京都府立医科大学 前期 2011年度 問4

このコンテンツをご覧いただくためにはJavaScriptをONにし、最新のFlash Playerが必要です。

最新のFlash Playerのインストールはこちら

入試情報

大学名 京都府立医科大学
学科・方式 前期
年度 2011年度
問No 問4
学部 医学部
カテゴリ
状態 解答なし 解説なし ウォッチリスト

コメントをつけるにはログインが必要です。

コメントはまだありません。

% % すうじあむはアップするファイルで完結したTeXソースでないといけない. % このひな形で表示確認をしよう. % \documentclass[b5paper,fleqn,papersize]{jsarticle} %=========================================================== % ■余白の設定 B5 %----------------------------------------------------------- \setlength{\oddsidemargin}{-5.4truemm} % 左マージン \setlength{\topmargin}{-15.4truemm} % 上マージン \setlength{\textwidth}{14.2cm} % B5 サイズ用 \setlength{\headheight}{2zw} \setlength{\headsep}{2zw} \setlength{\textheight}{217mm} %=========================================================== \usepackage[dvipdfm]{graphicx} \usepackage{wrapfig} % \usepackage{amsmath,amsthm,amssymb,cases} \usepackage{amsmath,amsthm,amssymb}% すうじあむはcasesが使えない? \usepackage{multicol} \usepackage{ascmac} \usepackage{fancybox} \usepackage{framed} \usepackage{ifthen} \usepackage{setspace} % setspaceパッケージのインクルード \usepackage{amssymb}% ≒を表示するために必要 \usepackage{comment}% コメント \usepackage{booktabs} \usepackage{ascmac} % \usebackage{eclbkbox}% すうじあむはこれもダメ \usepackage{itembbox} % % \usepackage{ custom_suseum}% すうじあむ %========================================================= %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % enumerate の自動ラベルの変更 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\labelenumi{(\theenumi)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 増減表の矢印 % http://www.biwako.shiga-u.ac.jp/sensei/kumazawa/tex/arrow.html % % $\cvinc$ increase % $\ccinc$ concavity 凹状の % $\cvdec$ convex 凸形の % $\ccdec$ decrease % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \newcommand{\ccinc}{\ifx\@currsize\small \setlength{\unitlength}{1.1pt} \begin{picture}(10,10)(1,2) \put(1,2){\line(0,1){3}} \put(5,5){\oval(8,8)[lt]} \put(5,9){\vector(1,0){4}} \end{picture}\else \setlength{\unitlength}{1.2pt} \begin{picture}(10,10)(1,2) \put(1,2){\line(0,1){3}} \put(5,5){\oval(8,8)[lt]} \put(5,9){\vector(1,0){4}} \end{picture}\fi} \newcommand{\cvinc}{\ifx\@currsize\small \setlength{\unitlength}{1.1pt} \begin{picture}(10,10)(2,1) \put(2,1){\line(1,0){3}} \put(5,5){\oval(8,8)[rb]} \put(9,5){\vector(0,1){4}} \end{picture}\else \setlength{\unitlength}{1.2pt} \begin{picture}(10,10)(2,1) \put(2,1){\line(1,0){3}} \put(5,5){\oval(8,8)[rb]} \put(9,5){\vector(0,1){4}} \end{picture}\fi} \newcommand{\ccdec}{\ifx\@currsize\small \setlength{\unitlength}{1.1pt} \begin{picture}(10,10)(2,1) \put(2,9){\line(1,0){3}} \put(5,5){\oval(8,8)[rt]} \put(9,5){\vector(0,-1){4}} \end{picture}\else \setlength{\unitlength}{1.2pt} \begin{picture}(10,10)(2,1) \put(2,9){\line(1,0){3}} \put(5,5){\oval(8,8)[rt]} \put(9,5){\vector(0,-1){4}} \end{picture}\fi} \newcommand{\cvdec}{\ifx\@currsize\small \setlength{\unitlength}{1.1pt} \begin{picture}(10,10)(1,0) \put(1,8){\line(0,-1){3}} \put(5,5){\oval(8,8)[lb]} \put(5,1){\vector(1,0){4}} \end{picture}\else \setlength{\unitlength}{1.2pt} \begin{picture}(10,10)(1,0) \put(1,8){\line(0,-1){3}} \put(5,5){\oval(8,8)[lb]} \put(5,1){\vector(1,0){4}} \end{picture}\fi} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %====================================================================== \newcommand{\解説}{\noindent\ovalbox{\bf 解説}\hspace{0.5em}} \newcommand{\解答}{\noindent\fbox{\bf 解答}\hspace{0.5em}} \newcommand{\別解}{\noindent\fbox{\bf 別解}\hspace{0.5em}} %====================================================================== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %■ベクトルの矢印の高さを揃える定義 % \overrightarrow の矢印の高さを揃える % \vec, \Vec で定義する %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\vec#1{\overrightarrow{\mathstrut #1}} \def\Vec#1{\overrightarrow{\mathstrut {\rm #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %=========================================================== % ■二重根号 %----------------------------------------------------------- \def\tsqrt#1{\textstyle\sqrt{#1}} %=========================================================== %%% 平行 // の記号%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\para{% \setlength{\unitlength}{1pt}% \thinlines % \begin{picture}(10, 12)% \put(1,0){/}% \put(4,0){/}% \end{picture}% }% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %■ 答えを太文字 \newcommand{\kotae}[1]{\mbox{\boldmath$#1$}} % \def\kotae#1{\mbox{\boldmath $#1$}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %=========================================================== % ■枠囲み %----------------------------------------------------------- \newcommand{\枠囲}[1]{\,\fboxsep1pt\fbox{ #1 }\,}% 文章中でも数式中でも使える? % \newcommand{\枠}[1]{\fboxsep1pt\fbox{ #1 }\,} %=========================================================== %=========================================================== % ■≦と≧の定義 %----------------------------------------------------------- \def\le{\leqq} \def\ge{\geqq} \newcommand{\LEQQ}{\leqq} \newcommand{\GEQQ}{\geqq} % \newcommand{\LEQQ}{\mathrel{\mathpalette\gl@align<}} % \newcommand{\GEQQ}{\mathrel{\mathpalette\gl@align>}} % \newcommand{\gl@align}[2]{\lower.6ex\vbox{\baselineskip\z@skip\lineskip\z@ % \ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}} % エラーになる。