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2019年7月12日 星期五

108年高考三級工程數學詳解


108 年公務人員高等考試三級考試試題

類 科 :電力工程、電子工程、電信工程、醫學工程
科 目:工程數學
甲、申論題部分:(50分)

{a=(1,1,1)b=(3,4,5)c=(2,3,4)d=(1,2,2){ab=(2,3,4)ac=(1,2,3)ad=(0,1,1)=16234123011=16|4+4+0036|=16




(一)Cz=0Ccoszz(z2+8)dzCf(z)zdz=2πi×f(0)=2πi×cos002+8=2πi×18=πi4
(二)Cz=0Ccoshzz4dzCf(z)z4dz=2πi3!×f(3)(0)=2πi3!×sinh0=0


2sin(y2)dx+xycos(y2)dy=0{M(x,y)=2sin(y2)N(x,y)=xycos(y2){My=4ycos(y2)Nx=ycos(y2)MyNxF=F(x)(MF)y=(NF)x4ycos(y2)F=ycos(y2)F+xycos(y2)dFdx3ycos(y2)F=xycos(y2)dFdx1FdFdx=3x1FdFdxdx=3xdxlnF=3lnxF=x3{ux=MF=2x3sin(y2)uyNF=x4ycos(y2){u(x,y)=2x3sin(y2)dx=12x4sin(y2)+p(y)u(x,y)=x4ycos(y2)dy=12x4sin(y2)+q(x)u(x,y)=012x4sin(y2)+C=0y(2)=π28sin(π/2)+C=0C=8x4sin(y2)=16


:(一)A=[002121103]det(AλI)=0|λ0212λ1103λ|=0λ(λ2)(λ3)2(λ2)=0(λ2)(λ2+3λ2)=0(λ2)2(λ1)=0λ=1,2A12(二)λ=1AλI=[102111102][102111102][x1x2x3]=0{x12x3=0x1+x2+x3=0x1+2x3=0[x1x2x3]=a[211]λ=2AλI=[202101101][202101101][x1x2x3]=0x1+x3=0[x1x2x3]=b[101]+c[010][211],[101],[010]

乙、測驗題部分:(50分)

(A)T(ax)=aT(x)(B)T(D)T(C)


(D){T(2,2)=(0,0,0)T(1,1)=(0,0,0)T(1,1)=T(2,2)(D)


ATB=[12121212][13572468]=[5111723511172351117235111723]r1+r2,r1+r3,r1+r4,[5111723000000000000]rank(ATB)=1(A)


[32121232]=[cosπ6sinπ6sinπ6cosπ6]π6[32121232]100=[cos100π6sin100π6sin100π6cos100π6]=[cos2π3sin2π3sin2π3cos2π3]=[12323212](D)


L(x,y,z)=(2x+3y+z,3x+3y+z,2x+4y+z)=[231331241][xyz]=AXA1=[110101623]L1(x,y,z)=(x+y,x+z,6x2y3z)(A)


A=[532170122027](A)det(A5I)=|03217072022|=0(B)det(A8I)=|33217042021|=|33217000021|=0(C)det(A11I)=|63217012024|=|63217012000|=0(D)det(A14I)=|93217022027|0(D)當然也可以按部就班的計算det(AλI)=0λ324λ2+183λ440=0(λ5)(λ8)(λ11)=0λ=5,8,11


z=1+i=2(12+i12)=2(cos9π4+isin9π4)=2ei9π4f(z)=r13cosθ+2π3+ir13sinθ+2π3=213cos9π4+2π3+i213sin9π4+2π3=62(cos17π12+isin17π12)=62ei17π12(C)


C={t+t2it:02}{x=ty=t2{dx=dtdy=2tdtCz2dz=C(x+yi)2d(x+yi)=C(x2y2)dx2xydy+iC(x2y2)dy+2xydx=20(t2t4)dt2t32tdt+i202(t2t4)tdt+2t3dt=20(t25t4)dt+i20(4t32t5)dt=[13t3t5]|20+i[t413t6]|20=(831632)+i(16643)=883163i(A)


nnn!(z+i)2n=nnn!(z(i))2ncn=nnn!R=limncncn+1=limnnnn!×(n+1)!(n+1)n+1=limn(nn+1)n×n+1n+1=limn(nn+1)n=limn(nn+1)n=1e=1e{iR=1e(C)


y=acos(3x)+bsin(3x)+ccos(4x)y=3asin(3x)+3bcos(3x)4csin(4x){y(0)=0y(0)=3{a+c=03b=3{a+c=0b=1a+b+c=1(A)


