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2023年1月14日 星期六

112年大學學測-數學A詳解

112學年度學科能力測驗試題-數學A考科

第壹部分、選擇(填)題(占85分 )
一、單選題(占 30 分 )

解答:N=2N=22N=(22)2=24N=(24)2=28(4)
解答:


{COE+θ=90COE+COD=90COD=θtanθ=tanCOD=¯CD¯CO=¯CD1=¯CD(5)
解答:(sk,logtk)(0,0)(1,1/2)y=x/2logt=s/2 t=10s/2t2=10s(4)
解答:9ABCDEFGHI滿A<B<C<D<EE>F>G>H>IE=9;84(C84)A,B,C,D;4F,G,H,I;C84=8!4!4!(1)
解答:(PQ×PR)(2,3,5)PQ×PR=k(2,3,5)=(2k,3k,5k),k0(1)111a1b1c1a2b2c2=|(2k,3k,5k)(1,1,1)|=0(2)111a1b1c1a2b2c2=|(2k,3k,5k)(1,1,1)|=|10k|(3)111a1b1c1a2b2c2=|(2k,3k,5k)(1,1,1)|=|6k|(4)111a1b1c1a2b2c2=|(2k,3k,5k)(1,1,1)|=|6k|(5)111a1b1c1a2b2c2=|(2k,3k,5k)(1,1,1)|=|4k|(2)
解答:

OAaOAOBab{a{c,f,g}=0a{b,d,e}=1,{bg=0b{c,d,f}=1be=2,{c{d,g}=0c{e,f}=1,{de=2d{f,g}=1,{ef=2eg=1,fg=1{6012132=0+12+3221=1821=67(3)

二、多選題( 占30 分)

解答:{iai1ibi1,a0=b0{ak=a0+200m,m=k/3bk=a0+1000m,m=k/12,kN(1)×:8/3=2a8=a0+2002=a0+400400600(2)×:{a12=a0+2004=a0+800b12=a0+10001=a0+1000a12<b12(3):{a18=a0+2006=a0+1200b18=a0+10001=a0+1000a18<b18(4)×:{17i=0ai=3(a0+(a0+200)+(a0+400)++(a0+1000))=18a0+900317i=0bi=12a0+(a0+1000)6=18a0+6000(5):{13a0+160046:a0+180079:a0+2000,1012:a0+2200:a0+200010,11,12,19(35)
解答:pn=1(10.1)n=10.9n(1):{pn=10.9npn+1=10.9n+1pn+1>pn(2)×:p3=10.93=10.729=0.2710.3(3)×:{p2p1=(10.92)(10.9)=0.90.92p3p2=(10.93)(10.92)=0.920.93p2p1p3p2(4):{0.90.1=0.09p2p1=10.92(10.9)=0.90.92=0.09(5)×:=10.9nCn10.10.9n12pn(14)
解答:S(n)log3a1log3a2+log3a3+(1)n+1log3an(1)×:n=23S(23)=log3a1a3a23a2a4a22=log3312(33)2+4++22311(33)1+3++21=log33(33)11=log3335/2=35218(2)×:S(24)=S(23)log3a24=352log33(33)23<S(23)<18(3):S(25)=S(23)log3a24+log3a25=352+log3a25a24=352+32=19>18(4)×:S(26)=S(25)log3a26=19log331+3225=19772<18(5):S(27)=S(25)log3a26+log3a27=19+32>18(35)
解答:(1):k=4L:5y+4x=40A(10,0)(2)×:C(0,6)L3010k=0k=35y+2x=30y=25x+62552(3):{Ck=3Ok=00k3(4)×:k=12L:5y3x=5x=10(10,7)¯AB(5):L42k5=310k=54L:5y32x=252x=10(10,112)¯AB(135)
解答:A=[cos(90)sin(90)sin(90)cos(90)]=[0110]B[cos90sin90sin90cos90]=[0110]x=y1tan45C[cos90sin90sin90cos90]=[0110]x=y1tan(45)D=[cos90sin90sin90cos(90)]=[0110](1)×:{[0110][10]=[01][0110][10]=[01][01][01](2):B=[0110]=[0110]=AA=B(3)×:D1=[0110]1=[0110]=DC(4)×:{AB=[1001]CD=[1001]ABCD(5):AC=[1001]=BD(25)
解答:(1):f(x)=2(cosπ3sinx+sinπ3cosx)=2sin(x+π3)x+π3=π2x=π6(2)×:{x+π/3=π/2x+π/3=0{a=π/6b=π/3f(a)f(b)(3)×:f(x)2π,x滿f(x)=3(4)×:f(x)=12sin(x+π3)=14,sin(x1+π/3)=sin(x2+π/3)=1/4,0<x1<x2<2π{sin(π/2+π/3)=12>14f(x)x[π/2,2π/3]x1>π/2sin(3π/2+π/2)=0<14f(x)x[3π/2,2π]x2>3π/2x1+x2>2π(5):4sin2x2=2(1cosx)=2(1sin(π2x))=2+2sin(xπ2)22sin(xπ2)5π/62sin(xπ2+5π6)=2sin(x+π3)=f(x)(15)

