02-2-数学活用能力検査_問題_H30.indd

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きみのしんふじた
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1 Instructions nswer all questions in sections to. The mathematics test written in English is from page five to page eight. The contents of both tests are the same in Japanese and English. The examination duration is 50 minutes. o not read anything aloud. e sure to answer in the language for which you applied. If you answer in other languages, your answer sheet will not be marked. Write your examinee number in the designated space. If any fraction appear in a solution, write the solution in a fully simplified form. If any radicals appear in a solution, write the solution with the radicals but do not include any radicals in the denominator. dditionally, leave the smallest possible integer inside the radicals. If you change answers, erase the original answers neatly, and write the new answers. Write clearly all your answers in the designated places on the answer sheet, and submit only the answer sheet.

2 1 次の各問に答えよ問 1 3 ( 3-6 () ) を計算せよ問 2 連立方程式 2y+x =25 0.6( y -x)=0.4x +0.1y を解け問 3 二次方程式 (x-4) 2 +1 =- x 2 - x +12 を解け問 4 1 から6までの目が出るさいころを 1 個と, 表と裏のどちらかが出る硬貨 1 枚を同時に 1 回投げるとき, さいころの出る目の数が偶数で, 硬貨は表が出る確率を求めよただし, さいころは 1 から 6 までのどの目が出ることも同様に確からしいものとし, 硬貨は表と裏のどちらが出ることも同様に確からしいものとする問 5 右の図で, 円は, の辺をの方向に延ばした直線と, 辺をの方向に延ばした直線, さらに辺のそれぞれと接する円である解答欄に示した図をもとにして, 円を定規とコンパスを用いて作図し, 円の中心の位置を示す文字も書けただし, 作図に用いた線は消さないでおくこと 1

3 2 右の図 1で, 点は原点, 曲線 lは図 1 関数 y = 1 4 x2 のグラフを表している y l 点, 点はともに曲線 l 上にあり, 点の座標は (-2, 1 ), 点の x 座標は正の数である点と点, 点と点をそれぞれ結ぶ点を通り, 線分と平行な直線と, 点を通り, x 線分と平行な直線の交点をとし, 点と点, 点と点をそれぞれ結ぶ原点から点 (1,0) までの距離, および原点から点 (0,1) までの距離をそれぞれ 1 cm として, 次の各問に答えよ問 1 点の x 座標が 2 のとき, 四角形の面積は何 cm 2 か問 2 右の図 2は, 図 1において, 点の x 座標が 2 より大きいとき, 線分と y 軸の交点をRとした場合を図 2 y l 表している R 次の ( 1 ),( 2 ) に答えよ ( 1 ) R=R のとき, 点の座標を求めよ x ( 2 ) = 90 のとき, 直線 R の式を求めよただし, 解答欄には, 答えだけでなく, 答えを求める過程が分かるように, 途中の式や計算なども書け 2

4 3 右の図 1において, 四角形は, >の長方形である辺上にあり,=となる点をとし, 頂点と点を結ぶ辺上にあり, 頂点と頂点のいずれにも一致しない点をとし, 頂点と点, 点と点をそれぞれ結ぶ図 1 次の各問に答えよ問 1 =, = 65 の場合を考えるの大きさを a とするとき, の大きさを a を用いた式で表せ問 2 右の図 2 は, 図 1 において, 点から線分に図 2 引いた垂線と, 線分の交点を H とした場合を表している =H のとき, 次の (1),(2) に答えよ H ( 1 ) H であることを証明せよ (2) 右の図 3 は, 図 2 において, 辺をの方向図 3 に延ばした直線と線分をの方向に延ばした直線の交点をとした場合を表している = 6 cm,= 4 cm のとき, H 線分の長さは何 cm か 3

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6 1 nswer the following questions. uestion 1 Simplify 3 ( () ) uestion 2 Solve the simultaneous equations 2y+x =25 0.6( y -x)=0.4x + 0.1y for x and y. uestion 3 Solve the quadratic equation (x-4) 2 +1=- x 2 -x+12 for x. uestion 4 fair die with sides numbered 1 to 6 is rolled once; simultaneously a fair coin is tossed once. land flat. Find the probability of rolling an odd number and getting heads, assuming the coin will always uestion 5 The figure on the right shows a triangle and a circle. When the side is extended away from vertex, the line is tangent to circle and when the side is extended away from vertex, the line is tangent to circle. Side is also tangent to circle. onstruct circle on the answer sheet, and label center of the circle with the letter. Use a ruler and a compass to construct the answer. o not erase the lines you have drawn in the process of your construction. 5

7 2 Figure 1 on the right shows a graph where the curve l Figure 1 1 represents the function y= x 2. The point represents 4 the origin of the graph. y l oint with coordinates (-2, 1)and point with positive x-coordinate both lie on curve l. onnect points and, and points and. x Let the intersection of the line which passes through point and is parallel to line segment and the line which passes through point and is parallel to line segment be. onnect points and, and points and. ssume the distance between the origin and the point (1, 0), and the distance between the origin and the point (0, 1), are both 1cm. nswer the following questions. uestion 1 Find the area of the quadrilateral when the x-coordinate of point is 2. uestion 2 Figure 2 on the right shows the case in Figure 1 where the x-coordinate of point is greater than 2. Figure 2 y l Let the intersection of the line segment and y-axis be R. R nswer (1)and (2). x (1) Find the coordinates of point when R=R. (2) Find the equation of line R when angle =90. You must show your working in the answer space. 6

8 3 Figure 1 on the right shows a rectangle Figure 1 where >. Let point be a point on side such that =. onnect vertex and point. Let point be a point on side. oint is neither vertex nor vertex. onnect vertex and point, and points and. nswer the following questions. uestion 1 onsider the case where = and angle =65. Find the magnitude of angle in terms of a when the magnitude of angle is a. uestion 2 Figure 2 on the right shows the case in Figure 2 Figure 1 where a perpendicular line to line segment is drawn from point. Let H be the intersection of the perpendicular H line and the line segment. nswer (1)and (2)when =H. (1) rove triangle H triangle. (2) Figure 3 on the right shows the case in Figure 3 Figure 2 where side is extended away from vertex and line segment is extended away from point. H Let the intersection of the two extended lines be. Find the length of line segment when =6 cm and =4 cm. 7

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25 II :30 16:00 (1),. Do not open this problem booklet until the start of the examination is announced. (2) 3.. Answer the following 3 proble

25 II :30 16:00 (1),. Do not open this problem booklet until the start of the examination is announced. (2) 3.. Answer the following 3 proble 25 II 25 2 6 13:30 16:00 (1),. Do not open this problem boolet until the start of the examination is announced. (2) 3.. Answer the following 3 problems. Use the designated answer sheet for each problem.