大阪大学 後期理系 1991年度 問4

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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 後期理系
年度 1991年度
問No 問4
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 確率 ・ 図形と方程式
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} \def\cdotss{{\cdots\cdots}} \def\cdotsss{{\cdotss\,\cdotss\,\cdotss}} \def\ans{{\cdotss\,\mbox{(答)}}} \def\cdotssp{{\cdotss\,}} \def\cdott{{\cdotss\,\cdotss\,}} \def\cdottt{{\cdott\cdott}} \def\cdotttt{{\cdott\cdott\cdott}} \def\cdottttt{{\cdott\cdott\cdott\cdott}} \def\cdotttttt{{\cdott\cdott\cdott\cdott\cdott}} \def\Ans{{\cdott(答)}} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{graphicx} \usepackage{delarray} \usepackage{multicol} \usepackage{amscd} \usepackage{pifont} \usepackage{color} \ExecuteOptions{usename} \usepackage{fancybox} \usepackage{vector3} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME} \begin{enumerate} \item[(1)]  曲線 $y = x^2$ 上の点$\mathrm{P}(a,\,\,a^2)\,\,\,(a \neq 0)$における接線と直交し,\smallskip 点Pを通る直線 $y = \fbox{ }$ と,\smallskip 直線 $y = x^2$ とのPと異なる交点Qの座標は \smallskip$\boxed{ }$ となる. 点Qを一つの頂点とし, Pにおける接線上に他の2頂点をもつ 正三角形の面積 $S$ は \smallskip\fbox{ } であり, 点Pが曲線 $y = x^2\,\,\,(x > 0)$ 上を動くとき, $S$ を最小にする $a$ の値は \fbox{ } である. \item[(2)]  赤玉1個,白玉2個,\smallskip 青玉$n$個を一列に並べる順列の総数は \fbox{ } である.  いま,赤玉1個,白玉2個,青玉$n$個のはいった箱から無作為に 玉を1個取り出し, 箱に戻すという操作を$n+3$回くり返す. このとき赤玉が1回,白玉が2回,\smallskip 青玉が$n$回取り出される確率を $P_n$ とすると, $P_n = \fbox{ }$ であり, $\lim\limits_{n \to \infty} P_n = \fbox{ }$ となる. \end{enumerate} \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $(x^2)' = 2x$ よりPにおける $y = x^2$ の接線は \begin{gather*} y = 2a(x - a) + a^2 \qquad \therefore \,\,\, y = 2ax - a^2 \tag*{$\cdott\MARU{1}$} \end{gather*} またPを通り\MARU{1}に直交する直線は, \begin{gather*} y = -\frac{1}{2a}(x - a) + a^2 \qquad \therefore \,\,\, \color{red}{\boldsymbol{ y = -\frac{1}{2a}x + a^2 + \frac{1}{2} }} \tag*{$\ans\,\,\,\MARU{2}$} \end{gather*} Qは $y = x^2$ と\MARU{2}のPと異なる交点だから, Qの$x$座標 $q$ は \begin{gather*} x^2 = -\frac{1}{2a}x + a^2 + \frac{1}{2} \qquad \therefore \,\,\, x^2 + \frac{1}{2a}x - a^2 - \frac{1}{2} = 0 \end{gather*} の $a$ と異なる実数解である. 解と係数の関係より \begin{gather*} q + a = -\frac{1}{2a} \qquad \therefore \,\,\, q = -a - \frac{1}{2a} \end{gather*} ゆえに \begin{gather*} \color{red}{\boldsymbol{ \Q\!\left(-a - \frac{1}{2a},\,\, \left(a + \frac{1}{2a} \right)^{\!\! 2} \right) }} \tag*{$\Ans$} \end{gather*} PQを用いて $S$ を表すと, \begin{align*} S = \frac{1}{2} \cdot \mathrm{P}\Q \cdot \frac{2}{\sqrt{\vphantom{b} 3}} \mathrm{P}\Q = \frac{1}{\sqrt{\vphantom{b} 3}}\mathrm{P}\Q^2 \end{align*} \vskip 2mm \begin{center} %\input{osaka91t4_1s_zu_3} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 53.3800, 18.9500)( 1.7000,-22.9500) % STR 2 0 3 0 % 3 1070 2118 1070 2130 2 0 % {\small O} \put(10.7000,-21.3000){\makebox(0,0)[lb]{{\small O}}}% % STR 2 0 3 0 % 3 1167 387 1167 400 4 3000 % $y$ \put(11.6700,-4.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2070 2028 2070 2040 4 3000 % $x$ \put(20.7000,-20.4000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1196 2295 1196 400 % \special{pn 8}% \special{pa 1196 2296}% \special{pa 1196 400}% \special{fp}% \special{sh 1}% \special{pa 1196 400}% \special{pa 1176 468}% \special{pa 1196 454}% \special{pa 1216 468}% \special{pa 1196 400}% \special{fp}% % VECTOR 2 0 3 0 % 2 321 2004 2071 2004 % \special{pn 8}% \special{pa 322 2004}% \special{pa 2072 2004}% \special{fp}% \special{sh 1}% \special{pa 2072 2004}% \special{pa 