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数学博物館『すうじあむ』にようこそ。 『すうじあむ』は数学の問題・解答・解説のデータベースをはじめ 数学に関するあらゆる情報が詰まったポータルサイトです。
京都府立医科大学 前期 1995年度 問2
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入試情報
| 大学名 | 京都府立医科大学 |
|---|---|
| 学科・方式 | 前期 |
| 年度 | 1995年度 |
| 問No | 問2 |
| 学部 | 医学部 |
| カテゴリ | 積分法 ・ 積分法の応用 |
| 状態 |    |
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%
% 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,color}
%\usepackage{wallpaper}
\usepackage{wrapfig}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{multicol}
\usepackage{longtable}
\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% mebio.styと競合する
\usepackage{booktabs}
\usepackage{ascmac}
\usepackage{itembbox}
%
% \usepackage{ custom_suseum}% すうじあむ
%=========================================================
%============================================================
% ■ 分数の横棒を長く，分数を立てに狭く（emath.styの古い記述）
%------------------------------------------------------------
% 分数記号(分数罫を少し長めに)
\newcommand{\bunsuu}[2]{\dfrac{\,#1\,}{\,#2\,}}%
% さらに罫線の上下を狭く．
\newcommand\Dfrac[2]{\bunsuu{\lower.6ex\hbox{$#1$}}{\lower-.3ex\hbox{$#2$}}}%
%============================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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]}
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\setlength{\unitlength}{1.2pt}
\begin{picture}(10,10)(1,2)
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\setlength{\unitlength}{1.1pt}
\begin{picture}(10,10)(2,1)
\put(2,1){\line(1,0){3}}
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\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}}
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\begin{picture}(10,10)(1,0)
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\put(5,5){\oval(8,8)[lb]}
\put(5,1){\vector(1,0){4}}
\end{picture}\fi}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%======================================================================
\newcommand{\解説印}{\fboxsep=1.3pt\noindent\ovalbox{\bf \,解\,説\,}\hspace{0.5em}}
% \newcommand{\解答印}{\fboxsep=1.3pt\noindent\fbox{\bf \,解\,答\,}\hspace{0.5em}}
\newcommand{\解答印}{\shadowsize=0.15\shadowsize{\fboxsep=1.3pt\noindent\shadowbox{\bf \,解\,答\,}\hspace{0.5em}}}
\newcommand{\別解印}{\fboxsep=1.3pt\noindent\fbox{\bf \,別\,解\,}\hspace{0.5em}}
\newcommand{\注意印}{\noindent{\bf [注意]}\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)
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===========================================================
% ■△ABCとか
%
% 「$\∠{ABC}=\Dfrac12\pi$」のような使い方をするので「$」は定義に含めない
%
%-----------------------------------------------------------
\def\△#1{\triangle{\rm #1}}
\def\∠#1{\angle{\rm #1}}
%===========================================================
%===========================================================
% ■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{\raisebox{-.25ex}{#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}
\newcommand{\Dsum}{\sum\limits}% ちっさいΣ
%===========================================================
%===========================================================
% ■ nCr マクロ
%-----------------------------------------------------------
\newcommand{\nCr}[2]{%
{}_{#1}\mathrm{C}_{#2}%
}
%===========================================================
%===========================================================
% ■ローマ数字 (i) (ii) (iii) (iv) (v) (vi) (vii) まで定義
%-----------------------------------------------------------
\providecommand{\RMi}{\mbox{(\hspace{.1em}i\hspace{.1em})}}
