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2023年5月16日 星期二

112年嘉義高中教甄-數學詳解

國立嘉義高級中學 112 學年度第 1 學期第 1 次教師甄選

解答:m,n=m+(m+1)++(m+n1)=nm+n(n1)2=2nm+n(n1)2=2023n(2m+n1)=40464046=2×7×172{4046=1×4046{n=1m=2023,n24046=2×2023{n=2m=10114046=7×578,14×289,17×238,34×1195
解答:

|1z1|1|z||1z1||z||z1||z|(x1)2+y2x2+y22x+10x12Ω1={x12}|z1|1Ω2={(x1)2+y21};Ω1Ω2,x=12(x1)2+y2=1{A(1/2,3/2)B(1/2,3/2)¯AB=3cosAOB=1+132=12AOB=120{OAB=3/4OAB=π/3Ω1Ω2=π334=4π3312

解答:{A(2,4,5)C(4,4,7)G(3,5,6)F(a,b,c){AC=(2,0,2)AG=(1,1,1)H=¯AC(3,4,6)n=AC×AG=(2,0,2)AnEx+z=3¯AC=22=2{ACHFAGHF{(2,0,2)(a3,b4,c6)=0(1,1,1)(a3,b4,c6)=0{a+c=9b=4¯HF=d(H,E)=2|a+c3|2=2a+c=51{a=2,c=7a=4,c=5{F=(2,4,7)F=(4,4,5){FH=(1,0,1)=2nFH=(1,0,1)=2nF=(2,4,7)
解答:n5=aCn5+bCn4+cCn3+dCn2+eCn1n=11=0+0+0+0+ee=1n=232=d+2ed=30n=3243=c+3d+3ec=150n=41024=b+4c+6d+4eb=240n=53125=a+5b+10c+10d+5ea=120a+b+c+d+e=120+240+150+30+1=541
解答:f(n)=4f(n1)+4n2=4(4f(n2)+4n3)+4n2=42f(n2)+24n2==4n2f(2)+(n2)4n2=4n2(n1)(f(2)=1)f(9)=478=21717118
解答:n2abcd11,1,1,11224,1,1,142,2,1,16323,3,1,16424,4,1,164,2,2,1122,2,2,21525,5,1,16626,6,1,166,3,2,1243,3,4,1123,3,2,26824,4,2,264,4,4,14923,3,3,311025,5,2,265,5,4,1121226,6,4,1126,6,2,266,4,3,2244,4,3,361525,5,3,361624,4,4,411826,6,3,362025,5,4,462426,6,4,462525,5,5,513026,6,5,563626,6,6,61200p=20064=251621p=16225
解答:

{3x5f(x)=x35tdt=12x2+5x212x5g(x)=535tdt+x5t5dt=12x25x+292Ly=2x2023k,L{y=f(x)y=g(x){2x2023k=f(x),0k=4046/12337.22x2023k=g(x),=0k=4046/20=202.3k=203337,337203+1=135
解答:

{A(1,0,1)B(0,1,1)C(1,1,0)O(0,0,0)OABC=2{D=¯AO=(1/2,0,1/2)E=(2A+B)/3=(2/3,1/3,1)F=(2B+A)/3=(1/3,2/3,1)G=¯BC=(1/2,1,1/2)CDEOFG,{E2=CDEE3=OFG{CD=(1/2,1,1/2)CE=(1/3,2/3,1)n=CD×CE=(2/3,1/3,0)CDE12|n|=56AB=(1,1,0)cosθ=ABn|AB||n|=310d(E1,E4)=352=352/310=15CDE12(15)2=560
解答:
解答::23R3tanθ=2363tan30=483
解答:=16k=1k(k+2)C16kxky16k,x=34,y=14f(x,y)=(x+y)16=16k=0C16kxky16kfx=16k=1kC16kxk1y16kx3fx=16k=1kC16kxk+2y16kx(x3fx)=3x2fx+x3fxx=16k=1k(k+2)C16kxk+1y16k1x(3x2fx+x3fxx)=3xfx+x2fxx=16k=1k(k+2)C16kxky16kf=(x+y)16fx=16(x+y)15{fxx=240(x+y)143xfx=48x(x+y)15g(x,y)=3xfx+x2fxx=48x(x+y)15+240x2(x+y)14g(34,14)=36+135=171
解答:44441B3WBWWW14B1WBWW2WWBW2WWWB12B2WBBWW13B1W4BWBW3BWWB2WBBW2WBWB3WWBB13B1WBBBW12B2W6BBWB2BWBB2WBBB1=1(1+2+2+1)+4(1+3+2+2+3+1)+6(1+2+2+1)=90=90C83=9056=4528
解答:P(1,a)asin(π3+θ)=aπ3+θ=π2θ=π6f(x)=asin(π3x+π6)Qπ3x+π6=32πx=4Q(4,a){P(1,a)R(1,0)Q(4,a){RP=(0,a)RQ=(3,a)cosPRQ=RPRQ|RP||RQa2a9+a2=12a2=3a=3