2004 1984}% \special{pa 2018 2004}% \special{pa 2004 2024}% \special{pa 2072 2004}% \special{fp}% % FUNC 2 0 3 0 % 10 321 400 2071 2295 1196 2004 1342 2004 1196 1858 321 400 2071 2295 0 3 0 0 1 0 % 1/2x^2 \special{pn 8}% \special{pa 512 400}% \special{pa 516 416}% \special{pa 520 440}% \special{pa 526 462}% \special{pa 530 486}% \special{pa 536 508}% \special{pa 540 530}% \special{pa 546 554}% \special{pa 550 576}% \special{pa 556 598}% \special{pa 560 620}% \special{pa 566 640}% \special{pa 570 662}% \special{pa 576 684}% \special{pa 580 704}% \special{pa 586 726}% \special{pa 590 746}% \special{pa 596 768}% \special{pa 600 788}% \special{pa 606 808}% \special{pa 610 828}% \special{pa 616 848}% \special{pa 620 868}% \special{pa 626 888}% \special{pa 630 908}% \special{pa 636 926}% \special{pa 640 946}% \special{pa 646 964}% \special{pa 650 984}% \special{pa 656 1002}% \special{pa 660 1020}% \special{pa 666 1038}% \special{pa 670 1056}% \special{pa 676 1074}% \special{pa 680 1092}% \special{pa 686 1110}% \special{pa 690 1128}% \special{pa 696 1144}% \special{pa 700 1162}% \special{pa 706 1178}% \special{pa 710 1196}% \special{pa 716 1212}% \special{pa 720 1228}% \special{pa 726 1244}% \special{pa 730 1260}% \special{pa 736 1276}% \special{pa 740 1292}% \special{pa 746 1308}% \special{pa 750 1324}% \special{pa 756 1338}% \special{pa 760 1354}% \special{pa 766 1368}% \special{pa 770 1384}% \special{pa 776 1398}% \special{pa 780 1412}% \special{pa 786 1426}% \special{pa 790 1440}% \special{pa 796 1454}% \special{pa 800 1468}% \special{pa 806 1480}% \special{pa 810 1494}% \special{pa 816 1508}% \special{pa 820 1520}% \special{pa 826 1534}% \special{pa 830 1546}% \special{pa 836 1558}% \special{pa 840 1570}% \special{pa 846 1582}% \special{pa 850 1594}% \special{pa 856 1606}% \special{pa 860 1618}% \special{pa 866 1630}% \special{pa 870 1640}% \special{pa 876 1652}% \special{pa 880 1662}% \special{pa 886 1674}% \special{pa 890 1684}% \special{pa 896 1694}% \special{pa 900 1704}% \special{pa 906 1714}% \special{pa 910 1724}% \special{pa 916 1734}% \special{pa 920 1744}% \special{pa 926 1752}% \special{pa 930 1762}% \special{pa 936 1772}% \special{pa 940 1780}% \special{pa 946 1788}% \special{pa 950 1798}% \special{pa 956 1806}% \special{pa 960 1814}% \special{pa 966 1822}% \special{pa 970 1830}% \special{pa 976 1838}% \special{pa 980 1844}% \special{pa 986 1852}% \special{pa 990 1860}% \special{pa 996 1866}% \special{pa 1000 1872}% \special{pa 1006 1880}% \special{pa 1010 1886}% \special{pa 1016 1892}% \special{pa 1020 1898}% \special{pa 1026 1904}% \special{pa 1030 1910}% \special{pa 1036 1916}% \special{pa 1040 1922}% \special{pa 1046 1926}% \special{pa 1050 1932}% \special{pa 1056 1936}% \special{pa 1060 1942}% \special{pa 1066 1946}% \special{pa 1070 1950}% \special{pa 1076 1954}% \special{pa 1080 1958}% \special{pa 1086 1962}% \special{pa 1090 1966}% \special{pa 1096 1970}% \special{pa 1100 1972}% \special{pa 1106 1976}% \special{pa 1110 1980}% \special{pa 1116 1982}% \special{pa 1120 1984}% \special{pa 1126 1988}% \special{pa 1130 1990}% \special{pa 1136 1992}% \special{pa 1140 1994}% \special{pa 1146 1996}% \special{pa 1150 1998}% \special{pa 1156 1998}% \special{pa 1160 2000}% \special{pa 1166 2002}% \special{pa 1170 2002}% \special{pa 1176 2002}% \special{pa 1180 2004}% \special{pa 1186 2004}% \special{pa 1190 2004}% \special{pa 1196 2004}% \special{pa 1200 2004}% \special{pa 1206 2004}% \special{pa 