1981 年

In the figure, $AT$ touches the circle at $A$. In $\triangle ABC$, $\angle A:\angle B:\angle C=2:3:4$. $\theta=$

In the figure, $AB$ is a diameter of the circle with centre at $O$. The length of the minor arc $AC$ is twice the length of the minor arc $CD$. $\angle BOD=$

In the figure, two circles both with radius $2\text{ cm}$ touch each other externally. $AP$ and $AQ$ are equal tangents to the two circles. $AP=$ ?

In the figure, $ABCD$ is a quadrilateral. The shaded portions are four sectors with centres at $A$, $B$, $C$ and $D$. Their radii are all equal to $a$. What is the total area of the four sectors?

$2x^2-2\le0$ is equivalent to

In the figure, which point in the shaded region will make the value of $x-2y$ a minimum?

$6x^2+kx+6=0$ is a quadratic equation in which $k$ is a constant. Its roots $\alpha$ and $\beta$ are positive. $\log_{10}\alpha+\log_{10}\beta=$

The H.C.F. and L.C.M. of three expressions are $a^2b^2c$ and $a^4b^6c^4$ respectively. Two of the expressions are $a^2b^3c^4$ and $a^3b^2c^2$. The third expression is

The sum of the first five terms of an arithmetic progression is $15$. If the fourth term is $7$, the first term is

Which of the following can be summed to infinity? I. The arithmetic progression $4, 3, 2, 1, \ldots$ II. The geometric progression $27, 9, 3, 1, \ldots$ III. The geometric progression $16, -8, 4, -2, \ldots$