一橋大学 前期 2004年度 問2

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解答作成者: 森 宏征

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入試情報

大学名 一橋大学
学科・方式 前期
年度 2004年度
問No 問2
学部 商 ・ 経済 ・ 法 ・ 社会
カテゴリ 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,dvipdfmx]{jreport} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{vector3} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{color} \ExecuteOptions{usename} \def\Op{{\mathrm{O}}} \def\LLL{{\lll}} \def\RRR{{\rrr}} \def\LL{{\ll}} \def\RR{{\rr}} \usepackage{graphicx} \usepackage{pifont} \usepackage{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  $a,\ b,\ c$ は整数で, $a < b < c$ をみたす. 放物線 $y = x^2$ 上に3点$\A(a,\ a^2),\\ \B(b,\ b^2),\enskip \C(c,\ c^2)$をとる. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $\angle\B\A\C = 60^\circ$ とはならないことを示せ. ただし,$\sqrt{\vphantom{b}3}$ が無理数であることを証明なしに用いてよい. \item  $a = -3$ のとき, $\angle\B\A\C = 45^\circ$ となる組$(b,\,\,c)$をすべて求めよ. \end{enumerate} \end{FRAME} \vskip 2mm \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \parbox{240pt}{ \item  図のように半直線AB,ACと$x$軸の正の向きのなす角を順に $\theta$, $\pphi$ とする. ただし反時計回りを正の向きと考える.  $\angle\B\A\C = \pphi - \theta$ であり{\small (右図では$\theta < 0$)}, \begin{align*} \tan\theta &= \resizebox{!}{1.6zw}{$\begin{pmatrix} 直線\A\B \\ の傾き \end{pmatrix}$} = \dfrac{b^2 - a^2}{b - a} \\[1mm] &= \dfrac{\>(b-a)(b+a)\>}{b - a} \\[1mm] &= a + b \end{align*} 同様に } \parbox{180pt}{ \hspace*{1zw} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 17.3000, 18.1800)( 15.4000,-27.4800) % STR 2 0 3 0 % 3 2397 2599 2397 2611 4 2800 % O \put(23.9700,-26.1100){\makebox(0,0)[rt]{O}}% % STR 2 0 3 0 % 3 2375 1008 2375 1020 4 2800 % $y$ \put(23.7500,-10.2000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 3268 2621 3268 2633 4 2800 % $x$ \put(32.6800,-26.3300){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 2404 2748 2404 1020 % \special{pn 8}% \special{pa 2404 2748}% \special{pa 2404 1020}% \special{fp}% \special{sh 1}% \special{pa 2404 1020}% \special{pa 2384 1088}% \special{pa 2404 1074}% \special{pa 2424 1088}% \special{pa 2404 1020}% \special{fp}% % VECTOR 2 0 3 0 % 2 1540 2604 3268 2604 % \special{pn 8}% \special{pa 1540 2604}% \special{pa 3268 2604}% \special{fp}% \special{sh 1}% \special{pa 3268 2604}% \special{pa 3202 2584}% \special{pa 3216 2604}% \special{pa 3202 2624}% \special{pa 3268 2604}% \special{fp}% % FUNC 2 0 3 0 % 9 1540 1020 3268 2748 2404 2604 2836 2604 2404 2172 1540 1020 3268 2748 0 1 0 0 % x^2 \special{pn 8}% \special{pa 1578 1020}% \special{pa 1580 1032}% \special{pa 1586 1052}% \special{pa 1590 1070}% \special{pa 1596 1090}% \special{pa 1600 1108}% \special{pa 1606 1126}% \special{pa 1610 1146}% \special{pa 1616 1164}% \special{pa 1620 1182}% \special{pa 1626 1200}% \special{pa 1630 1218}% \special{pa 