1210 2004}% \special{pa 1216 2004}% \special{pa 1220 2002}% \special{pa 1226 2002}% \special{pa 1230 2000}% \special{pa 1236 2000}% \special{pa 1240 1998}% \special{pa 1246 1996}% \special{pa 1250 1994}% \special{pa 1256 1992}% \special{pa 1260 1990}% \special{pa 1266 1988}% \special{pa 1270 1986}% \special{pa 1276 1984}% \special{pa 1280 1980}% \special{pa 1286 1978}% \special{pa 1290 1974}% \special{pa 1296 1970}% \special{pa 1300 1968}% \special{pa 1306 1964}% \special{pa 1310 1960}% \special{pa 1316 1956}% \special{pa 1320 1952}% \special{pa 1326 1948}% \special{pa 1330 1944}% \special{pa 1336 1938}% \special{pa 1340 1934}% \special{pa 1346 1928}% \special{pa 1350 1924}% \special{pa 1356 1918}% \special{pa 1360 1912}% \special{pa 1366 1906}% \special{pa 1370 1900}% \special{pa 1376 1894}% \special{pa 1380 1888}% \special{pa 1386 1882}% \special{pa 1390 1876}% \special{pa 1396 1868}% \special{pa 1400 1862}% \special{pa 1406 1854}% \special{pa 1410 1848}% \special{pa 1416 1840}% \special{pa 1420 1832}% \special{pa 1426 1824}% \special{pa 1430 1816}% \special{pa 1436 1808}% \special{pa 1440 1800}% \special{pa 1446 1792}% \special{pa 1450 1784}% \special{pa 1456 1774}% \special{pa 1460 1766}% \special{pa 1466 1756}% \special{pa 1470 1748}% \special{pa 1476 1738}% \special{pa 1480 1728}% \special{pa 1486 1718}% \special{pa 1490 1708}% \special{pa 1496 1698}% \special{pa 1500 1688}% \special{pa 1506 1678}% \special{pa 1510 1666}% \special{pa 1516 1656}% \special{pa 1520 1644}% \special{pa 1526 1634}% \special{pa 1530 1622}% \special{pa 1536 1610}% \special{pa 1540 1600}% \special{pa 1546 1588}% \special{pa 1550 1576}% \special{pa 1556 1564}% \special{pa 1560 1550}% \special{pa 1566 1538}% \special{pa 1570 1526}% \special{pa 1576 1512}% \special{pa 1580 1500}% \special{pa 1586 1486}% \special{pa 1590 1472}% \special{pa 1596 1460}% \special{pa 1600 1446}% \special{pa 1606 1432}% \special{pa 1610 1418}% \special{pa 1616 1404}% \special{pa 1620 1388}% \special{pa 1626 1374}% \special{pa 1630 1360}% \special{pa 1636 1344}% \special{pa 1640 1330}% \special{pa 1646 1314}% \special{pa 1650 1298}% \special{pa 1656 1282}% \special{pa 1660 1268}% \special{pa 1666 1252}% \special{pa 1670 1236}% \special{pa 1676 1218}% \special{pa 1680 1202}% \special{pa 1686 1186}% \special{pa 1690 1168}% \special{pa 1696 1152}% \special{pa 1700 1134}% \special{pa 1706 1118}% \special{pa 1710 1100}% \special{pa 1716 1082}% \special{pa 1720 1064}% \special{pa 1726 1046}% \special{pa 1730 1028}% \special{pa 1736 1010}% \special{pa 1740 992}% \special{pa 1746 972}% \special{pa 1750 954}% \special{pa 1756 934}% \special{pa 1760 916}% \special{pa 1766 896}% \special{pa 1770 876}% \special{pa 1776 856}% \special{pa 1780 836}% \special{pa 1786 816}% \special{pa 1790 796}% \special{pa 1796 776}% \special{pa 1800 756}% \special{pa 1806 734}% \special{pa 1810 714}% \special{pa 1816 692}% \special{pa 1820 672}% \special{pa 1826 650}% \special{pa 1830 628}% \special{pa 1836 606}% \special{pa 1840 584}% \special{pa 1846 562}% \special{pa 1850 540}% \special{pa 1856 518}% \special{pa 1860 494}% \special{pa 1866 472}% \special{pa 1870 448}% \special{pa 1876 426}% \special{pa 1880 402}% \special{pa 1880 400}% \special{sp}% % LINE 2 0 3 0 % 2 2005 400 1274 2295 % \special{pn 8}% \special{pa 2006 400}% \special{pa 1274 2296}% \special{fp}% % DOT 0 0 3 0 % 1 1565 1541 % \special{pn 20}% \special{sh 1}% \special{ar 1566 1542 10 10 0 