1988 年

Simplify $\dfrac{2^{n+4}-2(2^n)}{2(2^{n+3})}$

If $x=\dfrac{1+y}{1-y}$, then $y=$

$\dfrac{x^2-2x}{x^3-25x}\times\dfrac{x^2-2x-15}{x^2+x-6}=$

If $\alpha$ and $\beta$ are the two roots of $x^2-8x-4=0$, then the value of $\dfrac{1}{\alpha}+\dfrac{1}{\beta}$ is

$\dfrac{1}{2x-x^2}+\dfrac{1}{x^2+x-6}=$

Which of the following is an identity/are identities?

I. $\dfrac{1}{x}-1=\dfrac{1-x}{x}$ II. $(ax+b)(x-b)=ax^2-b^2$ III. $2x^2-3x+1=0$

If the roots of a quadratic equation are $a+\sqrt{b}$ and $a-\sqrt{b}$, then the equation is

Which of the following is a G.P./are G.P.'s?

I. $5, 0.5, 0.05, 0.005, 0.0005$ II. $\log 5, \log 50, \log 500, \log 5000, \log 50000$ III. $5, 5\sin 70^\circ, 5(\sin 70^\circ)^2, 5(\sin 70^\circ)^3, 5(\sin 70^\circ)^4$

A solid iron sphere of radius $r$ is melted and recast into a circular cone and a circular cylinder. If both of them have the same height $h$ and the same base radius $r$, find $h$ in terms of $r$.

In the figure, $PQRS$ is a rectangle with $PQ=24$ and $PS=d$. $T$ is the mid-point of $PQ$. $V$ is a point on $SR$ and $\dfrac{\text{area of }PTVS}{\text{area of }TQRV}=2$. $SV=$