\providecommand{\RMii}{\mbox{(ii)}}
\providecommand{\RMiii}{\mbox{(i\hspace{-.1em}i\hspace{-.1em}i)}}
\providecommand{\RMiv}{\mbox{(i\hspace{-.1em}v)}}
\providecommand{\RMv}{\mbox{(\hspace{.1em}v\hspace{.1em})}}
\providecommand{\RMvi}{\mbox{(v\hspace{-.1em}i)}}
\providecommand{\RMvii}{\mbox{(\hspace{-.1em}v\hspace{-.1em}i\hspace{-.1em}i)}}
%===========================================================
%===========================================================
% ■minipage マクロ
%
% #1 左幅% #2 左内容 #3 右幅% #4 右内容
%
% \ミニ頁{左幅%}{左内容}{右幅%}{右内容}
%
%-----------------------------------------------------------
\providecommand{\ミニ頁}[4]{%
%
\noindent
\begin{minipage}[t]{.#1\linewidth}{\setlength{\parindent}{1\Cwd}%
\vspace{0mm}% ← これはない方がいいのかも
#2}
\end{minipage}
\begin{minipage}[t]{.#3\linewidth}{\setlength{\parindent}{1\Cwd}
\vspace{0mm}
#4}
\end{minipage}
}
% http://www2s.biglobe.ne.jp/~nuts/labo/inti/inti05.html
% minipage 環境では、変数 \parindent が 0\Cwd になるために起こる。
% インデントをつけたい minipage の始まり （\begin{minipage} の後）に、
% \setlength{\parindent}{1\Cwd} （これが通常時の値）と書けばよい。
% 私の記憶によれば、breakbox 環境内でも同じことが起こったように思う。
%===========================================================
%===========================================================
% ■点A(t, t^3)とか
% #1 は点の名前（「A」とか）
% #2 は「(t,\,t^3)」とか「(s,\,t,\,u)」と（ ）内を全部書く
% （）の大きさとか，の後のスペースとか微調整ができるように
%
%-----------------------------------------------------------
\providecommand{\点}[2]{${\rm#1} #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% \usepackage{ custom_suseum} をゆうこうにすること
\begin{framed}
%
%----------------------------------------------------------------
%
$n$ は正の整数とする．円 $x^2+y^2-2y=0$ の中心をCとおく．\\
$k=0,\,1,\,2,\,\cdots,\,n$ に対して，点${\rm A}_k\left(\Dfrac{k}{n},\,0\right)$ をとり，
$\alpha_k=\angle{\rm A}_k{\rm CO}$ とする．\\
$\theta_k=\alpha_k-\alpha_{k-1}=\angle{\rm A}_{k-1}{\rm C}{\rm A}_k \ (k=1,\,2,\,\cdots,\,n)$
と定めるとき，\\
%
\ミニ頁{67}{%
\begin{enumerate}
\item $\Dfrac12\Dsum_{k=1}^n \sin\theta_k < \Dfrac{\pi}8<\Dfrac12\Dsum_{k=1}^n\tan\theta_k$ を示せ．
\item $k=0,\,1,\,2,\,\cdots,\,n$ に対して，次の2つの不等式を示せ．
\[\Dfrac1n\cdot\Dfrac1{1+\left(\Dfrac{k}{n}\right)^2}<\sin\theta_k\]
\[\tan\theta_k<\Dfrac1n\cdot\Dfrac1{1+\left(\Dfrac{k-1}{n}\right)^2}\]
\item 上の結果を用いて，$\dint_0^1\Dfrac1{1+x^2}dx$ を求めよ．
\end{enumerate}
}{26}{
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}}%
% LINE 2 0 3 0 Black White
% 4 1120 1120 1360 1600 1120 1120 1480 1600
%
{\color[named]{Black}{%
\special{pn 8}%
\special{pa 1120 1120}%
\special{pa 1360 1600}%
\special{fp}%
\special{pa 1120 1120}%
\special{pa 1480 1600}%
\special{fp}%
}}%
% CIRCLE 2 0 3 0 Black White
% 4 1120 1120 1198 1324 1708 2290 1960 2248
%
{\color[named]{Black}{%
\special{pn 8}%
\special{ar 1120 1120 218 218 0.9307082 1.1050995}%
}}%
% STR 2 0 3 0 Black White
% 4 1496 1682 1496 1742 2 0 0 0
% ${\rm A}_k$
\put(14.9600,-17.4200){\makebox(0,0)[lb]{${\rm A}_k$}}%
% STR 2 0 3 0 Black White
% 4 1204 1686 1204 1746 2 0 0 0
% ${\rm A}_{k-1}$
\put(12.0400,-17.4600){\makebox(0,0)[lb]{${\rm A}_{k-1}$}}%
% STR 2 0 3 0 Black White
% 4 1014 1120 1014 1180 2 0 0 0
% C
\put(10.1400,-11.8000){\makebox(0,0)[lb]{C}}%
\end{picture}}%
\end{center}
}
%--------------------------------------------------------------
\end{framed}
%\end{FRAME}
%--- 解答 ------------------------------------------------------------------------
{\footnotesize
}% footnotesize
%%%%%%% ■ 本文終了 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
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