6.28318530717959E+0000}% % LINE 2 0 3 0 % 2 321 1061 2071 1738 % \special{pn 8}% \special{pa 322 1062}% \special{pa 2072 1738}% \special{fp}% % LINE 2 0 3 0 % 2 1754 1049 715 1214 % \special{pn 8}% \special{pa 1754 1050}% \special{pa 716 1214}% \special{fp}% % LINE 2 0 3 0 % 2 715 1214 1375 2030 % \special{pn 8}% \special{pa 716 1214}% \special{pa 1376 2030}% \special{fp}% % DOT 0 0 3 0 % 1 715 1214 % \special{pn 20}% \special{sh 1}% \special{ar 716 1214 10 10 0 6.28318530717959E+0000}% % POLYLINE 2 0 3 0 % 4 1587 1483 1525 1459 1501 1521 1501 1521 % \special{pn 8}% \special{pa 1588 1484}% \special{pa 1526 1460}% \special{pa 1502 1522}% \special{pa 1502 1522}% \special{fp}% % STR 2 0 3 0 % 3 1630 1503 1630 1540 2 0 % {\footnotesize P} \put(16.3000,-15.4000){\makebox(0,0)[lb]{{\footnotesize P}}}% % STR 2 0 3 0 % 3 590 1333 590 1370 2 0 % {\footnotesize Q} \put(5.9000,-13.7000){\makebox(0,0)[lb]{{\footnotesize Q}}}% % STR 2 0 3 0 % 3 1982 554 1982 635 2 0 % {\footnotesize \MARU{1}} \put(19.8200,-6.3500){\makebox(0,0)[lb]{{\footnotesize \MARU{1}}}}% % STR 2 0 3 0 % 3 170 999 170 1080 2 0 % {\footnotesize \MARU{2}} \put(1.7000,-10.8000){\makebox(0,0)[lb]{{\footnotesize \MARU{2}}}}% % LINE 2 2 3 0 % 2 715 1540 1565 1540 % \special{pn 8}% \special{pa 716 1540}% \special{pa 1566 1540}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 715 1540 715 1215 % \special{pn 8}% \special{pa 716 1540}% \special{pa 716 1216}% \special{dt 0.045}% % POLYLINE 2 0 3 0 % 4 715 1470 785 1470 785 1540 785 1540 % \special{pn 8}% \special{pa 716 1470}% \special{pa 786 1470}% \special{pa 786 1540}% \special{pa 786 1540}% \special{fp}% % DOT 0 0 3 0 % 1 5077 1661 % \special{pn 20}% \special{sh 1}% \special{ar 5078 1662 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 3802 1171 % \special{pn 20}% \special{sh 1}% \special{ar 3802 1172 10 10 0 6.28318530717959E+0000}% % STR 2 0 3 0 % 3 5090 1565 5090 1620 2 0 % {\footnotesize P} \put(50.9000,-16.2000){\makebox(0,0)[lb]{{\footnotesize P}}}% % STR 2 0 3 0 % 3 3770 1064 3770 1120 2 0 % {\footnotesize Q} \put(37.7000,-11.2000){\makebox(0,0)[lb]{{\footnotesize Q}}}% % LINE 2 2 3 0 % 2 3802 1659 5077 1659 % \special{pn 8}% \special{pa 3802 1660}% \special{pa 5078 1660}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 3802 1659 3802 1172 % \special{pn 8}% \special{pa 3802 1660}% \special{pa 3802 1172}% \special{dt 0.045}% % POLYLINE 2 0 3 0 % 4 3802 1555 3907 1555 3907 1659 3907 1659 % \special{pn 8}% \special{pa 3802 1556}% \special{pa 3908 1556}% \special{pa 3908 1660}% \special{pa 3908 1660}% \special{fp}% % STR 2 0 3 0 % 3 4210 1820 4210 1920 2 0 % {\scriptsize$\zettaiti{a-q}$} \put(42.1000,-19.2000){\makebox(0,0)[lb]{{\scriptsize$\zettaiti{a-q}$}}}% % STR 2 0 3 0 % 3 3180 1350 3180 1450 2 0 % {\scriptsize$\frac{1}{2a}\zettaiti{a-q}$} \put(31.8000,-14.5000){\makebox(0,0)[lb]{{\scriptsize$\frac{1}{2a}\zettaiti{a-q}$}}}% % ELLIPSE 3 0 3 0 % 4 4443 1073 5706 1760 3439 2001 5557 2110 % \special{pn 4}% \special{ar 4444 1074 1264 688 1.0418741 2.1027253}% % CIRCLE 3 0 3 0 % 4 4360 1411 4530 2001 3360 991 3240 1891 % \special{pn 4}% \special{ar 4360 1412 614 614 2.7367009 3.5392206}% % LINE 0 0 3 0 % 2 3800 1171 5075 1661 % \special{pn 20}% \special{pa 3800 1172}% \special{pa 5076 1662}% \special{fp}% % STR 2 0 3 0 % 3 4150 1160 4150 1260 2 0 % {\scriptsize$\sqrt{1+\frac{1}{4a^2}}\zettaiti{a-q}$} \put(41.5000,-12.6000){\makebox(0,0)[lb]{{\scriptsize$\sqrt{1+\frac{1}{4a^2}}\zettaiti{a-q}$}}}% % ELLIPSE 