1991 年

In the figure, $ABCD$ is a square of side $a$ and $MNPQ$ is a square of side $b$. The four trapeziums are identical. The area of the shaded region is

In the figure, $TB$ touches the semi-circle at $B$. $TA$ cuts the semi-circle at $P$ such that $TP=PA$. If the radius of the semi-circle is $2$, find the area of the shaded region.

An equilateral triangle and a square have equal perimeters. $\dfrac{\text{Area of the triangle}}{\text{Area of the square}}=$

A man borrows $\$10\ 000$ from a bank at $12\%$ per annum compounded monthly. He repays the bank $\$2000$ at the end of each month. How much does he still owe the bank just after the second repayment?

$\left[\dfrac{1}{\cos\theta}+\tan\theta\right](1-\sin\theta)=$

$\dfrac{\sin(\theta-90^\circ)}{\tan(\theta+180^\circ)}=$

For $0\le\theta<2\pi$, how many roots does the equation $\tan\theta+2\sin\theta=0$ have?

In the figure, $XPY$ and $YQZ$ are semi-circles with areas $A_1$ and $A_2$ respectively. $\angle YXZ=60^\circ$ and $\angle YZX=45^\circ$. The ratio $A_1:A_2=$

In the figure, $\angle A=30^\circ$ and $\angle B=120^\circ$. The ratio of the altitudes of the triangle $ABC$ from $A$ and from $B$ is

In the figure, $O$ is the centre of the circle. Find $a+c$.