2010 年

Let $k$ be a constant. If $x^3+5x^2+3kx-k$ is divisible by $x-1$, find the value of $k$.

In the figure, the equations of $AB$ and $BC$ are $x+3y=18$ and $2x+y=16$ respectively. If $(x,y)$ is a point lying in the shaded region $OABC$ (including the boundary), then the greatest value of $3x-y+16$ is

If $h, 5, k$ are the first $3$ terms of an arithmetic sequence and $h, 4, k$ are the first $3$ terms of a geometric sequence, then $h^2+k^2=$

The sum of the first $2$ terms of a geometric sequence is $8$ and the $3$rd term of the sequence is $18$. Find the 1st term of the sequence.

The figure shows

$\cos^2 1^\circ+\cos^2 2^\circ+\cos^2 3^\circ+\cdots+\cos^2 89^\circ+\cos^2 90^\circ=$

In the figure, $AD$ is produced to meet $BC$ at $E$. If $AB=BC=AC$, $BE=CE$ and $AD=DE$, find $\sin\theta$.

In the figure, $ABCDEFGH$ is a cube. If $M$ and $N$ are the mid-points of $AB$ and $EH$ respectively, then $\cos\theta=$

In the figure, $AD$ is a diameter of the circle $ABCD$. It is given that $XBCY$ is a straight line. If $AD=20\ \text{cm}$ and $BC=12\ \text{cm}$, then $AX+DY=$

In the figure, $XY$ and $XZ$ are the tangents to the circle $ABCD$ at $A$ and $B$ respectively. If $\angle AXB=50^\circ$ and $\angle DAY=30^\circ$, then $\angle BCD=$