2 2 3 0 % 4 1150 1360 1830 1750 1830 1750 1830 1750 % \special{pn 8}% \special{ar 1150 1360 680 390 0.0000000 0.0224299}% \special{ar 1150 1360 680 390 0.0897196 0.1121495}% \special{ar 1150 1360 680 390 0.1794393 0.2018692}% \special{ar 1150 1360 680 390 0.2691589 0.2915888}% \special{ar 1150 1360 680 390 0.3588785 0.3813084}% \special{ar 1150 1360 680 390 0.4485981 0.4710280}% \special{ar 1150 1360 680 390 0.5383178 0.5607477}% \special{ar 1150 1360 680 390 0.6280374 0.6504673}% \special{ar 1150 1360 680 390 0.7177570 0.7401869}% \special{ar 1150 1360 680 390 0.8074766 0.8299065}% \special{ar 1150 1360 680 390 0.8971963 0.9196262}% \special{ar 1150 1360 680 390 0.9869159 1.0093458}% \special{ar 1150 1360 680 390 1.0766355 1.0990654}% \special{ar 1150 1360 680 390 1.1663551 1.1887850}% \special{ar 1150 1360 680 390 1.2560748 1.2785047}% \special{ar 1150 1360 680 390 1.3457944 1.3682243}% \special{ar 1150 1360 680 390 1.4355140 1.4579439}% \special{ar 1150 1360 680 390 1.5252336 1.5476636}% \special{ar 1150 1360 680 390 1.6149533 1.6373832}% \special{ar 1150 1360 680 390 1.7046729 1.7271028}% \special{ar 1150 1360 680 390 1.7943925 1.8168224}% \special{ar 1150 1360 680 390 1.8841121 1.9065421}% \special{ar 1150 1360 680 390 1.9738318 1.9962617}% \special{ar 1150 1360 680 390 2.0635514 2.0859813}% \special{ar 1150 1360 680 390 2.1532710 2.1757009}% \special{ar 1150 1360 680 390 2.2429907 2.2654206}% \special{ar 1150 1360 680 390 2.3327103 2.3551402}% \special{ar 1150 1360 680 390 2.4224299 2.4448598}% \special{ar 1150 1360 680 390 2.5121495 2.5345794}% \special{ar 1150 1360 680 390 2.6018692 2.6242991}% \special{ar 1150 1360 680 390 2.6915888 2.7140187}% \special{ar 1150 1360 680 390 2.7813084 2.8037383}% \special{ar 1150 1360 680 390 2.8710280 2.8934579}% \special{ar 1150 1360 680 390 2.9607477 2.9831776}% \special{ar 1150 1360 680 390 3.0504673 3.0728972}% \special{ar 1150 1360 680 390 3.1401869 3.1626168}% \special{ar 1150 1360 680 390 3.2299065 3.2523364}% \special{ar 1150 1360 680 390 3.3196262 3.3420561}% \special{ar 1150 1360 680 390 3.4093458 3.4317757}% \special{ar 1150 1360 680 390 3.4990654 3.5214953}% \special{ar 1150 1360 680 390 3.5887850 3.6112150}% \special{ar 1150 1360 680 390 3.6785047 3.7009346}% \special{ar 1150 1360 680 390 3.7682243 3.7906542}% \special{ar 1150 1360 680 390 3.8579439 3.8803738}% \special{ar 1150 1360 680 390 3.9476636 3.9700935}% \special{ar 1150 1360 680 390 4.0373832 4.0598131}% \special{ar 1150 1360 680 390 4.1271028 4.1495327}% \special{ar 1150 1360 680 390 4.2168224 4.2392523}% \special{ar 1150 1360 680 390 4.3065421 4.3289720}% \special{ar 1150 1360 680 390 4.3962617 4.4186916}% \special{ar 1150 1360 680 390 4.4859813 4.5084112}% \special{ar 1150 1360 680 390 4.5757009 4.5981308}% \special{ar 1150 1360 680 390 4.6654206 4.6878505}% \special{ar 1150 1360 680 390 4.7551402 4.7775701}% \special{ar 1150 1360 680 390 4.8448598 4.8672897}% \special{ar 1150 1360 680 390 4.9345794 4.9570093}% \special{ar 1150 1360 680 390 5.0242991 5.0467290}% \special{ar 1150 1360 680 390 5.1140187 5.1364486}% \special{ar 1150 1360 680 390 5.2037383 5.2261682}% \special{ar 1150 1360 680 390 5.2934579 5.3158879}% \special{ar 1150 1360 680 390 5.3831776 5.4056075}% \special{ar 1150 1360 680 390 5.4728972 5.4953271}% \special{ar 1150 1360 680 390 5.5626168 5.5850467}% \special{ar 1150 1360 680 390 5.6523364 5.6747664}% \special{ar 1150 1360 680 390 5.7420561 