三、選填題( 占25 分)

解答:{xyz,{60x+80y+50z=1290030x+40y+30z=685050x+70y+40z=10800{6x+8y+5z=12903x+4y+3z=6855x+7y+4z=1080z=|68129034685571080||685343574|=801=80
解答:ax2+(2a+b)x12=a(x2+(2a)x2a)+6{2a+b=a(2a)12=2a2+6{a2+b=0(1)a2=9(2),(2)a=3(a>0,a3)(1)b=9(a,b)=(3,9)
解答:35+25=1{C¯AB¯AC:¯CB=2:3{¯AC=2k¯BC=3k¯BD=a,3¯AD=8¯BD3(2k+3k+a)=8aa=3kB¯CD¯OBOCD{¯OC2+¯OD2=2(¯BC2+¯OB2)¯OC2+¯OD2=¯CD22(9k2+¯OB2)=(6k)2¯OB=3k;¯OA2+¯OB2=¯AB2¯OA2+9k2=(2k+3k)2¯OA=4k¯OB¯OA=3k4k=34
解答:PEQQ=(a,b,2a)¯QA=¯QB=¯QC(a2)2+(b+1)2+(2a)2=a2+(b1)2+(a)2=(a+2)2+(b1)2+(a2)2{a=1b=4Q(1,4,3)Q(1,0,1)L:(1,4,3)+t(1,0,1)=(1+t,4,3+t)Lz=1P=(3,4,1)PE|221+1|=22
解答:{TL1T=(1+t,1t,2+t)SL2S=(2+2s,5+s,6s)n=TS=(2st+1,s+t+4,st+4){n(1,1,1)=0n(2,1,1)=0{s=1/3t=1/3{T=(4/3,2/3,7/3)S=(4/3,14/3,19/3){¯PT=3¯QS=3{P=T3(1,1,1)Q=S+62(2,1,1)¯PQ=50=52

解答:cosθ=¯OA2+¯OP2¯AP22¯OA¯OP=1+¯OP212¯OP=¯OP2¯OP=2cosθ(4)
解答:cosQOB=cos(90θ)=sinθ=3535=¯OB2+¯OQ2¯BQ22¯OQ¯OB=¯OQ4¯OQ=125Q(¯OQcosQOB,¯OQsinQOB)=(12535,12545)=3625,4825){A=1802θB=1802(90θ)=2θA+B=180¯AP¯BQ,{¯BQ=¯OB=2¯AP=¯OA=1,¯BQ=2¯AQ;{¯AP¯BQ¯BQ=2¯AQBQ=2AP
解答:BQ4825÷(23625)=247BQ:y=247(x+2)24x7y+48=0d(A,BQ)=24+48242+72=7225PABQOAP+OPQ+OBQ=12(¯OA¯OPsinθ+¯OP¯OQ+¯OB¯BQsin(2θ))=12(8535+85125+22(23545))=10825


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解題僅供參考,其他升大學歷屆試題及詳解


4 則留言:

  1. 20題BQ的斜率有問題

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  2. 選填題13題的z,分子的行列式第(1,3)有問題

    回覆刪除