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Let \( \displaystyle I(m) = \int_0^\pi \ln (1 - 2m \cos x + m^2) \, dx \). Find the values of \(\dfrac{I(9)}{I(3)} \) and \(I(81) \).

Let \(A\) be a region in the \(xy\)-plane given by \( A=\{(x,y):x=u+v,y=v,u^2+v^2<=1 \} . \)

Derive the length of the longest ...

Consider \(n>1\) lotus leaves placed around a circle. A frog jumps from one leaf to another in the following manner. It starts from some selected leaf. From there it skips ...

Consider a triangle \(PQR\) in int the cartesian plane. Let \(A\) be a point lying on triangle \(PQR\) or inside the region enclosed by it . Prove that, for any function ...

Let \( f: [0,\infty] \rightarrow \mathbb R\) be a non-decreasing continuous function. Show then that the inequality.

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