Processing math: 100%

2021年9月6日 星期一

107年新竹市建功高中國中部教甄-數學詳解

 新竹市立建功高中107學年度第一次正式教師甄試國中數學科

一、填充題:【20 題,每題 4 分,共計 80 分】

解答
¯AB:¯AC:¯BC=3:4:5{¯AB=3k¯AC=4k¯BC=5k,kR¯BDB¯AD¯DC=¯AB¯BC=3k5k{¯AD=¯AB¯AB+¯BCׯAC=38×4k=32k¯DC=4k32k=52k;¯AF¯FB=¯CA¯CB=4k5k{¯AF=¯CA¯CA+¯CBׯAB=49×3k=43k¯FB=3k43k=53k;{CEDCABBGFBAC(AAA){CED:ABC=¯CD2:¯BC2=25/4:25BGF:ABC=¯BF2:¯BC2=25/9:25DEC+FGBABC=25/4+25/925=1336
解答12+14+15=1920{a=2b=4c=6a+b+c=12
解答{A(1,1)B(2018,20182)C(2,4)D(a,a2)AB:y=2017(x+1)+1AByP(0,2018)CP:y=1007(x+2)+4y=x2x21007x2018=0(x1009)(x+2)=0Dx1009
解答160=180×(1602)=28440160n(160n)18028440<160n+180(160n)20n<360n<18n=17
解答x=n+y,nN,0b<1x2+y2=2018x2=2018y22017<x2<20182017<x<2018n=44x=44+y(44+y)2+y2=2018y2+44y41=0y=22+521x=44+(22+521)=22+521
解答{anrabnrban+bn=(ra+rb)(an1+bn1)rarb(an2+bn2){86=(ra+rb)(4)rarb2471=(ra+rb)86rarb(4){ra+rb=1rarb=154a5+b5=171+15486=39312

解答:51321(12)5+C43(12)4+C32(12)3=2132
解答

{¯BP=¯CQ=a¯PQ=b¯AQ=h¯AP=w¯ABR使¯AR=¯AC=12AQRAQC(SAS)¯QR=¯QC=a¯AQA¯AB:¯AC=¯BQ:¯QC20:12=a+b:ab=23aa+b=53a{BQR:cosB=(5a/3)2+82a216(5a/3)=64+(16a2/9)80a/3(1)BAP:cosB=202+a2w240a(2)BAQ:cosB=202+(5a/3)2h240(5a/3)=400+(25a2/9)h2200a/3(3){(2)=(3)h2=24053a2(1)=(2)w2=30453a2w2h2=304240=64=8
解答3x5x7x9x=11x13x111<x<19,f(x)=(13)+(15x)+(17x)+(19x)+(11x1)+(13x1)=2{a=1/11b=1/9c=2a×b×c=299
解答n=9+99+999++999999=(1011)+(1021)+(1031)++(101001)=109(101001)100=1091009999100=101001111100=10011110100=981111010991
解答an=2nmod100=2,4,8,16,32,64,28,56,12,24,48,96,92,84,68,36,72,44,88,76,52,4,8,16,32,64,28,56,12,24,48,96,92,84,68,36,72,44,88,76,52,n=2202018=1+20×100+171744A=220181441=43
解答n(99+9+9<313){abc0a,b,c9100a+10b+c+a+b+c=313101a+11b+2c=313a=211b+2c=111{(b,c)=(9,6)(b,c)=(7,17)n=296
解答{a+b>cca<b(a,b,c)=(1,9,9),(2,8,9),(3,7,9),(3,8,8),(4,6,9),(4,7,8),(5,5,9),(5,6,8),(5,7,7),(6,6,7)10
解答
解答
ABCD¯AC¯BD¯AC¯BDQ{¯BQ=¯QD=a¯AQ=¯QC=b¯CP=a{AQD:a2+b2=36(1)PQD:a2+(b+c)2=81(2)(1)(2)36+2bc+c2=812bc+c2=c(2b+c)=¯PCׯAP=45

解答{x2A2xyBy2C(xπ+2y){x2102xy28y250(x,y)=(1,1)(1,7),(2,1)(2,7),(3,1)(3,4)(x,y)=(2,7)2π+14
解答
F=360909045=135{E(0,0)F(0,1)C(a,0)D(1,2)C=45CD:y=(x1)+2a=3{¯CE=3¯CD=22+22=22{CDF=12×2×22=2CEF=12×1×3=32DFEC=2+32=72
解答f(x)=(x+1)(2x+1)(3x1)(4x1)36x4=12x4+22x37x24x+1f(1)=0f(x)=(x1)(12x3+10x2+3x1)=(x1)g(x)g(1)=0f(x)=(x1)2(12x22x+1)=(x1)2(12x+2x1)
解答{yx+1xy=203xy+xy=53y2+1y2=8299y482y2+9=0(y29)(9y21)=0{y=3x=1/2y=3x=1/2y=1/3x=1/2y=1/3x=1/2(x,y)=(12,3),(12,3),(12,13),(12,13)
解答y=(x2)(x4)(x6)(x8)+12=((x2)(x8))((x4)(x6))+12=(x210x+16)(x210x+24)+12=(x210x)2+40(x210x)+396=(x210x+20)24x210x+20=0y1(x,y)=(5±5,4)
解答11+2++n=1n(n+1)/2=2(1n1n+1)11+11+2+11+2+3++11+2+3++100=1+2((1213)+(1314)++(11001101))=1+2(121101)=1+99101=200101

二、計算題:【4 題,每題 5 分,依計算過程,給予適當分數】

解答x(xx)={x0xx+x=0+xx0=0+x=x2018(20172017)={20182017+201720180=201820182017+2017=201820182017=20182017=1
解答(x2+x+1)(1+x+x2++x9+x10)=(1+x+x2++x6)2(x1)(x2+x+1)(x1)(1+x+x2++x9+x10)=((x1)(1+x+x2++x6))2(x31)(x111)=(x71)2x112x7+x3=0x3(x82x4+1)=0x3(x41)2=0x3(x2+1)2(x21)2=0x3(x1)2(x+1)2(x2+1)2=0x=0,1
解答f(x,y)=x1y2+y1x2{fx=1y2xy1x2fy=xy1y2+1x2{fx=0fy=0xy=1x21y2x2y2=(1x2)(1y2)=1x2y2+x2y2x2+y2=1f(x,y)=xx2+yy2=x2+y2=1
解答
{¯BD=a¯BE=b{cosBDA=a2+16368acosBDE=a2+16b28acosBDA=cosBDE2a2b2=4;{cosBED=b2+16a28bcosBEC=b2+16648bcosBED=cosBECa22b2=32{2a2b2=4(1)a2b2=32(2)(1)+(2)3a2b2=283(3)(1)(3)a2=403a=2303

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