The value of C for which Mean value Theorem holds for the function $\mathrm{f}(x)=\log _e x$ on the interval $[1,3]$ is

The shaded region in the following figure is the solution set of the inequations

The diagonals of a parallelogram $A B C D$ are along the lines $x+3 y=4$ and $6 x-2 y=7$. Then ABCD must be a

If $x=\sec \theta-\cos \theta, y=\sec ^{10} \theta-\cos ^{10} \theta$ and $\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2=k\left(y^2+4\right)$, then the value of $k$ is