理由はよく解らん。 %=========================================================== %=========================================================== % ■行列 2×2 %----------------------------------------------------------- \newcommand{\matrixTT}[4]{ \begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix} } %=========================================================== %=========================================================== % ■行列 3×3 %----------------------------------------------------------- \newcommand{\matrixTTT}[9]{ \begin{pmatrix} #1 & #2 & #3\\ #4 & #5 & #6\\ #7 & #8 & #9 \end{pmatrix} } %=========================================================== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ■ 3*1行列の定義 \newcommand{\matrixthreeone}[3]{ \left(\begin{array}{c} #1\\ #2\\ #3 \end{array}\right) } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %=========================================================== % ■角括弧 複数行 #1 は横幅%,#2は内容 % %----------------------------------------------------------- \newcommand{\角括弧}[2]{% $% \left[ \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right]% $% } %=========================================================== %=========================================================== % ■丸括弧 複数行 #1 は横幅%,#2は内容 %----------------------------------------------------------- \newcommand{\丸括弧}[2]{% $% \left( \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right)% $% } %=========================================================== %=========================================================== % ■丸囲み数字 ①とか %----------------------------------------------------------- \newcommand{\tyoMaru}[1]{\mbox{{\normalsize \textcircled{\scriptsize #1}}}} %=========================================================== %=========================================================== % ■ 微分 df/dt %----------------------------------------------------------- \newcommand{\dd}[2]{% \dfrac{{\rm d}#1}{{\rm d}#2}% } %=========================================================== %=========================================================== % ■ ∫記号のdisplaystyleマクロ %----------------------------------------------------------- % \newcommand{\dint}{\displaystyle\int}% emathと干渉しないように\defで定義 \def\dint{\displaystyle\int} %=========================================================== %=========================================================== % ■ Σ記号のdisplaystyleマクロ %----------------------------------------------------------- \newcommand{\dsum}{\displaystyle\sum} %=========================================================== %=========================================================== % ■ lim記号のdisplaystyleマクロ %----------------------------------------------------------- \newcommand{\dlim}{\displaystyle\lim} %=========================================================== %=========================================================== % ■ nCr マクロ %----------------------------------------------------------- \newcommand{\nCr}[2]{% {}_{#1}\mathrm{C}_{#2}% } %=========================================================== %=========================================================== % ■ 証明終了記号 ■ %----------------------------------------------------------- \newcommand{\■}{{\tiny \text{■}}} %=========================================================== \begin{document} %%%%% ■ 本文開始 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{FRAME}% この間に設問を書く $n$を5以上の整数とする.平面上に点Oをとる.Oを通る直線上に${\rm OA}_0=1$となる点${\rm A}_0$を一つとる. 点Oを中心として直線${\rm OA}_0$を正の向きに角$\dfrac{2\pi}{n}$だけ回転した直線上に ${\rm OA}_1\perp {\rm A}_0{\rm A}_1$となる点A$_1$をとる. 次に,点Oを中心として直線OA$_1$を正の向きに角$\dfrac{2\pi}{n}$だけ回転した直線上に ${\rm OA}_2\perp {\rm A}_1{\rm A}_2$となる点A$_2$をとる. 以下同様にして$k=3,\ 4,\ \cdots,\ n$について,点Oを中心として直線OA$_{k-1}$を正の向きに角$\dfrac{2\pi}{n}$だけ回転した直線上に ${\rm OA}_k\perp {\rm A}_{k-1}{\rm A}_k$となる点A$_k$をとる. 特に,点A$_n$は線分OA$_0$上の点となる. \begin{enumerate} \item 不等式$1-\dfrac{x^2}{2}\LEQQ\cos{x}$を証明せよ. \item 線分OA$_n$の長さを$r_n$とする.極限値$\dlim_{n\to\infty}r_n$を求めよ. \item 線分${\rm A}_0{\rm A}_1,\ {\rm A}_1{\rm A}_2,\ \cdots ,\ {\rm A}_{n-1}{\rm A}_n$の長さの和を$L_n$とする.極限値$\dlim_{n\to\infty}L_n$を求めよ. \end{enumerate} % \end{FRAME} %--- 解答 ------------------------------------------------------------------------ %%%%%%% ■ 本文終了 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}