The uniform lamina $$\(A B C D\)$$ is a square of side $$\(6 a\)$$. The template $$\(T\)$$, shown shaded in Figure 1, is formed by removing the right-angled triangle $$\(E F G\)$$ and the circle, centre $$\(H\)$$ and radius $$\(a\)$$, from the square lamina. Triangle $$\(E F G\)$$ has $$\(E F=E G=4 a\)$$, with $$\(E F\)$$ parallel to $$\(A B\)$$ and $$\(E G\)$$ parallel to $$\(A D\)$$. The distance between $$\(A B\)$$ and $$\(E F\)$$ is $$\(a\)$$ and the distance between $$\(A D\)$$ and $$\(E G\)$$ is $$\(a\)$$. The point $$\(H\)$$ lies on $$\(A C\)$$ and the distance of $$\(H\)$$ from $$\(B C\)$$ is $$\(2 a\)$$. (a) Show that the centre of mass of $$\(T\)$$ is a distance $$\(\frac{4(67-3 \pi)}{3(28-\pi)} a\)$$ from $$\(A D\)$$. (5) The template $$\(T\)$$ is suspended from the ceiling by two light inextensible vertical strings. One string is attached to $$\(T\)$$ at $$\(A\)$$ and the other string is attached to $$\(T\)$$ at $$\(B\)$$ so that $$\(T\)$$ hangs in equilibrium with $$\(A B\)$$ horizontal. The weight of $$\(T\)$$ is $$\(W\)$$. The tension in the string attached to $$\(T\)$$ at $$\(B\)$$ is $$\(k W\)$$, where $$\(k\)$$ is a constant. (b) Find the value of $$\(k\)$$, giving your answer to 2 decimal places. (3)

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