1990 年

From the table, a root of the equation $f(x)=0$ must be

$\dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+\dfrac{1}{\sqrt{4}+\sqrt{5}}=$

Let $f(x)=3x^3-4x+k$. If $f(x)$ is divisible by $x-k$, find the remainder when $f(x)$ is divided by $x+k$.

Let $m$ be a constant. Find the value of $x$ such that $\begin{cases}x^2+x+1=m\\x-1=\dfrac{26}{m}\end{cases}$

If $a<b<0$, which of the following must be true?

The H.C.F. and L.C.M. of three expressions are $xyz^2$ and $x^3y^5z^4$ respectively. If two of the expressions are $x^2y^3z^3$ and $x^3yz^2$, find the third expression.

In the figure, $AM=MB=MC=5$ and $BC=6$. The area of triangle $ABC=$

In the figure, an equilateral triangle is inscribed in a circle of radius 1. The circumference of the circle is greater than the perimeter of the triangle by

Three equal circles of radii 1 touch each other as shown in the figure. Find the area of the shaded region.

If $A$ is $30\%$ greater than $B$ and $B$ is $30\%$ less than $C$, then