大阪大学 文系 2010年度 問1

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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 文系
年度 2010年度
問No 問1
学部 文学部 ・ 人間科学部 ・ 外国語学部 ・ 法学部 ・ 経済学部
カテゴリ 図形と方程式 ・ 微分法と積分法
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{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  曲線 $C : y = -x^2 - 1$ を考える. \begin{enumerate} \item[(1)]  $t$ が実数全体を動くとき, 曲線 $C$ 上の点$(t,\ {-t^2} - 1)$を頂点とする放物線 \[ y = \frac{3}{4}(x - t)^2 - t^2 - 1 \] が通過する領域を$xy$平面上に図示せよ. \item[(2)]  $D$ を(1)で求めた領域の境界とする. $D$ が$x$軸の正の部分と交わる点を$(a,\ 0)$とし, $x = a$ での $C$ の接線を $\ell$ とする. $D$ と $\ell$ で囲まれた部分の面積を求めよ. \end{enumerate} \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $y = \dfrac{3}{4}(x - t)^2 - t^2 - 1$ を $t$ の式として整理すると, \begin{gather*} -\frac{1}{4}t^2 - \frac{3}{2}xt + \frac{3}{4}x^2 - y - 1 = 0 \\ \therefore \,\,\, t^2 - 6x \cdot t - 3x^2 + 4y + 4 = 0 \tag*{$\cdott\MARU{1}$} \end{gather*} したがって, \begin{align*} & (x,\ y) \in D \\ \Longleftrightarrow \,\,\,{} & 上手にtを選べば\> y = \frac{3}{4}(x - t)^2 - t^2 - 1\>をみたす \\ \Longleftrightarrow \,\,\,{} & tの2次方程式\MARU{1}が実数解をもつ \\ \Longleftrightarrow \,\,\,{} & \frac{1}{4}(\MARU{1}の判別式) = 9x^2 - (-3x^2 + 4y + 4) \geqq 0 \\ \Longleftrightarrow \,\,\,{} & y \leqq 3x^2 - 1 \tag*{$\cdott\MARU{2}$} \end{align*} ゆえに求めるべき通過領域は\MARU{2}によって定まる領域である. これを図示すれば下図の斜線部分のようになる. \vspace*{-1zw} \begin{center} %\input{osaka2010l1f_zu_1} %WinTpicVersion4.23 \unitlength 0.1in \begin{picture}( 16.0000, 19.9400)( 4.0000,-23.3900) % STR 2 0 3 0 Black White % 4 1168 558 1168 579 4 1200 0 0 % $y$ \put(11.6800,-5.7900){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 Black White % 4 2000 1710 2000 1731 4 1200 0 0 % $x$ \put(20.0000,-17.3100){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 Black White % 2 1200 2339 1200 579 % {\color[named]{Black}{% \special{pn 8}% \special{pa 1200 2340}% \special{pa 1200 580}% \special{fp}% \special{sh 1}% \special{pa 1200 580}% \special{pa 1180 646}% \special{pa 1200 632}% \special{pa 1220 646}% \special{pa 1200 580}% \special{fp}% }}% % VECTOR 2 0 3 0 Black White % 2 400 1699 2000 1699 % {\color[named]{Black}{% \special{pn 8}% \special{pa 400 1700}% \special{pa 2000 1700}% \special{fp}% \special{sh 1}% \special{pa 2000 1700}% \special{pa 1934 1680}% \special{pa 1948 1700}% \special{pa 1934 1720}% \special{pa 2000 1700}% \special{fp}% }}% % CIRCLE 3 0 3 0 Black White % 4 620 480 675 725 660 945 1335 545 % {\color[named]{Black}{% \special{pn 4}% \special{ar 620 480 252 252 0.0906599 1.4849861}% }}% % FUNC 1 0 3 0 Black White % 9 400 579 2000 2339 1200 1699 1680 1699 1200 1379 400 579 2000 2339 0 2 0 0 % 3x^2-1 {\color[named]{Black}{% \special{pn 13}% \special{pa 612 580}% \special{pa 616 594}% \special{pa 636 690}% \special{pa 640 712}% \special{pa 646 736}% \special{pa 650 760}% \special{pa 656 782}% \special{pa 660 804}% \special{pa 666 826}% \special{pa 670 850}% \special{pa 676 872}% \special{pa 680 892}% \special{pa 686 914}% \special{pa 710 1020}% \special{pa 730 1100}% \special{pa 756 1194}% \special{pa 760 1212}% \special{pa 766 1232}% \special{pa 770 1250}% \special{pa 776 1266}% \special{pa 780 1284}% \special{pa 786 1302}% \special{pa 