1993 年

Find the H.C.F. and L.C.M. of $ab^2c$ and $abc^3$.

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2-3x-1=0$, find the value of $\dfrac{1}{\alpha}+\dfrac{1}{\beta}$.

If the simultaneous equations $\begin{cases}y=x^2-k\\y=x\end{cases}$ have only one solution, find $k$.

The price of a cylindrical cake of radius $r$ and height $h$ varies directly as the volume. If $r=5\text{ cm}$ and $h=4\text{ cm}$, the price is $\$30$. Find the price when $r=4\text{ cm}$ and $h=6\text{ cm}$.

Find the perimeter of the sector in the figure.

In the figure, the base of the conical vessel is inscribed in the bottom of the cubical box. If the box and the conical vessel have the same capacity, find $h:r$.

The figure shows a solid consisting of a cylinder of height $h$ and a hemisphere of radius $r$. The area of the curved surface of the cylinder is twice that of the hemisphere. Find the ratio volume of cylinder : volume of hemisphere.

A merchant marks his goods 25% above the cost. He allows 10% discount on the marked price for a cash sale. Find the percentage profit the merchant makes for a cash sale.

$\dfrac{\cos\theta}{1-\sin^2\theta}\times\dfrac{1-\cos^2\theta}{\sin\theta}=$

$\cos^4\theta-\sin^4\theta+2\sin^2\theta=$