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2024年4月22日 星期一

113年身障三等-工程數學詳解

113年身心障礙人員考試

等 別: 三等考試
類 科: 電力工程
科 目: 工程數學

甲、 申論題部分:( 50 分)

解答: ()2xcos(3y)3x2sin(3y)dydx=02xcos(3y)dx3x2sin(3y)dy=0{P(x,y)=2xcos(3y)Q(x,y)=3x2sin(3y){Py=6xsin(3y)Qx=6xsin(3y)Px=Qy,()Φ(x,y)=2xcos(3y)dx=3x2sin(3y)dyΦ(x,y)=x2cos(3y)+ϕ(y)=x2cos(3y)+ρ(x)+c1x2cos(3y)+c1=0:y=13cos1c1x2,c1()y(1)=011+c1=0c1=1x2cos(3y)=1cos(3y)=1x2y=13cos11x2
解答: ()det(A)=14201241556=930A,()():det(A)=93()[AI]=[732100524010121001]R1/7R1[137271700524010121001]R25R1R2,R3R1R3[13727170001738757100177571701]7R2R2[13727170001385700177571701]R1+(3/7)R2R1,R3(17/7)R2R3[10162300138570009312171]R3/93R3[101623001385700014311793193]R116R3R1,R238R3R2[100231793169301033159338930014311793193]A1=[231793169333159338934311793193](四)AB=Idet(B)=1det(A)=193det(B4)=1934

乙、 測驗題部分:( 50 分)


解答: rref(A)=[10010100001000000000]rank(A)=3,(C)
解答: y=xm{y=mxm1y=m(m1)xm2x2yxy3y=m(m1)xmmxm3xm=(m22m3)xm=0m22m3=(m3)(m+1)=0m=1,3y=c1x1+c2x3,(A)
解答: T(x,y)=(2x+y,3x+4y)[2134][xy]=0Ax=0A=[2134]A1=[45153525]rank(T)=2nullity(T)=0=rank(T)=20,(D)
解答: AA=AT[023a03bc0]=[0ab20c330]{a=2b=3c=3(D)×:bc=3×(3)=99,(D)
解答: [1+x23412+x34123+x41234+x]R1R2R1,R3R2R3,R4R2R4[xx0012+x340xx00x0x]det(A)=x|2+x34xx0x0x||x00xx0x0x|=x(x3+9x2)(x3)=x4+10x3=x3(10+x),(A)
解答: det(AλI)=0(λ+2)(λ2)(λ4)=0λ=2,2,4λ1=2(Aλ1I)v=0v1=k(111)λ2=2(Aλ2I)v=0v2=s(111)λ3=4(Aλ3I)v=0v3=t(111){(A)I=v3(B)J=v2(C)×(D)L=v1,(C)
解答: D=[3417]det(DλI)=0(λ5)2=0λ=5,(AλI)v=0v=k[21],D,(D)
解答: {A(1,0,1)B(2,x,4)C(5,5,7)D(8,8,10){AC=(4,5,6)AD=(7,8,9)n=AC×AD=(3,6,3)E:3(x1)+6y3(z1)=0x2y+z=2BE22x+4=2x=2,(B)
解答: (y+x3)dx=xdyy=dydx=yx+x2y=ax3+bxc{y(1)=a+b=2y=3ax2+bcxc1=yx+x2=ax2+bxc1+x2=(a+1)x2bxc1{3a=a+1bc=b{a=1/2b=3/2c=1,(D)
解答: f(x)an=0bn=01sin(nπx)dx+10sin(nπx)dx=[1nπcos(nπx)]|01+[1nπcos(nπx)]|10=2nπ(1(1)n)f(x)=n=12nπ(1(1)n)sin(nπx)=4πsin(πx)+0sin(2πx)+43πsin(3πx)+0sin(4πx)+=4π(sin(πx)+0sin(2πx)+13sin(3πx)+0sin(4πx)+){a=1b=0c=13d=0a+b+c+d=43,(B)
解答: {P(x,y)=4x2y3xy2Q(x,y)=x32x2y{Py=4x26xyQx=3x24xyPyQxQ=1xu=1xuu(x)=x=xmyn{m=1n=0m+n=1,(C)
解答: {lnx2=2lnxlnx3=3lnxlnx,lnx2,lnx3,(B)
解答: y(t)=t0y(τ)cos(tτ)dτL{y(t)}=L{t0y(τ)cos(tτ)dτ}sY(s)1=L{y(t)}L{cost}=Y(s)ss2+1(sss2+1)Y(s)=1Y(s)=s2+1s3=1s+1s3y(t)=L1{1s}+L1{1s3}=1+12t2,(D)
解答: Res(f,z=0)=z+1(z2)2(z+4)|z=0=144=116Res(f,z=2)=ddzz+1z(z+4)|z=2=1z(z+4)(z+1)(2z+4)(z2+4z)2|z=2=112Cf(z)dz=2πi(116112)=124πi,(C)
解答: f(z)=1(z1)(z2)=1z21z1=1211z21z111z=12(1+z2+z24+z38+)1z(1+1z+1z2++1z3+)=1z21z12z4+{a=1b=1c=1/2d=1/4,(B)
解答: z=x+iyz2=(x2y2)+i(2xy)f(z)=Re(z2)iIm(z2)=(x2y2)i(2xy){u(x,y)=x2y2v(x,y)=2xy{ux=2xvy=2xuxvyf(z),(B)
解答: z=2+2iz2=8iz4=64z46iz2+16=646i×(8i)+16=64+48+16=02+2i,(D)
解答: Var(Z)=Var(3X2Y+3)=9Var(x)+4Var(Y)12Cov(X,Y)=5492+4Var(Y)12(2)=544Var(Y)=12Var(Y)=σ2Y=3,(C)
解答: fX(x)=1xxydy=12x12x3E[X]=10(12x12x3)dx=115,(C)
解答: a,b,c=(10.4)(10.5)(10.4)=0.60.50.6=0.12=10.12=0.88,(A)

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解題僅供參考,其他歷年試題及詳解



 

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