SI. 1 If $a=-1+2-3+4-5+6-\ldots+100$, find $a$.
SI. 2 The sum of the first $b$ positive odd numbers is $2 a$. Find $b$.
SI. 3 A bag contains $b$ white balls and 3 black balls. Two balls are drawn from the bag at random.
If the probability of getting 2 balls of different colours is $\frac{c}{13}$, find $c$.

SI. 4 If the lines $c x+10 y=4$ and $d x-y=5$ are perpendicular to each other, find $d$.
$$
\begin{array}{l}
a=(-1+2-3+4)+(-5+6-7+8)+\ldots+(-97+98-99+100) \\
\quad=2+2+\ldots+2(25 \text { terms })=50 \\
1+3+\ldots+(2 b-1)=2 a=100 \\
\frac{b}{2}[2+2(b-1)]=100 \\
b^{2}=100 \\
b=10
\end{array}
$$

The bag contains 10 white balls and 3 black balls.
$$
\begin{array}{l}
\mathrm{P}(2 \text { different colours })=2 \times \frac{10}{13} \times \frac{3}{12}=\frac{5}{13}=\frac{c}{13} \\
c=5 \\
-\frac{5}{10} \times \frac{d}{1}=-1 \\
\Rightarrow d=2
\end{array}
$$