

If $a^2 x^4 + b^2 y^4 = c^6$,then the maximum value of $xy$ is: