2002 年

The figure shows part of a page torn off from a mathematics book. According to the information shown, which of the following is a root of the equation $\mathrm{f}(x)=0$ ?

How many different triangles can be constructed so that the lengths of the three sides are $x\text{ cm}$, $2x\text{ cm}$ and $12\text{ cm}$, where $x$ is an integer?

If the geometric mean of two positive numbers $a$ and $b$ is $100$, then the arithmetic mean of $\log a$ and $\log b$ is

In the figure, $ABCD$ is a parallelogram. $E$ and $F$ are points on $AD$ and $BC$ respectively such that $AB/\!/EF$. $EF$ meets $AC$ at $G$. If $AG:GC=1:2$, then area of $ABFG$ : area of $EGCD=$

In the figure, $P$ and $Q$ are two right cylindrical vessels each containing some water. The two vessels are placed on the same horizontal surface. The internal base radii of $P$ and $Q$ are in the ratio $1:3$. $A$ and $B$ are two cubes with sides in the ratio $1:2$. $A$ and $B$ are put into $P$ and $Q$ respectively. Suppose both cubes are totally immersed in water without any overflow. If the rise in water level in $P$ is $1\text{ cm}$, then the rise in water level in $Q$ is

If $\sin\theta=\dfrac{3}{5}$ and $\theta$ lies in the first quadrant, then $\sin(90^\circ-\theta)+\sin(180^\circ+\theta)=$

$[1+\cos(\pi+\theta)][1-\cos(\pi-\theta)]=$

For $0^\circ\le x\le360^\circ$, how many roots does the equation $\tan x=2\sin x$ have?

In the figure, $ABCDEFGH$ is a rectangular block with a square base $ABCD$. Find $\angle FBH$ correct to the nearest degree.

In the figure, $E$ and $F$ are the mid-points of $AB$ and $AC$ respectively. $G$ and $H$ are points on $BD$ and $CD$ respectively such that $\dfrac{DG}{GB}=\dfrac{DH}{HC}=\dfrac{3}{5}$. If $EF=6\text{ cm}$, then $GH=$