If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then $\sin \left(\frac{\pi}{4}+\theta\right)=$

The line MN whose equation is $x-y-2=0$ cuts the X -axis at M and co-ordinates of N are $(4,2)$. The line MN is rotated about M through $45^{\circ}$ in anticlockwise direction. The equation of the line MN in the new position is

The money invested in a company is compounded continuously. If $Rs\, 400$ invested today becomes ₹800 in 6 years, then at the end of 30 years, it will become (in ₹)

The least distance of the point $\mathrm{A}(10,7)$ from the circle $x^2+y^2-4 x-2 y-20=0$ is length of seg AM . If $\mathrm{MM}^{\prime}$ is the diameter of the circle, then the lengths of AM and $\mathrm{AM}^{\prime}$ are respectively ___________ , ____________units