江西省2020届高三下学期调研考试(三)数学(理)(扫描版)
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1.【/0】C
【12】3 sinπ
3=sin2π
3 =sin8π
3 =槡3
2,sin4π
3 =-槡3
2,45 A∩B={π
3,2π
3,8π
3},67 C.
2.【/0】B
【12】a+i
1-2i=(a+i)(1+2i)
5 =a-2
5 +2a+1
5 i,89 a+i
1-2i(a∈R):;<=>?@AB:CD y
=xE,FGa-2
5 =2a+1
5 ,FG a=-3,67 B.
3.【/0】A
【12】H(x-2)2 +(y-1)2 =1IJKLD C:x2
a-y2
3=1(a>0)AMNOPD?Q,RHS
(2,1):OPD y= 3
槡axE,FG 1=2 3
槡a,a=12,FGKLD CATUV9 2槡a 槡=4 3,
67 A.
4.【/0】A
【12】89 a=2log32=log34<log3 槡27=3
2,c=log23=log2槡9>log2槡8=3
2,FG a<b<c,6
7 A.
5.【/0】B
【12】3WX,X~B(3,1
5),FG EX=3×1
5=3
5,67 B.
6.【/0】C
【12】3 f(x)=
1,x>0
0,x=0
-1,x{ <0
,g(x)=sinπx,45Y x>0Z,g(f(x))=g(1)=sinπ=0,Y x=0
Z,g(f(x))=g(0)=sin0=0,Y x<0Z g(f(x))=g(-1)=sin(-π)=0,FG A[;Y x
>0Z,f(x)=1,f(f(x))=f(1)=1,f(f(x))=f(x)]^,Y x=0Z,f(0)=0,f(f(0))=f(0)=
0,f(f(x))=f(x)]^,Y x<0Z,f(x)=-1,f(f(x))=f(-1)=-1,f(f(x))=f(x)]^,F
G B[,3 f(3
2)g(3
2)=-1,4_ C`a,3 g(x)≥ -1,g(x)+2≥1,4_ f(g(x)+2)=1
[,67 C.
7.【/0】D
【12】89 f(x)=ex,g(x)=x3-ax,FG f′(x)=ex,g′(x)=3x2 -a,3 f(x)=ex: x=0bA
cDdLDg(x)=x3-ax: x=1bAcD
4vkANtw i>6?,67 D.
10.【/0】C
【12】x BCyB O,BOyB F,z{ OD,OE,FE,DF,R∠ODE|wCD DEd ACF]}.~
AB=4,R OD=2,OF=1,OE=2,DF 槡= 3,EF= OE2+OF槡 2 槡= 5,DE= DF2+EF槡 2 槡=2 2,F
G∠ODE=π
4,fCD DEd ACF]}A9槡2
2,67 C.
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11.【/0】B
【12】3 AB⊥xU45 →AB· →AF= →AB 2,FG →AB =2b
3,
→AB
→FB
=tan∠BFA=b
c,FG
→FB =2c
3,FGA(5c
3,-2b
3),kH CAi
525c2
9a2 +4
9=1,FG e=槡5
5,67 B.
12.【/0】A
【12】f(x)A[9 π,①`a;y=f(x)+f(x+π
2)=|sin(x+π
4)|+|cos(x+π
4)|=
(|sin(x+π
4)|+|cos(x+π
4)|)槡 2 = 1+|sin(2x+π
2)槡 |= 1+|cos2x槡 |,3 1≤1+|cos2x|
≤2,45 y=f(x)+f(x+π
2):(-∞,+∞)EA9[1,槡2],②`a; m=3
4,R x1 +x2
=π
2,x2+x3=3π
2,x3+x4=5π
2,5 x1+2x2+2x3+x4=9π
2,③[;m=1
2,Ri
f(x)=1
2
:[0,+∞)EA 5T97π
12,11π
12,19π
12,23π
12,31π
12,FG23π
12≤α<31π
12,④`a.6
7 A.
13.【/0】3
2
【12】89 f(x)=logax,FG f(4t)-f(t)=loga4=3,FG f(6t)-f(3t)=loga2=1
2loga4
=3
2.
14.【/0】30
【12】(xy+x+1)5y, x4y2A#9 C2
5C2
3=30.
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15.【/0】6
【12】3 a-b = c2槡 -6,5 c2=a2+b2-2ab+6.3 C=π
3,5 c2 =a2 +b2 -ab.g5
ab=6.FG S△ ABC =1
2absinC=1
2×6×槡3
2 = 槡33
2 .FG S△ABC =1
2OA·OBsin120°+1
2OB·
OCsin120°+1
2OA·OCsin120°= 槡33
2 ,5 OA·OB+OB·OC+OC·OA=6.
16.【/0】112π
3
【12】AD⊥DF,EC⊥CF,89∠AFD=∠EFC,FG tan∠AFD=tan∠EFC,fAD
FD=EC
FC,89
AD=2EC,FG FD=2FC, FG⊥CDJ G,~ DG=x,FG=y,3 FD=2FC5 x2+y槡 2 =
2 (4-x)2+y槡 2,!5 y2=-x2+32
3x-64
3,FGY x=4Z,yx5 槡43
3 ,oZB F:
CC1E, CF= 槡43
3 ,) A-CDFA( AD=4,FGY△CDF= ,f yZ¡
,FG) A-CDF¡ Z,AD⊥DF,AC⊥CF,x AFyB O,R OA=OF=OC=
OD,FG O9¢S,AF9C£,89(2R)2=AF2=42+42+( 槡43
3 )2=112
3,FG) A-CDF
¡ Z,¤¥{¢A¦= S=4πR2=112π
3 .
17.1:(1)~§¨#©{an}Aª¨9 q,3 S6=-7S3,5(1+q3)S3=-7S3,
S3≠0,FG 1+q3=-7,q=-2,
3 a2,1,a3]§«#©,45 a2+a3=2,f -2a1+4a1=2,FG a1=1,(3)
FG an=a1qn-1=(-2)n-1.(6)
(2)Y n9¬#Z,an-1=-2n-1-1<0,Y n9#Z an-1=2n-1-1≥0,
FG T2n=(a1-1)-(a2-1)+(a3-1)-(a4-1)+… +(a2n-1-1)-(a2n-1)
=a1-a2+a3-a4+… +a2n-1-a2n
=1+2+22+23+… +22n-1=1-22n
1-2 =22n-1.(12)
18.1:(1)~ AC1AyB9 F,ACAyB9 G,z{ DF,FG,BG,
R FG∥CC1, FG=1
2CC1. D9 BB1AyB,∴DB∥FG, DB=FG,
FG®h¯ BDFG9
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