790 1320}% \special{pa 796 1336}% \special{pa 800 1352}% \special{pa 806 1370}% \special{pa 830 1450}% \special{pa 850 1510}% \special{pa 876 1580}% \special{pa 880 1592}% \special{pa 886 1606}% \special{pa 890 1620}% \special{pa 896 1632}% \special{pa 900 1644}% \special{pa 906 1656}% \special{pa 910 1670}% \special{pa 916 1682}% \special{pa 920 1692}% \special{pa 926 1704}% \special{pa 950 1760}% \special{pa 970 1800}% \special{pa 996 1844}% \special{pa 1000 1852}% 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1065 1785 1065 1835 2 0 0 0 % {\small O} \put(10.6500,-18.3500){\makebox(0,0)[lb]{{\small O}}}% \end{picture}% \end{center} \item  (1)の結果より $D : y = 3x^2 - 1$ である. $D$ と$x$軸の交点の$x$座標は \begin{gather*} 3x^2 - 1 = 0 \qquad \therefore \,\,\, x = \pm\frac{1}{\sqrt{\vphantom{b} 3}} \end{gather*} したがって $a = \dfrac{1}{\sqrt{\vphantom{b} 3}}$ であり, $\ell$ は $C$ の点$\left(\dfrac{1}{\sqrt{\vphantom{b} 3}},\ -\dfrac{4}{3} \right)$における接線である. \smallskip  $(-x^2 - 1)' = -2x$ より $\ell$ の式は \begin{gather*} y = -\frac{2}{\sqrt{\vphantom{b} 3}} \left(x - \frac{1}{\sqrt{\vphantom{b} 3}} \right) - \frac{4}{3} \qquad \therefore \,\,\, y = -\frac{2}{\sqrt{\vphantom{b} 3}}x - \frac{2}{3} \end{gather*} $D$ と $\ell$ の交点の$x$座標は, \begin{gather*} 3x^2 - 1 = -\frac{2}{\sqrt{\vphantom{b} 3}}x - \frac{2}{3} \\ \qquad 9x^2 + 2\sqrt{\vphantom{b} 3}\,x - 1 = 0 \qquad (3\sqrt{\vphantom{b} 3}\,x - 1)(\sqrt{\vphantom{b} 3}\,x + 1) = 0 \displaybreak[0] \\ \therefore \,\,\, x = \frac{1}{3\sqrt{\vphantom{b} 3}},\enskip {-\frac{1}{\sqrt{\vphantom{b} 3}}} \end{gather*} ゆえに $D$ と $\ell$ で囲まれた部分は右図の斜線部分であり, その面積は,\\ \begin{minipage}{260pt} \begin{gather*} \int_{-\frac{1}{\sqrt{3}}}^\frac{1}{3\sqrt{3}} \left\{ -\frac{2}{\sqrt{\vphantom{b} 3}}x - \frac{2}{3} - (3x^2 - 1) \right\} dx \\[1mm] = -3\int_{-\frac{1}{\sqrt{3}}}^\frac{1}{3\sqrt{3}} \left(x + \frac{1}{\sqrt{\vphantom{b} 3}} \right)\!\! \left(x - \frac{1}{3\sqrt{\vphantom{b} 3}} \right) dx \\[1mm] = -3 \cdot \left(-\frac{1}{6} \right)\! \left(\frac{1}{3\sqrt{\vphantom{b} 3}} + \frac{1}{\sqrt{\vphantom{b} 3}} \right)^{\!\! 3} \\[1mm] = {\color{red}{\boldsymbol{\frac{32}{81\sqrt{\vphantom{b} 3}}}}} \tag*{$\Ans$} \end{gather*} \end{minipage} \begin{minipage}{140pt} \vspace*{0.5zw} %\input{osaka2010l1f_zu_2} %WinTpicVersion4.10 \unitlength 0.1in \begin{picture}( 16.0000, 20.0000)( 10.0000,-24.0000) % STR 2 0 3 0 Black White % 4 1760 387 1760 400 4 2800 0 0 % $y$ \put(17.6000,-4.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 Black White % 4 2600 1027 2600 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\special{pa 2100 726}% \special{pa 2116 656}% \special{pa 2126 608}% \special{pa 2140 534}% \special{pa 2156 456}% \special{pa 2166 402}% \special{pa 2166 400}% \special{fp}% }}% % FUNC 2 0 3 0 Black White % 10 1000 400 2600 2400 1800 1000 2200 1000 1800 600 1000 400 2600 2400 0 2 0 0 0 0 % -x^2-1 {\color[named]{Black}{% \special{pn 8}% \special{pa 1168 2400}% \special{pa 1170 2392}% \special{pa 1176 2378}% \special{pa 1180 2362}% \special{pa 1186 2346}% \special{pa 1190 2330}% \special{pa 1210 2270}% \special{pa 1216 2256}% \special{pa 1220 2242}% \special{pa 1226 2228}% \special{pa 