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2025年5月9日 星期五

114年新北高中教甄聯招-數學詳解

 新北市公立高級中等學校 114 學年度教師聯合甄選

一、填充題: 共 10 題,每題 7 分。

解答:{log2(x+1)+log2(y+1)=4xyxy=1{log2(x+1)(y+1)=4xyxy+1=0{(x+1)(y+1)=16(1)(x1)(y1)=0(2)eqn.(2){x=1y+1=8y=7y=1x+1=8x=7{(x,y)=(1,7)(x,y)=(7,1)x+y=8
解答:(9,11,13)(10,12),4+43+54+4C84/212C84×2=70;3+5C83C83×2=11270+112=182
解答:a,bx46x2+αx+(5α)=(xa)3(xb)=x4(3a+b)x3+(3ab+3a2)x2(a3+3a2b)x+a3b{3a+b=03ab+3a2=6b=3aa2=1{a=1b=3{α=85α=3a=1b=3{α=85α=3α=8
解答:

z=x+yi2x2+y2<(x1)2+y2x2+23x+y2<13(x+13)2+y2<49Γ:(x+13)2+y2=49{A(0,1/3)r=2/3ΓyB(0,3/3),C(0,3/3)O(0,0)OAB=60{ABC=3/9ABC=4π/27=4π2739
解答:{1=1/42=1/162=((1/4)×(1/2))2=1/643=((1/4)×(1/2))2=1/644=((1/4)×(1/2))2=1/645=((1/4)×(1/2))2=1/646=1/42=1/16=2×116+4×164=316=2316=2916=1.8125
解答:f(x)=x5+x2+1=(xr1)(xr2)(xr5)P(x)=x22=(x+2)(x2)=(2x)(2x)P(r1)P(r2)P(r5)=(2r1)(2r1)×(2r2)(2r2)××(2r5)(2r5)=[(2r1)(2r2)(2r5)]×[(2r1)(2r2)(2r5)]=f(2)×f(2)=(3+42)(342)=932=23
解答:y=x+bx2=1+b+2x2x=b+2y1+2=2y+by1f(y)=x+a2x+1=(a+2)y+ba5y+2b1{limyf(y)=(a+2)/5=1a=3limy0f(y)=(ba)/(2b1)=0a=ba=b=3a+b=6
解答:n5+2n2+1n2+3=n33n+2+9n5n2+39n5n2+3Zn=9,8,,0,1,,9n=1,8
解答:1,3,5,74!=243124
解答:12cosπ3=12eiπ/3z=12eiπ/3zn=12nenπi/3f(z)=n=1zn=z1zzf(z)=z(1z)2=n=1nzn=n=1n2nenπi/3z=12eiπ/3=14(1+3i)zf(z)=14(1+3i)(343i4)2=231+3i13i=13+33in=1n2ncosnπ3=zf(z)=13

二、計算題: 共 3 題,每題 10 分。

解答:Cn2Cm2=12(n(n1)m(m1))=2025n2m2n+m=4050(nm)(n+m)(nm)=4050(nm)(n+m1)=4050=2×34×52n,mNnm<n+m1,nm40504050=54×75nm54
解答:a1+2a2++nan<2025+(a1+a22++ann)(a1a11)+(2a2a22)++(nanann)<2025nk=1(kakakk)<2025f(x)=kxxkf(x)=kkxk1f(x)=k(k1)xk2f(k)=0x=1f(1)=k(k1)<0,k=2,3,f(1)=k1nk=1(k1)=n(n+1)2n=n2n2<2025n(n1)<4050n=64
解答:
PAB=PBC=PCA=θPDA=PFA=90PDAF:{FAP=PDF=1PAD=PFD=θ,{PDBE{PBD=PED=2PBE=PDE=θPECF{PCE=PFE=3PCF=PEF=θ{A=θ+1=EDFB=θ+2=DEFC=θ+3=DFEABCDEFQED

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解題僅供參考,其他教甄試題及詳解




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