G6.1 An $n$-sided convex polygon has 20 diagonals. Find $n$.
G6.2 Two dice are thrown. The probability of getting a total of $n$ is $\frac{k}{36}$. Find $k$.
G6.3 A man drives at $25 \mathrm{~km} / \mathrm{h}$ for 3 hours and then at $50 \mathrm{~km} / \mathrm{h}$ for 2 hours.
His average speed for the whole journey is $u \mathrm{~km} / \mathrm{h}$. Find $u$.

G6.4 If $a \Delta b=a b+1$ and $(2 \Delta a) \Delta 3=10$, find $a$.
$\begin{array}{l}\text { Number of diagonals }=C_{2}^{n}-n=\frac{n(n-1)}{2}-n=20 \\ n^{2}-3 n-40=0 \\ (n-8)(n+5)=0 \\ \Rightarrow n=8 \\ \text { Total }=8 \\ \text { Favourable outcomes }=\{(2,6),(3,5),(4,4),(5,3),(6,2)\} \\ \text { P(total }=8)=\frac{5}{36} \\ k=5 \\ u=\frac{25 \times 3+50 \times 2}{3+2}=35 \\ 2 \Delta a=2 a+1 \\ (2 \Delta a) \Delta 3=(2 a+1) \Delta 3=3(2 a+1)+1=10 \\ 6 a+4=10 \\ a=1\end{array}$