5.7644860}% \special{ar 1150 1360 680 390 5.8317757 5.8542056}% \special{ar 1150 1360 680 390 5.9214953 5.9439252}% \special{ar 1150 1360 680 390 6.0112150 6.0336449}% \special{ar 1150 1360 680 390 6.1009346 6.1233645}% \special{ar 1150 1360 680 390 6.1906542 6.2130841}% \special{ar 1150 1360 680 390 6.2803738 6.2832853}% % ELLIPSE 2 2 3 0 % 4 4284 1402 5508 2104 5508 2104 5508 2104 % \special{pn 8}% \special{ar 4284 1402 1224 702 0.0000000 0.0124611}% \special{ar 4284 1402 1224 702 0.0498442 0.0623053}% \special{ar 4284 1402 1224 702 0.0996885 0.1121495}% \special{ar 4284 1402 1224 702 0.1495327 0.1619938}% \special{ar 4284 1402 1224 702 0.1993769 0.2118380}% \special{ar 4284 1402 1224 702 0.2492212 0.2616822}% \special{ar 4284 1402 1224 702 0.2990654 0.3115265}% \special{ar 4284 1402 1224 702 0.3489097 0.3613707}% \special{ar 4284 1402 1224 702 0.3987539 0.4112150}% \special{ar 4284 1402 1224 702 0.4485981 0.4610592}% \special{ar 4284 1402 1224 702 0.4984424 0.5109034}% \special{ar 4284 1402 1224 702 0.5482866 0.5607477}% \special{ar 4284 1402 1224 702 0.5981308 0.6105919}% \special{ar 4284 1402 1224 702 0.6479751 0.6604361}% \special{ar 4284 1402 1224 702 0.6978193 0.7102804}% \special{ar 4284 1402 1224 702 0.7476636 0.7601246}% \special{ar 4284 1402 1224 702 0.7975078 0.8099688}% \special{ar 4284 1402 1224 702 0.8473520 0.8598131}% \special{ar 4284 1402 1224 702 0.8971963 0.9096573}% \special{ar 4284 1402 1224 702 0.9470405 0.9595016}% \special{ar 4284 1402 1224 702 0.9968847 1.0093458}% \special{ar 4284 1402 1224 702 1.0467290 1.0591900}% \special{ar 4284 1402 1224 702 1.0965732 1.1090343}% \special{ar 4284 1402 1224 702 1.1464174 1.1588785}% \special{ar 4284 1402 1224 702 1.1962617 1.2087227}% \special{ar 4284 1402 1224 702 1.2461059 1.2585670}% \special{ar 4284 1402 1224 702 1.2959502 1.3084112}% \special{ar 4284 1402 1224 702 1.3457944 1.3582555}% \special{ar 4284 1402 1224 702 1.3956386 1.4080997}% \special{ar 4284 1402 1224 702 1.4454829 1.4579439}% \special{ar 4284 1402 1224 702 1.4953271 1.5077882}% \special{ar 4284 1402 1224 702 1.5451713 1.5576324}% \special{ar 4284 1402 1224 702 1.5950156 1.6074766}% \special{ar 4284 1402 1224 702 1.6448598 1.6573209}% \special{ar 4284 1402 1224 702 1.6947040 1.7071651}% \special{ar 4284 1402 1224 702 1.7445483 1.7570093}% \special{ar 4284 1402 1224 702 1.7943925 1.8068536}% \special{ar 4284 1402 1224 702 1.8442368 1.8566978}% \special{ar 4284 1402 1224 702 1.8940810 1.9065421}% \special{ar 4284 1402 1224 702 1.9439252 1.9563863}% \special{ar 4284 1402 1224 702 1.9937695 2.0062305}% \special{ar 4284 1402 1224 702 2.0436137 2.0560748}% \special{ar 4284 1402 1224 702 2.0934579 2.1059190}% \special{ar 4284 1402 1224 702 2.1433022 2.1557632}% \special{ar 4284 1402 1224 702 2.1931464 2.2056075}% \special{ar 4284 1402 1224 702 2.2429907 2.2554517}% \special{ar 4284 1402 1224 702 2.2928349 2.3052960}% \special{ar 4284 1402 1224 702 2.3426791 2.3551402}% \special{ar 4284 1402 1224 702 2.3925234 2.4049844}% \special{ar 4284 1402 1224 702 2.4423676 2.4548287}% \special{ar 4284 1402 1224 702 2.4922118 2.5046729}% \special{ar 4284 1402 1224 702 2.5420561 2.5545171}% \special{ar 4284 1402 1224 702 2.5919003 2.6043614}% \special{ar 4284 1402 1224 702 2.6417445 2.6542056}% \special{ar 4284 1402 1224 702 2.6915888 2.7040498}% \special{ar 4284 1402 1224 702 2.7414330 2.7538941}% \special{ar 4284 1402 1224 702 2.7912773 2.8037383}% \special{ar 4284 1402 1224 702 2.8411215 