1636 1236}% \special{pa 1640 1254}% \special{pa 1646 1270}% \special{pa 1650 1288}% \special{pa 1656 1306}% \special{pa 1660 1324}% 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\special{pa 2080 2362}% \special{pa 2086 2368}% \special{pa 2090 2376}% \special{pa 2096 2384}% \special{pa 2100 2390}% \special{pa 2106 2398}% \special{pa 2110 2404}% \special{pa 2116 2412}% \special{pa 2120 2418}% \special{pa 2126 2424}% \special{pa 2130 2430}% \special{pa 2136 2436}% \special{pa 2140 2444}% \special{pa 2146 2450}% \special{pa 2150 2456}% \special{pa 2156 2460}% \special{pa 2160 2466}% \special{pa 2166 2472}% \special{pa 2170 2478}% \special{pa 2176 2484}% \special{pa 2180 2488}% \special{pa 2186 2494}% \special{pa 2190 2498}% \special{pa 2196 2504}% \special{pa 2200 2508}% \special{pa 2206 2512}% \special{pa 2210 2518}% \special{pa 2216 2522}% \special{pa 2220 2526}% \special{pa 2226 2530}% \special{pa 2230 2534}% \special{pa 2236 2538}% \special{pa 2240 2542}% \special{pa 2246 2546}% \special{pa 2250 2550}% \special{pa 2256 2554}% \special{pa 2260 2556}% \special{pa 2266 2560}% \special{pa 2270 2562}% \special{pa 2276 2566}% \special{pa 2280 2568}% \special{pa 2286 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\special{pa 2496 2586}% \special{pa 2500 2584}% \special{pa 2506 2580}% \special{pa 2510 2578}% \special{pa 2516 2576}% \special{pa 2520 2574}% \special{pa 2526 2570}% \special{pa 2530 2568}% \special{pa 2536 2564}% \special{pa 2540 2562}% \special{pa 2546 2558}% \special{pa 2550 2556}% \special{pa 2556 2552}% \special{pa 2560 2548}% \special{pa 2566 2544}% \special{pa 2570 2540}% \special{pa 2576 2536}% \special{pa 2580 2532}% \special{pa 2586 2528}% \special{pa 2590 2524}% \special{pa 2596 2520}% \special{pa 2600 2516}% \special{pa 2606 2510}% \special{pa 2610 2506}% \special{pa 2616 2502}% \special{pa 2620 2496}% \special{pa 2626 2492}% \special{pa 2630 2486}% \special{pa 2636 2480}% \special{pa 2640 2476}% \special{pa 2646 2470}% \special{pa 2650 2464}% \special{pa 2656 2458}% \special{pa 2660 2452}% \special{pa 2666 2446}% \special{pa 2670 2440}% \special{pa 2676 2434}% \special{pa 2680 2428}% \special{pa 2686 2422}% \special{pa 2690 2416}% \special{pa 2696 2408}% \special{pa 2700 2402}% \special{pa 2706 2394}% \special{pa 2710 2388}% \special{pa 2716 2380}% \special{pa 2720 2374}% \special{pa 2726 2366}% \special{pa 2730 2358}% \special{pa 2736 2350}% \special{pa 2740 2344}% \special{pa 2746 2336}% \special{pa 2750 2328}% \special{pa 2756 2320}% \special{pa 2760 2312}% \special{pa 2766 2302}% \special{pa 2770 2294}% \special{pa 2776 2286}% \special{pa 2780 2278}% \special{pa 2786 2268}% \special{pa 2790 2260}% \special{pa 2796 2250}% \special{pa 2800 2242}% \special{pa 2806 2232}% \special{pa 