L{tsin(at)}=2as(s2+a2)2L{t2ωsin(ωt)}=12ω2ωs(s2+ω2)2=s(s2+ω2)2(C)



y4xy+4x2y=x2+1,,y4xy+4x2y=0y=xmy=mxm1,y=m(m1)xm2m(m1)xm24mxm2+4xm2=0m(m1)4m+4=0m25m+4=0(m4)(m1)=0yh=c1x+c2x4(C),(D)(C)yp=a1+a2x2yp=2a2x,yp=2a2(D)若要按部就班,可用參數變換法求解:yh=c1x+c2x4yp=ϕ1x+ϕ2x4[xx4ddxxddxx4][ϕ1ϕ2]=[0x2+1][xx414x3][ϕ1ϕ2]=[0x2+1]{xϕ1+x4ϕ2=0ϕ1+4x3ϕ2=x2+1{ϕ1=x2/31/3ϕ2=1/3x+1/3x3{ϕ1=(x2/31/3)dx=19x313xϕ2=(1/3x+1/3x3)dx=13lnx16x2yp=ϕ1x+ϕ2x4=19x413x2+13x4lnx16x2=19x412x2+13x4lnxy=yh+yp=c1x+(c21/9)x412x2+13x4lnxC1x+C2x4+A1x2+A2x4lnx


y=e2xcosx{y=2e2xcosxe2xsinxy=3e2xcosx+4e2xsinxy=2e2xcosx11e2xsinxAy+By+Cy+Dy=0(2A+3B2C+D)e2xcosx+(11A+4BC)e2xsinx=0{2A+3B2c+D=011A+4BC=0(A)×:A=1,B=7,C=16,D=1011A+4BC=11+2816=10(B):A=B=1,C=7,D=15{2A+3B2c+D=011A+4BC=0(C)×:A=0,B=1,C=8,D=1711A+4BC=0+48=40(D)×:A=3,B=2,C=8,D=1611A+4BC=33+8+8=170(B)


(A)y=xy+Ay+By=0+A+BX=0A=0B=0(B)y=x2y+Ay+By=2+2Ax+Bx2=0A,B滿(C)y=ex+1y+Ay+By=ex+1+Aex+1+Bex+1=(A+B+1)ex+1=0A+B=1(D)y=excos(2x+3)y+Ay+By=(A+B3)excos(2x+3)(2A+4)exsin(2x+3)=0{A+B3=02A+4=0(B)


(A):u(t,x)=sin2tsinx{ux=sin2tcosxut=2cos2tsinx{2ux2=sin2tsinx2ut2=4sin2tsinx2ut2=222ux2(B):u(t,x)=e4tcos3x{ux=3e4tsin3x2ux2=9etcos3xut=4e4tcos3xut=(3/2)22ux2(C)×:u(t,x)=etsin3x{ux=3etcos3tut=etsin3x{2ux2=9etsin3x2ut2=etsin3x2ut2=(1/3)22ux2(D):u(t,x)=etsin3x{ux=3etcos3x2ux2=9etsin3xut=etsin3xut=322ux2(C)


L{te2t}=L{t}L{e2t}=1s21s2=1s2(s2)(C)


bn=1555F(x)sin(nπ5x)dx=15503sin(nπ5x)dx=35[5nπcos(nπ5x)]|50=3nπ(cos(nπ)1)=3(1cos(nπ))nπ(C)


P(0<X1)=10x23dx=[19x3]|10=19(A)


P[4<X12,0<Y<]=0124112e(x/4)(y/3)dxdy=0[13e(x/4)(y/3)]|124dy=13(0e3y/3e1y/3dy)=13[3e3y/3+3e1y/3]|0=13(0(3e3+3e1))=13(3e33e1)=e1e3(B)


E[X]=xf(x)dx=102x2dx=[23x3]|10=23(C)


考選部未公布非選答案,解題僅供參考

5 則留言:

  1. 謝謝你的辛苦詳解,對在職考生幫助很大
    感謝

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    回覆
    1. 希望有幫助到大家,這也是寫詳解的原始初衷!!

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  2. 找到這個網站的詳解 如獲至寶 國家考試的選擇題都不給詳解的 不會寫的題目要畫很多時間思考才想得出來 感謝!

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    回覆
    1. 能提供幫助就是建站的初衷,若發現錯誤,記得留言提醒,謝謝

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  3. 第一題不太懂何謂"平行四邊體",如果是指三向量所形成的"四面體",應該不會"平行"啊?

    回覆刪除