全一卷
1.已知数列
满足
,记
。则下列结论正确的是( )。
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A.![]() | B.![]() |
C.![]() | D.![]() |
2.如图,正四面体ABCD中,E在棱AB上,F在棱CD上,使得
.记
,其中
表示EF与AC所成的角,
表示EF与BD所成的角,则( ).
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A.![]() | B.![]() |
C.![]() | D.![]() |
3.设等差数列的首项和公差都是非负的整数,项数不少于3,且各项和为
,则这样的数列共有( )
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A.2个 | B.3个 | C.4个 | D.5个 |
4.在平面直角坐标系中,若方程
表示的曲线为椭圆,则
的取值范围是( )
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A.![]() | B.![]() | C.![]() | D.![]() |
5.设
。则( )。
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A.![]() | B.![]() |
C.![]() | D.![]() |
6.如果空间三条直线a,b,c两两成异面直线,那么与a,b,c都相交的直线有
A.0条 | B.1条 | C.多于1条但为有限条 | D.无数条 |
7.设x、y为实数,且满足
则x+y=__________________。
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8.过双曲线
的右焦点作直线l与双曲线交于A、B两点.若实数
使得
的直线l恰有三条,则
=_______.
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9.已知复数z满足
,则z的辐角主值范围是________________________ .
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10.已知三棱锥S-ABC的底面是以AB为斜边的等腰直角三角形,SA=SB= SC=2,AB=2,设S、A、B、C四点均在以O为球心的某个球面上.则点O到平面ABC的距离为________________ .
11.设
为正六边形,一只青蛙开始在顶点
处,它每次可随意地跳到相邻两顶点之一,若在5次之内跳到
点,则停止跳动;若5次之内不能到达
点,则跳完5次也停止跳动,那么这只青蛙从开始到停止,可能出现的不同跳法共____ 种.
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12.设
,记a、b、c中的最大数为M.,则M的最小值为_________________________ .
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13.设
,且
,求乘积
的最大值和最小值.
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14.设双曲线
的两支为
(如图),正三角形PQR的三顶点位于此双曲线上。
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(1)求证:P、Q、R不能都在双曲线的同一支上;
(2)设P(-1,-1)在
上,Q、R在
上。求顶点Q、R的坐标。
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(1)求证:P、Q、R不能都在双曲线的同一支上;
(2)设P(-1,-1)在
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15.设非零复数
满足
其中S为实数且
。求证:复数
在复平面上所对应的点位于同一圆周上。
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16.如图,已知两个半径不相等的
与
相交于M、N两点,且
、
分别与
内切于S、T两点。求证:OM⊥MN的充分必要条件是S、N、T三点共线。
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17.试问:当且仅当实数
满足什么条件时,存在实数
使得
成立,其中
为虚数单位,
。证明你的结论。
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18.在100x25的长方形表格中每一格填入一个非负实数,第
行第
列中填入的数为
(如表 1)。然后将表1每列中的数按由大到小的次序从上到下重新排列为
,
。(如表2)求最小的自然数k,使得只要表1中填入的数满足
则当i≥k时,在表2中就能保证
成立。
表1 表2
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表1 表2
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