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\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-06_652_716_258_230} The diagram shows the shaded region R bounded by the curve \(\mathrm { y } = \sqrt { 3 \mathrm { x } + 4 }\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \(( k , 0 )\) and \(( 4,4 )\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.

1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac { \mathrm { a } + \mathrm { bk } } { \mathrm { c } + \mathrm { dk } }\), where \(a , b , c\) and \(d\) are integers to be determined.
2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac { 3 } { 2 }\).