1931　京都帝国大学　農学部MathJax

【１】　Show that, if $1>a>0\text{,}$ then ${\displaystyle \frac{1}{1-a}}>1+a+a^2+\cdots +a^n\text{.}$

【２】　Find the co-ordinates of the centre of the following ellipse $x^2+y^2-3(x+y)-xy=0\text{.}$

【３】　Given $y=\sin x\cos x\text{,}$ evaluate ${\displaystyle \frac{d^{4}y}{{\mathit{dx}}^4}}$

【４】　Integrate $r^2\mathit{dm}$ for a hollow circular cylinder, whose outer and innner radii are $r_1$ and $r_2$ and the total mass is $m\text{,}$ $\mathit{dm}$ being element of mass and $r$ being its distance from the axis of the cylinder.