2810 2222}% \special{pa 2816 2214}% \special{pa 2820 2204}% \special{pa 2826 2194}% \special{pa 2830 2184}% \special{pa 2836 2174}% \special{pa 2840 2164}% \special{pa 2846 2154}% \special{pa 2850 2144}% \special{pa 2856 2134}% \special{pa 2860 2124}% \special{pa 2866 2112}% \special{pa 2870 2102}% \special{pa 2876 2090}% \special{pa 2880 2080}% \special{pa 2886 2068}% \special{pa 2890 2058}% \special{pa 2896 2046}% \special{pa 2900 2036}% \special{pa 2906 2024}% \special{pa 2910 2012}% \special{pa 2916 2000}% \special{pa 2920 1988}% \special{pa 2926 1976}% \special{pa 2930 1964}% \special{pa 2936 1952}% \special{pa 2940 1940}% \special{pa 2946 1926}% \special{pa 2950 1914}% \special{pa 2956 1902}% \special{pa 2960 1888}% \special{pa 2966 1876}% \special{pa 2970 1862}% \special{pa 2976 1850}% \special{pa 2980 1836}% \special{pa 2986 1824}% \special{pa 2990 1810}% \special{pa 2996 1796}% \special{pa 3000 1782}% \special{pa 3006 1768}% \special{pa 3010 1754}% \special{pa 3016 1740}% \special{pa 3020 1726}% \special{pa 3026 1712}% \special{pa 3030 1698}% \special{pa 3036 1682}% \special{pa 3040 1668}% \special{pa 3046 1654}% \special{pa 3050 1638}% \special{pa 3056 1624}% \special{pa 3060 1608}% \special{pa 3066 1594}% \special{pa 3070 1578}% \special{pa 3076 1562}% \special{pa 3080 1546}% \special{pa 3086 1530}% \special{pa 3090 1516}% \special{pa 3096 1500}% \special{pa 3100 1484}% \special{pa 3106 1466}% \special{pa 3110 1450}% \special{pa 3116 1434}% \special{pa 3120 1418}% \special{pa 3126 1402}% \special{pa 3130 1384}% \special{pa 3136 1368}% \special{pa 3140 1350}% \special{pa 3146 1334}% \special{pa 3150 1316}% \special{pa 3156 1298}% \special{pa 3160 1282}% \special{pa 3166 1264}% \special{pa 3170 1246}% \special{pa 3176 1228}% \special{pa 3180 1210}% \special{pa 3186 1192}% \special{pa 3190 1174}% \special{pa 3196 1156}% \special{pa 3200 1138}% \special{pa 3206 1120}% \special{pa 3210 1100}% \special{pa 3216 1082}% \special{pa 3220 1064}% \special{pa 3226 1044}% \special{pa 3230 1026}% \special{pa 3232 1020}% \special{sp}% % DOT 0 0 3 0 % 1 1907 2028 % \special{pn 20}% \special{sh 1}% \special{ar 1908 2028 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 2714 2388 % \special{pn 20}% \special{sh 1}% \special{ar 2714 2388 10 10 0 6.28318530717959E+0000}% % LINE 2 0 3 0 % 2 2714 2388 1907 2028 % \special{pn 8}% \special{pa 2714 2388}% \special{pa 1908 2028}% \special{fp}% % LINE 2 0 3 0 % 2 1907 2028 3074 1574 % \special{pn 8}% \special{pa 1908 2028}% \special{pa 3074 1574}% \special{fp}% % DOT 0 0 3 0 % 1 3074 1574 % \special{pn 20}% \special{sh 1}% \special{ar 3074 1574 10 10 0 6.28318530717959E+0000}% % LINE 3 0 3 0 % 2 1540 2030 3270 2030 % \special{pn 4}% \special{pa 1540 2030}% \special{pa 3270 