1230 2212}% \special{pa 1250 2156}% \special{pa 1256 2144}% \special{pa 1260 2130}% \special{pa 1266 2116}% \special{pa 1270 2102}% \special{pa 1290 2050}% \special{pa 1296 2038}% \special{pa 1300 2026}% \special{pa 1306 2014}% \special{pa 1310 2000}% \special{pa 1330 1952}% \special{pa 1336 1942}% \special{pa 1340 1930}% \special{pa 1346 1918}% \special{pa 1350 1906}% \special{pa 1370 1862}% \special{pa 1376 1852}% \special{pa 1380 1842}% \special{pa 1386 1832}% \special{pa 1390 1820}% \special{pa 1410 1780}% \special{pa 1416 1772}% \special{pa 1420 1762}% \special{pa 1426 1752}% \special{pa 1430 1742}% \special{pa 1450 1706}% \special{pa 1456 1698}% \special{pa 1460 1690}% \special{pa 1466 1682}% \special{pa 1470 1672}% \special{pa 1490 1640}% \special{pa 1496 1634}% \special{pa 1500 1626}% \special{pa 1506 1618}% \special{pa 1510 1610}% \special{pa 1530 1582}% \special{pa 1536 1576}% \special{pa 1540 1570}% \special{pa 1546 1564}% \special{pa 1550 1556}% \special{pa 1570 1532}% \special{pa 1576 1528}% \special{pa 1580 1522}% \special{pa 1586 1516}% \special{pa 1590 1510}% \special{pa 1610 1490}% \special{pa 1616 1486}% \special{pa 1620 1482}% \special{pa 1626 1478}% \special{pa 1630 1472}% \special{pa 1650 1456}% \special{pa 1656 1454}% \special{pa 1660 1450}% \special{pa 1666 1446}% \special{pa 1670 1442}% \special{pa 1690 1430}% \special{pa 1696 1428}% \special{pa 1700 1426}% \special{pa 1706 1424}% \special{pa 1710 1420}% \special{pa 1730 1412}% \special{pa 1736 1412}% \special{pa 1740 1410}% \special{pa 1746 1408}% \special{pa 1750 1406}% \special{pa 1770 1402}% \special{pa 1786 1402}% \special{pa 1790 1400}% \special{pa 1810 1400}% \special{pa 1816 1402}% \special{pa 1830 1402}% \special{pa 1850 1406}% \special{pa 1856 1408}% \special{pa 1860 1410}% \special{pa 1866 1412}% \special{pa 1870 1412}% \special{pa 1890 1420}% \special{pa 1896 1424}% \special{pa 1900 1426}% \special{pa 1906 1428}% \special{pa 1910 1430}% \special{pa 1930 1442}% \special{pa 1936 1446}% \special{pa 1940 1450}% \special{pa 1946 1454}% \special{pa 1950 1456}% \special{pa 1970 1472}% \special{pa 1980 1482}% \special{pa 1986 1486}% \special{pa 2016 1516}% \special{pa 2020 1522}% \special{pa 2030 1532}% \special{pa 2050 1556}% \special{pa 2056 1564}% \special{pa 2060 1570}% \special{pa 2066 1576}% \special{pa 2070 1582}% \special{pa 2090 1610}% \special{pa 2096 1618}% \special{pa 2100 1626}% \special{pa 2106 1634}% \special{pa 2110 1640}% \special{pa 2130 1672}% \special{pa 2136 1682}% \special{pa 2140 1690}% \special{pa 2146 1698}% \special{pa 2150 1706}% \special{pa 2170 1742}% \special{pa 2176 1752}% \special{pa 2180 1762}% \special{pa 2186 1772}% \special{pa 2216 1832}% \special{pa 2220 1842}% \special{pa 2226 1852}% \special{pa 2230 1862}% \special{pa 2250 1906}% \special{pa 2256 1918}% \special{pa 2260 1930}% \special{pa 2266 1942}% \special{pa 2270 1952}% \special{pa 2290 2000}% \special{pa 2296 2014}% \special{pa 2300 2026}% \special{pa 2306 2038}% \special{pa 2310 2050}% \special{pa 2330 2102}% \special{pa 2336 2116}% \special{pa 2340 2130}% \special{pa 2346 2144}% \special{pa 2350 2156}% \special{pa 2370 2212}% \special{pa 