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2020年11月14日 星期六

108年關西高中教甄-數學詳解

國立關西高級中學108學年度第1次教師甄選數學科試題


{Γ1:yx2/4x24yΓ2:|x1|+|y1|=k{Γ1:()Γ2:(1,1)k(x,y)Γ1使(x=a,y=a2/4)k=|a1|+|a2/41|a=1(p,q)=(x,y)=(a,a2/4)=(1,1/4)

{zn+1=(1+3i)zn3i(1)zn=(1+3i)zn13i(2)(1)(2)zn+1zn=(1+3i)(znzn1)zn+1zn=(1+3i)2(zn1zn2)==(1+3i)n1(z2z1)=(1+3i)n1(3(1+3i)3i3)=(1+3i)n1(23i)|zn+1zn|=|(1+3i)n1(23i)|=|(1+3i)n1||23i|=2n1×23=2n3n=11|zn+1zn|=n=112n3=13×1/211/2=13

EY=2(Cn0p0(1p)n+Cn2p2(1p)n2+)(Cn1p1(1p)n1+Cn3p2(1p)n3+)=(12+32)(Cn0p0(1p)n+Cn2p2(1p)n2+)+(1232)(Cn1p1(1p)n1+Cn3p2(1p)n3+)=12nk=0Cnkpk(1p)nk+32nk=0Cnk(1)kpk(1p)nk=12(p+(1p))n+32(p+(1p))n=12+32(12p)n

{y2+xy7x=0x,yNy2=x(7y)>0y=1,2,,6y=636=x(76)x=36(x,y)=(36,6)y=1,2,,5x,yN

5. 將正整數1,2,3,4,5,6,7任意分成兩組,使得每組至少有一個數。則第一組數的和與第二組數的和相等的機率為________。
符合要求的分組方式(1,6,7)+(2,3,4,5),(3,4,7)+(1,2,5,6),(2,5,7)+(1,3,4,6), (3,5,6)+(1,2,4,7),共四種;七個數字分兩組有C71+C72+C73=7+21+ 35=63種分法;因此機率為463

6. 已知點A(3,1),B(53,2)是平行四邊形ABCD的頂點,已知ABCD四個頂點都落在函數f(x)=log2axbx1的圖形上,試求平行四邊形ABCD的面積=___
{f(x)=log2axbx1A(3,1)B(53,2)A,B,C,Dy=f(x){f(3)=1f(53)=2{log23ab2=1log25a/3b2/3=2{3ab=45a3b=8{a=1b=1f(x)=log2x+1x1f(x)=log2x+1x1=log2x1x+1=log2x+1x1=f(x)f(x)=f(x)y=f(x){A(3,1)A(3,1)B(53,2)B(53,2)ABABy=f(x){AB=(43,1)AB=(143,3)ABAB=|AB|2|AB|2(ABAB)2=259×2779(299)2=6769=263


Γ:y2=4cxL:4x+y20=0P;PLP(t,204t)Γ(204t)2=4ct4t2(40+c)t+100=0{t1=(40+c+c2+80c)/8t2=(40+cc2+80c)/8{B(t1,204t1)C(t2,204t1)A(xa,ya)F(c,0)=(A+B+C)÷3{c=(t1+t2+xa)÷3=(10+c/4+xa)÷30=(204t1+204t2+ya)÷3=(yac)÷3{xa=11c/410ya=cA(xa,ya)Γy2a=4cxac2=4c(114c10)=11c240c10c240c=0c=4Γ:y2=16x
{F1(1,0)F2(1,0)c=1x2a2+y2b2=1;{y=x3a2=b2+1x2b2+1+(x3)2b2=1(2b2+1)x223(b2+1)x+3(b2+1)b2(b2+1)=0012(b2+1)24(2b2+1)(3(b2+1)b2(b2+1))=08b68b2=08b2(b41)=0b=1a2=b2+1=2:x22+y21=1
{z1=a+biˉz1=abiz2=c+diˉz2=cdi,a,b,c,dR;x4+ax3+bx2+cx+d=(xz1)(xˉz1)(xz2)(xˉz2)=(x2(z1+ˉz1)x+z1ˉz1)(x2(z2+ˉz2)x+z2ˉz2)b=z1ˉz1+z2ˉz2+(z1+ˉz1)(z2+ˉz2)=(a+bi)(abi)+(c+di)(cdi)+(2a)×(2c)=a2+b2+c2+d2+4ac{z1z2=(acbd)+(ad+bc)i=13+iˉz1+ˉz2=(a+c)(b+d)i=3+4i{acbd=13ad+bc=1a+c=3b+d=4(a+c)2+(b+d)2=32+(4)2a2+b2+c2+d2+2ac+2bd=25a2+b2+c2+d2+2ac+2(ac13)=25a2+b2+c2+d2+4ac=25+26=51
{Γ1:y=1/exy=1/exΓ2:y=ex+1y=ex+1{LΓ1A(a,1/ea)LΓ2B(b,eb+1){A:y(a)=1/ea(1)B:y(b)=eb+1(2)¯AB:eb+1eaba(3){(1)=(2)ea=eb+1a=b+1(4)(1)=(3)eb+1eaba=ea(5)(4)(5)=2ea2a1=ea2a1=2a=3/2LAe3/2L:y=e3/2(x+32)+e3/2

(1)fn(x)={nn2|x||x|1n0|x|1n={nn2x0x1nn+n2x1nx00In=11fn(x)cosxdx=01/n(n+n2x)cosxdx+1/n0(nn2x)cosxdx=[nsinx+n2xsinx+n2cosx]|01/n+[nsinxn2xsinxn2cosx]|1/n0=(n2n2cos1n)+(n2cos1n+n2)=2n2(1cos1n)(2)limnIn=limn2n2(1cos1n)=2limn1cos(1/n)1/n2=2limn(1n2)sin(1/n)2/n3=2limnsin(1/n)2/n=2×12=1(limnsin(1/n)1/n=limx0sin(x)x=1)
(1){L:y=k(xa)+a33ay=f(x)=x33xk(xa)+a33a=x33xk(xa)+3(xa)=x3a3=(xa)(x2+ax+a2)(xa)(x2+ax+a23k)=0x2+ax+a23k=0>0a{a24(a23k)>0a2+aa+a23k0{k>34a23k3a33

a=3P(3,0);{Q(x1,y1)R(x2,y2)k<0{x1>x2y2>y1LPkL:y=k(x3)P,Q,Rx33x=k(x3)x33x=x(x+3)(x3)=k(x3)x(x+3)=kx2+3xk=0{x1+x2=3x1x2=k(x1x2)2=(x1+x2)24x1x2=3+4kx1x2=4k+3x2x1=4k+3y2y1=k(x23)k(x13)=k(x2x1)=k4k+3=4k3+3k214=12(g(k)=4k3+3k2g(k)=12k2+6kg(1/2)=1/4)S(k)=OQR=OPROPQ=123(y2y1)32×12=34S(k)34


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