Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of $$k$$ colours, such that the colour-pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $$k$$ that satisfies this requirement?

What is the maximum number of edges in an acyclic undirected graph with $$n$$ vertices?

What values of x, y and z satisfy the following system of linear equations? $$$\left[ {\matrix{ 1 & 2 & 3 \cr 1 & 3 & 4 \cr 2 & 3 & 3 \cr } } \right]\,\,\left[ {\matrix{ x \cr y \cr z \cr } } \right]\,\, = \,\left[ {\matrix{ 6 \cr 8 \cr {12} \cr } } \right]$$$

Let A, B, C, D be $$n\,\, \times \,\,n$$ matrices, each with non-zero determination. If ABCD = I, then $${B^{ - 1}}$$ is