2376 2228}% \special{pa 2380 2242}% \special{pa 2386 2256}% \special{pa 2390 2270}% \special{pa 2410 2330}% \special{pa 2416 2346}% \special{pa 2420 2362}% \special{pa 2426 2378}% \special{pa 2430 2392}% \special{pa 2432 2400}% \special{fp}% }}% % FUNC 2 0 3 0 Black White % 9 1000 400 2600 2400 1800 1000 2200 1000 1800 600 1000 400 2600 2400 0 2 0 0 % -2/sqrt(3)x-2/3 {\color[named]{Black}{% \special{pn 8}% \special{pa 1050 400}% \special{pa 1056 406}% \special{pa 1076 430}% \special{pa 1080 436}% \special{pa 1096 454}% \special{pa 1100 458}% \special{pa 1116 476}% \special{pa 1120 482}% \special{pa 1140 506}% \special{pa 1146 510}% \special{pa 1160 528}% \special{pa 1166 534}% \special{pa 1186 558}% \special{pa 1190 562}% \special{pa 1206 580}% \special{pa 1210 586}% \special{pa 1226 604}% \special{pa 1230 608}% \special{pa 1250 632}% \special{pa 1256 638}% \special{pa 1270 656}% \special{pa 1276 660}% \special{pa 1296 684}% \special{pa 1300 690}% \special{pa 1316 708}% \special{pa 1320 712}% \special{pa 1340 736}% \special{pa 1346 742}% \special{pa 1360 760}% \special{pa 1366 764}% \special{pa 1380 782}% \special{pa 1386 788}% \special{pa 1406 812}% \special{pa 1410 816}% \special{pa 1426 834}% \special{pa 1430 840}% \special{pa 1450 864}% \special{pa 1456 868}% \special{pa 1470 886}% \special{pa 1476 892}% \special{pa 1490 910}% \special{pa 1496 914}% \special{pa 1516 938}% \special{pa 1520 944}% \special{pa 1536 962}% \special{pa 1540 966}% \special{pa 1560 990}% \special{pa 1566 996}% \special{pa 1580 1014}% \special{pa 1586 1018}% \special{pa 1606 1042}% \special{pa 1610 1048}% \special{pa 1626 1066}% \special{pa 1630 1070}% \special{pa 1646 1088}% \special{pa 1650 1094}% \special{pa 1670 1118}% \special{pa 1676 1122}% \special{pa 1690 1140}% \special{pa 1696 1146}% \special{pa 1716 1170}% \special{pa 1720 1174}% \special{pa 1736 1192}% \special{pa 1740 1198}% \special{pa 1756 1216}% \special{pa 1760 1220}% \special{pa 1780 1244}% \special{pa 1786 1250}% \special{pa 1800 1268}% \special{pa 1806 1272}% \special{pa 1826 1296}% \special{pa 1830 1302}% \special{pa 1846 1320}% \special{pa 1850 1324}% \special{pa 1866 1342}% \special{pa 1870 1348}% \special{pa 1890 1372}% \special{pa 1896 1376}% \special{pa 1910 1394}% \special{pa 1916 1400}% \special{pa 1936 1424}% \special{pa 1940 1428}% \special{pa 1956 1446}% \special{pa 1960 1452}% \special{pa 1980 1476}% \special{pa 1986 1480}% \special{pa 2000 1498}% \special{pa 2006 1504}% \special{pa 2020 1522}% \special{pa 2026 1526}% \special{pa 2046 1550}% \special{pa 2050 1556}% \special{pa 2066 1574}% \special{pa 2070 1578}% \special{pa 2090 1602}% \special{pa 2096 1608}% \special{pa 2110 1626}% \special{pa 2116 1630}% \special{pa 2130 1648}% \special{pa 2136 1654}% \special{pa 2156 1678}% \special{pa 2160 1682}% \special{pa 2176 1700}% \special{pa 2180 1706}% \special{pa 2200 1730}% \special{pa 2206 1734}% \special{pa 2220 1752}% \special{pa 2226 1758}% \special{pa 2246 1782}% \special{pa 2250 1786}% \special{pa 2266 1804}% \special{pa 2270 1810}% \special{pa 2286 1828}% \special{pa 2290 1832}% \special{pa 2310 1856}% \special{pa 2316 1862}% \special{pa 2330 