1999 年

$A(-4,2)$ and $B(1,-3)$ are two points. $C$ is a point on the $y$-axis such that $AC=CB$. Find the coordinates of $C$.

In the figure, $OABC$ is a parallelogram. If the equation of $OC$ is $2x-y=0$ and the length of $CB$ is $3$, find the equation of $AB$.

Find the median and mode of the ten numbers $6,\ 8,\ 3,\ 3,\ 5,\ 5,\ 5,\ 7,\ 7,\ 11$.

A student scored $50$ marks in a test and the corresponding standard score is $-0.5$. If the mean of the test scores is $60$ marks, find the standard deviation of the scores.

Two cards are drawn randomly from four cards numbered $1$, $2$, $3$ and $4$ respectively. Find the probability that the sum of the numbers drawn is odd.

Tom and Mary each throws a dart. The probability of Tom's dart hitting the target is $\dfrac13$ while that of Mary's is $\dfrac25$. Find the probability of only one dart hitting the target.

In the figure, a right circular cone is divided into two parts $X$ and $Y$ by a plane parallel to the base such that the lengths of their slant edges are $4\text{ cm}$ and $3\text{ cm}$ respectively. Find the ratio of the curved surface areas of $X$ and $Y$.

It is given that $\mathrm{F}(x)=x^3-4x^2+ax+b$. $\mathrm{F}(x)$ is divisible by $x-1$. When it is divided by $x+1$, the remainder is $12$. Find $a$ and $b$.

If $\dfrac12\log y=1+\log x$, then

$\dfrac{2}{x^2-1}-\dfrac{x-1}{x^2-2x-3}=$