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2021年6月11日 星期五

105年中壢高中教甄-數學詳解

國立中央大學附屬中壢高級中學105學年度第 1次教師甄選

一、填充題

解答an=2an+1+anan+1ananan+1=2an+1anan+1+anan+1anan+11an+1=21an+1bn+1=2bn+1{bn=1anb1=1a1=1bn=2bn1+1=2(2bn2+1)+1=22bn2+2+1==2n1b1+2n2+2n3++1=1+2+22++2n1=2n11an=2n1an=1(2n1)2=122n2n+1+1
解答α=x[x]0α<1[x]=xα4x220(xα)+23=04x220x+23=20α4(x52)22=20α20<4(x52)220{4(x52)2218<4(x52)204(x52)225212x52+12[x]=1,2,3[x]{[x]=1[x]=2[x]=3{4x2+3=0()4x217=04x237=0x=172,372
解答


x2+(mx3m+2)2+x2+(mx3m+10)2=10x2+(mx3m(2))2+x2+(mx3m(10))2=10{L:y=m(x3)Γ:x2+(y+2)2+x2+(y+10)2=10Γ:{F1(0,2)F2(0,10)2a=10{O(0,6)a=5b=3c=4Γ:x29+(y+6)225=12:yk=m(xh)±b2m2+a2(h,k)y+6=mx±9m2+25(3,0)6=3m±9m2+259m2+25=(3m6)2m=1136m=x=3m>1136
解答
{A,B,C,D¯AB=¯BC=¯ADABCD{A(15/2,0)B(15/2,0)O(0,a)¯BP¯CD¯AQ¯CD¯CP=(¯CD¯AB)÷2=5/2BPC¯BP=152(5/2)2=5235C(10,5235);r=¯OB=¯OC(152)2+a2=102+(5235a)2a=3235r=(152)2+a2=315ABCD=(2r)×(2r)=(615)2=540

解答AB:y=m(x4)+1xy=1mx(x4)+x=1mx2+(14m)x1=0=4m1m=4×2(=4)m=14AB:x+4y=8xy=1C(t,1/t)ABd=|t+4/t8|17|48|17=41717(t+4t2t4t=4)
解答a=10100[101000010100+1]=[a100a+1]=[((a+1)1)100a+1]=[100k=0C100k(a+1)k1(1)100k]=[1a+1C1001+C1002(a+1)+C100100(a+1)99]=[1a+1+]==C1001+C1002(a+1)+C100100(a+1)99(1)C100m(a+1)n=C100m(Cnnan+Cnn1an1++Cn1a+1)=100k+C100m(1)=100K+(C100100C10099++C1002C1001+C1000)C1000,m,n,kN=100K+(1+(1))1001=100k199mod100,KN[101000010100+1]÷10099
解答limn(n1+32n+n32n+52n++n(2m1)2n+(2m+1)2n)=limn(32n(13)2n+1+52n(35)2n+1++(2m+1)2n(2m12m+1)2n+1)=32+52++(2m+1)2=mk=1(2k+1)2=4mk=1k2+4mk=1k+mk=11=23m(m+1)(2m+1)+2m(m+1)+mlimm23m(m+1)(2m+1)+2m(m+1)+mm3=43
解答


[x+12]=k,kZkx+12<k+1x[2k1,2k+1)x;log2{log2(|x[x+12]|)}<00<log2(|x[x+12]|)<11<|x[x+12]|<2{1<x[x+12]<22<x[x+12]<1{x+1<[x+12]<x+2x2<[x+12]<x1()4<x<22<x<4x±3
解答(a,b,c)3a,b,c(9,0,0)=1(8,1,0)C98=9,(7,2,0)C97=36,(6,3,0)C96=84,(5,4,0)C95=126(7,1,1)C97=36,(6,2,1)C96C32=252,(5,3,1)C95C43=504,(5,2,2)C95C42÷2=378;(4,4,1)C94C51÷2=315,(4,3,2)C94C53=1260;(3,3,3)C93C63÷3!=2801+9+36+84+126+36+252+504+378+315+1260+280=3281
解答
OAB:|z|sinα=|w|sin(πθ)=|w|sinθsinθ=|w||z|sinα=110sinαsin2θ=sin2α1001100tan2θ=sin2θcos2θ=sin2θ1sin2θ=11sin2θ1111/1001=199

二、計算題

解答
(1)x2f(x)=35x5+12ax413x3+2x0tf(t)dt2xf(x)+x2f(x)=3x4+2ax3x2+2xf(x)f(x)=3x2+2ax1f(x)=x3+ax2x+C,Cf(0)=0C=0f(x)=x3+ax2x(2)f(x)=x3+ax2x=0x(x2+ax1)=0x=0,α,β,{α=aa2+42β=a+a2+42{α2+β2=(α+β)22αβ=a2+2α3+β3=(α+β)33αβ(α+β)=a33aα4+β4=(α2+β2)22α2β2=(a2+2)22=a4+4a2+2S(a)=0αf(x)dxβ0f(x)dx=[14x4+13ax312x2]|0α[14x4+13ax312x2]|β0=14α413aα3+12α214β413aβ3+12β2=14(α4+β4)13a(α3+β3)+12(α2+β2)=14(a4+4a2+2)13a(a33a)+12(a2+2)=112a4+12a2+123S(a)=013a3+a=0a(a2+3)=0a=0S(a)S(0)=12
解答ω=cos2π7+isin2π7ω7=1(2ω)(2ω3)(2ω5)=(42ω2ω3+ω4)(2ω5)=82ωω24ω3+2ω44ω5+2ω6=82cos2π7cos4π74cos6π7+2cos8π74cos10π7+2cos12π7=82cos2π7cos4π7+4cosπ72cosπ74cos4π7+2cos2π7=8+2cosπ75cos4π7=2sin2π7sin4π74sin6π7+2sin8π74sin10π7+2sin12π7=2sin2π7sin4π74sinπ72sinπ7+4sin4π72sin2π7=6sinπ74sin2π7+3sin4π78+2cosπ75cos4π7+i(6sinπ74sin2π7+3sin4π7)
 
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