解答:¯AC=a¯AB=10a=f(a)=12a(10a)sin602af(a)=32(10a2a3)f(a)=32(20a3a2)f(a)=32(206a)f(a)=0{a=0a=20/3{f(0)>0f(20/3)<0f(203)=324009(10203)=2000327
解答:80(1+3.53%)n100(1+1.5%)n(1.03531.015)n10080(102100)n108nlog1.0213log2=130.301=0.097n0.0970.008611.312

解答:

y,{A(a,a2+4)B(a,a2+4)C(a,2a28)D(a,2a28){¯AB=2a¯AD=a2+4+82a2=3a2+12=f(a)=2a(3a2+12)=6a3+24af(x)=18a2+24=0a=23f(23)=163+483=3233
解答:I,XXI=aa+b+cXA+ba+b+cXB+ca+b+cXCAI=32+3+4AB+42+3+4AC=13AB+49ACAOAI=13AOAB+49AOAC=1312¯AB2+4912¯AC2=1616+299=143
解答:{E1:2x+y+2z+3=0E2:3x+4y5=0{E1u=(2,1,2)E2v=(3,4,0){n1=u/|u|=(2/3,1/3,2/3)n2=v/|v|=(3/5,4/5,0)cosθ=n1n2|n1||n2|>0n1n2s=n1+n2=(1/15,7/15,10/15)P(17/5,19/5,0)E1E2115(x+175)+715(y195)1015z=0x7y+10z+30=0
解答:x2m+y2=1{a=mb=1c=m1{F1(m1,0)F1(m1,0)x2ny23=1{a=nb=3c=n+3{F1(n+3,0)F2(n+3,0)m1=n+3mn=4{n=1m=5{F1(2,0)F2(2,0)x25+y2=x2y23x2=53y2y2=34y=32x=52P(5/2,3/2){¯PF1=5+1¯PF2=51cosF1PF2=(5+1)2+(51)2422(5+1)(51)=12F1PF2=120tanF1PF2=3
解答:

A(Rcosα,Rsinα)L:y=tanθx,A(Rcos(2θα),Rsin(2θα)){x=Rcosαy=Rsinα{x=Rcos(2θα)=Rcos2θcosα+Rsin2θsinα=xcos2θ+ysin2θy=Rsin(2θα)=R(sin2θcosαsinαcos2θ)=xsin2θycos2θ[cos2θsin2θsin2θcos2θ][xy]=[xy]L[cos2θsin2θsin2θcos2θ]
解答:{A=[cos2αsin2αsin2αcos2α]B=[cos2βsin2βsin2βcos2β]AB=[cos2αcos2β+sin2αsin2βcos2αsin2βsin2αcos2βsin2αcos2βcos2βsin2αsin2αsin2β+cos2αcos2β]=[cos(2α2β)sin(2α2β)sin(2α2β)cos(2α2β)]2
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解題僅供參考,其他教甄試題及詳解

1 則留言:

  1. 朱老師你好!請問第16題,我算出AI的長度與A B的夾角,AO同樣算出其長度與AB的夾角,而AO與AB的夾角餘弦值和A l與AB的夾角餘弦值相等,即AI和AO夾角為0,兩向量內積,即其長度相乘,但答案是16 /3,老師我那裏弄錯了,能否解迷?謝謝你

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