2030}% \special{fp}% % CIRCLE 3 0 3 0 % 4 1910 2030 2060 2170 2420 2260 2560 2030 % \special{pn 4}% \special{ar 1910 2030 206 206 6.2831853 6.2831853}% \special{ar 1910 2030 206 206 0.0000000 0.4236689}% % CIRCLE 3 0 3 0 % 4 1910 2030 2170 2240 2810 2030 2800 1680 % \special{pn 4}% \special{ar 1910 2030 334 334 5.9085041 6.2831853}% % CIRCLE 3 0 3 0 % 4 1910 2030 2220 2210 2690 2030 2660 1730 % \special{pn 4}% \special{ar 1910 2030 358 358 5.9026789 6.2831853}% % STR 2 0 3 0 % 3 2210 2050 2210 2150 2 0 % {\footnotesize$\theta$} \put(22.1000,-21.5000){\makebox(0,0)[lb]{{\footnotesize$\theta$}}}% % CIRCLE 3 0 3 0 % 4 2160 1830 2170 2000 2590 2360 2390 1410 % \special{pn 4}% \special{ar 2160 1830 170 170 5.2134024 6.2831853}% \special{ar 2160 1830 170 170 0.0000000 0.8891905}% % STR 2 0 3 0 % 3 2130 1610 2130 1710 2 0 % {\footnotesize$\pphi$} \put(21.3000,-17.1000){\makebox(0,0)[lb]{{\footnotesize$\pphi$}}}% % STR 2 0 3 0 % 3 1800 2070 1800 2170 2 0 % {\small A} \put(18.0000,-21.7000){\makebox(0,0)[lb]{{\small A}}}% % STR 2 0 3 0 % 3 2760 2410 2760 2510 2 0 % {\small B} \put(27.6000,-25.1000){\makebox(0,0)[lb]{{\small B}}}% % STR 2 0 3 0 % 3 3120 1530 3120 1630 2 0 % {\small C} \put(31.2000,-16.3000){\makebox(0,0)[lb]{{\small C}}}% % STR 2 0 3 0 % 3 1640 1000 1640 1100 2 0 % {\small $y=x^2$} \put(16.4000,-11.0000){\makebox(0,0)[lb]{{\small $y=x^2$}}}% \end{picture}% %\input{hit20042f_zu_1} } \begin{align*} \tan\pphi = \resizebox{!}{1.6zw}{$\begin{pmatrix} 直線\A\C \\ の傾き \end{pmatrix}$} = a + c \end{align*} $\tan$の加法定理より \begin{align*} \tan\angle\B\A\C &= \tan(\pphi - \theta) = \frac{\tan\pphi - \tan\theta} {\>1 + \tan\pphi\tan\theta\>} \\[1mm] &= \frac{(a+c) - (a+b)}{\>1 + (a+c)(a+b)\>} \\[1mm] &= \frac{c-b}{\>1 + (a+b)(a+c)\>} \end{align*} $\angle\B\A\C = 60^\circ$ であると仮定すると \begin{align*} \sqrt{\vphantom{b}3} = \tan 60^\circ = \frac{c-b}{\>1 + (a+b)(a+c)\>} \tag*{$\cdott\MARU{1}$} \end{align*} $a,\,\,b,\,\,c$ は整数だから\MARU{1}より $\sqrt{\vphantom{b}3}$ は 有理数となる. これは $\sqrt{\vphantom{b}3}$ が無理数であることに矛盾する. したがって $\angle\B\A\C = 60^\circ$ となることはない. \hfill ■ \item  (1)の計算より $a = -3$ のとき $\tan\theta = b - 3,\enskip \tan\pphi = c - 3$ だから, \begin{gather*} \tan\angle\B\A\C = \frac{c - b}{\>1 + (c-3)(b-3)\>} = \frac{\>(c-3) - (b-3)\>}{\>1 + (c-3)(b-3)\>} \end{gather*} $\angle\B\A\C = 45^\circ$ のとき \begin{gather*} \frac{\>(c-3) - (b-3)\>}{\>1 + (c-3)(b-3)\>} = 1 \\ \qquad (b-3)(c-3) + (b-3) - (c-3) + 1 = 0 \\ \qquad \{(b-3) - 1\}\{(c-3) + 1\} = -2 \\ \therefore \,\,\, (b-4)(c-2) = -2 \end{gather*} $b < c$ の仮定より $b - 4 < c - 2$ だから, \begin{align*} \begin{array}{c||c|c} b-4 & -2 & -1 \\ \hline c-2 & 1 & 2 \end{array} \qquad \therefore \,\,\, (b,\ c) = \textcolor{red}{\boldsymbol{(2,\ 3),\ (3,\ 4)}} \tag*{$\Ans$} \end{align*} \end{enumerate} \end{document}