2.8535826}% \special{ar 4284 1402 1224 702 2.8909657 2.9034268}% \special{ar 4284 1402 1224 702 2.9408100 2.9532710}% \special{ar 4284 1402 1224 702 2.9906542 3.0031153}% \special{ar 4284 1402 1224 702 3.0404984 3.0529595}% \special{ar 4284 1402 1224 702 3.0903427 3.1028037}% \special{ar 4284 1402 1224 702 3.1401869 3.1526480}% \special{ar 4284 1402 1224 702 3.1900312 3.2024922}% \special{ar 4284 1402 1224 702 3.2398754 3.2523364}% \special{ar 4284 1402 1224 702 3.2897196 3.3021807}% \special{ar 4284 1402 1224 702 3.3395639 3.3520249}% \special{ar 4284 1402 1224 702 3.3894081 3.4018692}% \special{ar 4284 1402 1224 702 3.4392523 3.4517134}% \special{ar 4284 1402 1224 702 3.4890966 3.5015576}% \special{ar 4284 1402 1224 702 3.5389408 3.5514019}% \special{ar 4284 1402 1224 702 3.5887850 3.6012461}% \special{ar 4284 1402 1224 702 3.6386293 3.6510903}% \special{ar 4284 1402 1224 702 3.6884735 3.7009346}% \special{ar 4284 1402 1224 702 3.7383178 3.7507788}% \special{ar 4284 1402 1224 702 3.7881620 3.8006231}% \special{ar 4284 1402 1224 702 3.8380062 3.8504673}% \special{ar 4284 1402 1224 702 3.8878505 3.9003115}% \special{ar 4284 1402 1224 702 3.9376947 3.9501558}% \special{ar 4284 1402 1224 702 3.9875389 4.0000000}% \special{ar 4284 1402 1224 702 4.0373832 4.0498442}% \special{ar 4284 1402 1224 702 4.0872274 4.0996885}% \special{ar 4284 1402 1224 702 4.1370717 4.1495327}% \special{ar 4284 1402 1224 702 4.1869159 4.1993769}% \special{ar 4284 1402 1224 702 4.2367601 4.2492212}% \special{ar 4284 1402 1224 702 4.2866044 4.2990654}% \special{ar 4284 1402 1224 702 4.3364486 4.3489097}% \special{ar 4284 1402 1224 702 4.3862928 4.3987539}% \special{ar 4284 1402 1224 702 4.4361371 4.4485981}% \special{ar 4284 1402 1224 702 4.4859813 4.4984424}% \special{ar 4284 1402 1224 702 4.5358255 4.5482866}% \special{ar 4284 1402 1224 702 4.5856698 4.5981308}% \special{ar 4284 1402 1224 702 4.6355140 4.6479751}% \special{ar 4284 1402 1224 702 4.6853583 4.6978193}% \special{ar 4284 1402 1224 702 4.7352025 4.7476636}% \special{ar 4284 1402 1224 702 4.7850467 4.7975078}% \special{ar 4284 1402 1224 702 4.8348910 4.8473520}% \special{ar 4284 1402 1224 702 4.8847352 4.8971963}% \special{ar 4284 1402 1224 702 4.9345794 4.9470405}% \special{ar 4284 1402 1224 702 4.9844237 4.9968847}% \special{ar 4284 1402 1224 702 5.0342679 5.0467290}% \special{ar 4284 1402 1224 702 5.0841121 5.0965732}% \special{ar 4284 1402 1224 702 5.1339564 5.1464174}% \special{ar 4284 1402 1224 702 5.1838006 5.1962617}% \special{ar 4284 1402 1224 702 5.2336449 5.2461059}% \special{ar 4284 1402 1224 702 5.2834891 5.2959502}% \special{ar 4284 1402 1224 702 5.3333333 5.3457944}% \special{ar 4284 1402 1224 702 5.3831776 5.3956386}% \special{ar 4284 1402 1224 702 5.4330218 5.4454829}% \special{ar 4284 1402 1224 702 5.4828660 5.4953271}% \special{ar 4284 1402 1224 702 5.5327103 5.5451713}% \special{ar 4284 1402 1224 702 5.5825545 5.5950156}% \special{ar 4284 1402 1224 702 5.6323988 5.6448598}% \special{ar 4284 1402 1224 702 5.6822430 5.6947040}% \special{ar 4284 1402 1224 702 5.7320872 5.7445483}% \special{ar 4284 1402 1224 702 5.7819315 5.7943925}% \special{ar 4284 1402 1224 702 5.8317757 5.8442368}% \special{ar 4284 1402 1224 702 5.8816199 5.8940810}% \special{ar 4284 1402 1224 702 5.9314642 5.9439252}% \special{ar 4284 1402 1224 702 5.9813084 5.9937695}% \special{ar 4284 1402 1224 702 6.0311526 6.0436137}% \special{ar 4284 1402 1224 702 6.0809969 6.0934579}% \special{ar 