大阪大学 前期理系 1997年度 問5

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 前期理系
年度 1997年度
問No 問5
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ 積分法の応用
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{graphicx} \usepackage{delarray} \usepackage{multicol} \usepackage{amscd} \usepackage{pifont} \usepackage{color} \ExecuteOptions{usename} \usepackage{vector3} %\usepackage{myhyper} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} 関数 $f(\theta) = \sqrt{\vphantom{b} 2}\,\sin^2\theta + \cos\theta$ に対し, 次の条件をみたす正の数 $a$ を考える. \begin{align*} \left\{ \begin{array}{lll} \smallskip \zettaiti{\theta} < a & ならば & f(\theta) > 0 \\ \zettaiti{\theta} = a & ならば & f(\theta) = 0 \end{array} \right. \end{align*} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $a$ の値を求めよ. \item  曲線 $C$ を媒介変数 $\theta\,\,\,(-a \leqq \theta \leqq a)$ を用いて \\ \begin{minipage}{250pt} \begin{align*} C : \left\{ \begin{array}{l} \smallskip x = f(\theta) \\ y = \sin\theta \end{array} \right. \end{align*} で定める. $x$軸に平行な直線 $y = t$ と曲線 $C$ が共有点をもつような 実数 $t$ の範囲を求め, 共有点の$x$座標を $t$ で表せ. \item  曲線 $C$ と$y$軸で囲まれる図形を, $y$軸のまわりに1回転してできる立体の体積を求めよ.\\ \hfill(配点率20%) \end{minipage} \begin{minipage}{180pt} %\input{osaka97s5f_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 16.3900, 17.3000)( 9.0600,-26.4400) % STR 2 0 3 0 % 3 1176 976 1176 985 4 3600 % $y$ \put(11.7600,-9.8500){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2550 1841 2550 1850 4 3600 % $x$ \put(25.5000,-18.5000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1197 2644 1197 985 % \special{pn 8}% \special{pa 1198 2644}% \special{pa 1198 986}% \special{fp}% \special{sh 1}% \special{pa 1198 986}% \special{pa 1178 1052}% \special{pa 1198 1038}% \special{pa 1218 1052}% \special{pa 1198 986}% \special{fp}% % VECTOR 2 0 3 0 % 2 990 1815 2545 1815 % \special{pn 8}% \special{pa 990 1816}% \special{pa 2546 1816}% \special{fp}% \special{sh 1}% \special{pa 2546 1816}% \special{pa 2478 1796}% \special{pa 2492 1816}% \special{pa 2478 1836}% \special{pa 2546 1816}% \special{fp}% % FUNC 2 0 3 0 % 9 990 985 2545 2644 1197 1815 1301 1815 1197 1711 990 985 2545 2644 50 3 0 2 % 6.2(sqrt(2)(sin(t))^2+cos(t))///6.2sin(t)///-3pi/4///3pi/4 \special{pn 8}% \special{pa 1198 2272}% \special{pa 1224 2280}% \special{pa 1250 2288}% \special{pa 1276 2296}% \special{pa 1302 2304}% \special{pa 1328 2312}% \special{pa 1354 2320}% \special{pa 1380 2328}% \special{pa 1406 2334}% \special{pa 1432 2342}% \special{pa 1458 2348}% \special{pa 1484 2356}% \special{pa 1510 2362}% \special{pa 1536 2368}% \special{pa 1562 2374}% \special{pa 1588 2380}% \special{pa 1612 2386}% \special{pa 1636 2392}% \special{pa 1662 2398}% \special{pa 1686 2402}% \special{pa 1710 2408}% \special{pa 1732 2412}% \special{pa 1756 2416}% \special{pa 1778 2420}% \special{pa 1802 2424}% \special{pa 1824 2428}% \special{pa 1844 2432}% \special{pa 1866 2436}% \special{pa 1886 2440}% 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\special{pa 2214 1190}% \special{pa 2210 1188}% \special{pa 2206 1186}% \special{pa 2200 1182}% \special{pa 2194 1180}% \special{pa 2188 1178}% \special{pa 2180 1176}% \special{pa 2174 1176}% \special{pa 2164 1174}% \special{pa 2156 1172}% \special{pa 2146 1172}% \special{pa 2136 1172}% \special{pa 2126 1170}% \special{pa 2114 1170}% \special{pa 2102 1170}% \special{pa 2088 1172}% \special{pa 2074 1172}% \special{pa 2060 1172}% \special{pa 2046 1174}% \special{pa 2030 1174}% \special{pa 2014 1176}% \special{pa 1998 1178}% \special{pa 1980 1180}% \special{pa 1962 1182}% \special{pa 1944 1184}% \special{pa 1926 1186}% \special{pa 1906 1188}% \special{pa 1886 1192}% \special{pa 1866 1196}% \special{pa 1844 1198}% \special{pa 1824 1202}% \special{pa 1802 1206}% \special{pa 1778 1210}% \special{pa 1756 1214}% \special{pa 1732 1220}% \special{pa 1710 1224}% \special{pa 1686 1228}% \special{pa 1662 1234}% \special{pa 1636 1240}% \special{pa 1612 1244}% \special{pa 1588 1250}% \special{pa 1562 1256}% \special{pa 1536 1262}% \special{pa 1510 1268}% \special{pa 1484 1276}% \special{pa 1458 1282}% \special{pa 1432 1290}% \special{pa 1406 1296}% \special{pa 1380 1304}% \special{pa 1354 1312}% \special{pa 1328 1318}% \special{pa 1302 1326}% \special{pa 1276 1334}% \special{pa 1250 1342}% \special{pa 1224 1352}% \special{pa 1198 1360}% \special{sp}% % STR 2 0 3 0 % 3 1405 1032 1405 1084 2 0 % {\scriptsize 曲線$C$の概形} \put(14.0500,-10.8400){\makebox(0,0)[lb]{{\scriptsize 曲線$C$の概形}}}% % STR 2 0 3 0 % 3 1070 1840 1070 1940 2 0 % {\small O} \put(10.7000,-19.4000){\makebox(0,0)[lb]{{\small O}}}% \end{picture}% \end{minipage} \end{enumerate} \end{document}