1880}% \special{pa 2336 1884}% \special{pa 2356 1908}% \special{pa 2360 1914}% \special{pa 2376 1932}% \special{pa 2380 1936}% \special{pa 2396 1954}% \special{pa 2400 1960}% \special{pa 2420 1984}% \special{pa 2426 1988}% \special{pa 2440 2006}% \special{pa 2446 2012}% \special{pa 2466 2036}% \special{pa 2470 2040}% \special{pa 2486 2058}% \special{pa 2490 2064}% \special{pa 2510 2088}% \special{pa 2516 2092}% \special{pa 2530 2110}% \special{pa 2536 2116}% \special{pa 2550 2134}% \special{pa 2556 2138}% \special{pa 2576 2162}% \special{pa 2580 2168}% \special{pa 2596 2186}% \special{pa 2600 2190}% \special{fp}% }}% % STR 2 0 3 0 Black White % 4 1825 1105 1825 1155 2 0 0 0 % {\small O} \put(18.2500,-11.5500){\makebox(0,0)[lb]{{\small O}}}% % LINE 3 0 3 0 Black White % 40 1810 1280 1730 1360 1825 1295 1745 1375 1840 1310 1765 1385 1850 1330 1785 1395 1865 1345 1815 1395 1795 1265 1715 1345 1785 1245 1705 1325 1770 1230 1690 1310 1755 1215 1680 1290 1740 1200 1670 1270 1725 1185 1660 1250 1715 1165 1650 1230 1700 1150 1640 1210 1685 1135 1635 1185 1670 1120 1625 1165 1655 1105 1615 1145 1645 1085 1610 1120 1630 1070 1600 1100 1615 1055 1595 1075 1600 1040 1585 1055 % {\color[named]{Black}{% \special{pn 4}% \special{pa 1810 1280}% \special{pa 1730 1360}% \special{fp}% \special{pa 1826 1296}% \special{pa 1746 1376}% \special{fp}% \special{pa 1840 1310}% \special{pa 1766 1386}% \special{fp}% \special{pa 1850 1330}% \special{pa 1786 1396}% \special{fp}% \special{pa 1866 1346}% \special{pa 1816 1396}% \special{fp}% \special{pa 1796 1266}% \special{pa 1716 1346}% \special{fp}% \special{pa 1786 1246}% \special{pa 1706 1326}% \special{fp}% \special{pa 1770 1230}% \special{pa 1690 1310}% \special{fp}% \special{pa 1756 1216}% \special{pa 1680 1290}% \special{fp}% \special{pa 1740 1200}% \special{pa 1670 1270}% \special{fp}% \special{pa 1726 1186}% \special{pa 1660 1250}% \special{fp}% \special{pa 1716 1166}% \special{pa 1650 1230}% \special{fp}% \special{pa 1700 1150}% \special{pa 1640 1210}% \special{fp}% \special{pa 1686 1136}% \special{pa 1636 1186}% \special{fp}% \special{pa 1670 1120}% \special{pa 1626 1166}% \special{fp}% \special{pa 1656 1106}% \special{pa 1616 1146}% \special{fp}% \special{pa 1646 1086}% \special{pa 1610 1120}% \special{fp}% \special{pa 1630 1070}% \special{pa 1600 1100}% \special{fp}% \special{pa 1616 1056}% \special{pa 1596 1076}% \special{fp}% \special{pa 1600 1040}% \special{pa 1586 1056}% \special{fp}% }}% % STR 2 0 3 0 Black White % 4 1580 930 1580 980 2 0 0 0 % \resizebox{!}{2.6mm}{$-\dfrac{1}{\sqrt{\vphantom{b} 3}}$} \put(15.8000,-9.8000){\makebox(0,0)[lb]{\resizebox{!}{2.6mm}{$-\dfrac{1}{\sqrt{\vphantom{b} 3}}$}}}% % STR 2 0 3 0 Black White % 4 2155 565 2155 615 2 0 0 0 % {\scriptsize $D$} \put(21.5500,-6.1500){\makebox(0,0)[lb]{{\scriptsize $D$}}}% % STR 2 0 3 0 Black White % 4 1275 2170 1275 2220 2 0 0 0 % {\scriptsize $C$} \put(12.7500,-22.2000){\makebox(0,0)[lb]{{\scriptsize $C$}}}% % STR 2 0 3 0 Black White % 4 2330 1780 2330 1830 2 0 0 0 % {\scriptsize $\ell$} \put(23.3000,-18.3000){\makebox(0,0)[lb]{{\scriptsize $\ell$}}}% \end{picture}% \end{minipage} \end{enumerate} \end{document}