4284 1402 1224 702 6.1308411 6.1433022}% \special{ar 4284 1402 1224 702 6.1806854 6.1931464}% \special{ar 4284 1402 1224 702 6.2305296 6.2429907}% \special{ar 4284 1402 1224 702 6.2803738 6.2832853}% % CIRCLE 3 0 3 0 % 4 2540 1860 2920 2740 3250 710 970 570 % \special{pn 4}% \special{ar 2540 1860 960 960 3.8293997 5.2654982}% % VECTOR 3 0 3 0 % 2 3045 1045 3155 1115 % \special{pn 4}% \special{pa 3046 1046}% \special{pa 3156 1116}% \special{fp}% \special{sh 1}% \special{pa 3156 1116}% \special{pa 3110 1062}% \special{pa 3110 1086}% \special{pa 3088 1096}% \special{pa 3156 1116}% \special{fp}% % STR 2 0 3 0 % 3 2400 1010 2400 1060 2 0 % {\scriptsize 拡大} \put(24.0000,-10.6000){\makebox(0,0)[lb]{{\scriptsize 拡大}}}% \end{picture}% \end{center} \vspace{-2mm} ここで \begin{align*} \mathrm{P}\Q &= \sqrt{\vphantom{b} 1 + \left(-\frac{1}{2a} \right)^{\!\! 2}} \zettaiti{a - q} = \sqrt{\vphantom{b} 1 + \frac{1}{4a^2}} \left(2a + \frac{1}{2a} \right) \\[1mm] &= 2a\!\left(1 + \frac{1}{4a^2} \right)^{\!\!\frac{3}{2}} \end{align*} $u = \dfrac{1}{4a^2}$ とおく.\smallskip Pが $y = x^2\,\,\,(x > 0)$ 上を動くとき $a > 0$. よって $u > 0$ であり, \begin{align*} S = \frac{1}{\sqrt{\vphantom{b} 3}} \cdot 4a^2\!\left(1 + \frac{1}{4a^2} \right)^{\!\!3} = \frac{1}{\sqrt{\vphantom{b} 3}} \cdot \frac{1}{u}(1 + u)^3 \end{align*} $f(u) = \dfrac{(1 + u)^3}{u}$ とおく.\\ \begin{minipage}{100pt} \begin{align*} f'(u) &= \frac{3(1 + u)^2u - (1 + u)^3}{u^2} \\[1mm] &= \frac{(1 + u)^2\{3u - (1 + u)\}}{u^2} \\[1mm] &= \frac{(1 + u)^2(2u - 1)}{u^2} \end{align*} \end{minipage} \begin{minipage}{100pt} \begin{align*} \begin{array}{|c||c|c|c|c|} \hline u & (0) & \cdots & 1 \slash 2 & \cdots \\ \hline f'(u) & & - & 0 & + \\ \hline f(u) & & \searrow & 最小 & \nearrow \\ \hline \end{array} \end{align*} \end{minipage} \vskip 0.5zw \noindent 増減表より \smallskip$u = \dfrac{1}{2}$ で $f(u)$ および $S$ は最小になる. このときの $a$ の値が求めるべきもので, \begin{gather*} \frac{1}{4a^2} = \frac{1}{2} \qquad \therefore \,\,\, a = \color{red}{\boldsymbol{\frac{1}{\sqrt{\vphantom{b} 2}}}} \tag*{$\Ans$} \end{gather*} \newpage \item  赤玉1個,白玉2個, 青玉$n$個を一列に並べる順列の総数は \begin{align*} {}_{n+3}\C_1 \cdot {}_{n+2}\C_2 &= (n+3) \cdot \frac{(n+2)(n+1)}{2} \\[1mm] &= \color{red}{\boldsymbol{\frac{(n+3)(n+2)(n+1)}{2}}} \tag*{$\Ans$} \end{align*}  赤玉,白玉,青玉を取り出す確率はそれぞれ \smallskip$\dfrac{1}{n+3},\,\, \dfrac{2}{n+3},\,\,\dfrac{n}{n+3}$ で一定である. したがって, \begin{align*} P_n &= {}_{n+3}\C_1 \cdot {}_{n+2}\C_2\, \frac{1}{n+3}\! \left(\frac{2}{n+3} \right)^{\!\! 2}\! \left(\frac{n}{n+3} \right)^{\!\! n} \\[1mm] &= \frac{(n+3)(n+2)(n+1)}{2} \cdot \frac{1}{n+3}\! \left(\frac{2}{n+3} \right)^{\!\! 2}\! \left(\frac{n}{n+3} \right)^{\!\! n} \tag*{$\cdott\MARU{1}$} \\[1mm] &= \color{red}{\boldsymbol{\frac{2(n+2)(n+1)n^n}{(n+3)^{n+2}}}} \tag*{$\Ans$} \end{align*} \MARU{1}より $\left(\dfrac{n}{n+3} \right)^{\!\! n} = \dfrac{1}{\left(1 + \dfrac{3}{n} \right)^{\!\! n}} = \dfrac{1}{\left\{\left(1 + \dfrac{3}{n} \right)^{\!\! \frac{n}{3}} \right\}^{\!3}}$ から \begin{align*} \lim_{n \to \infty} P_n &= \lim_{n \to \infty} \frac{\,2\!\left(1+\dfrac{1}{n} \right)\!\! \left(1+\dfrac{2}{n}\right)\,} {\left(1+\dfrac{3}{n}\right)^{\!\!2}} \cdot \frac{1}{ \left\{\left(1 + \dfrac{3}{n} \right)^{\!\!\frac{n}{3}} \right\}^{\!3}} = 2 \cdot \frac{1}{e^3} \\[1mm] &= \color{red}{\boldsymbol{\frac{2}{e^3}}} \tag*{$\Ans$} \end{align*} ここで $\lim\limits_{n \to \infty} \left(1 + \dfrac{3}{n} \right)^{\!\! \frac{n}{3}} = e$ を用いた. \end{enumerate} \end{document}