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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=lim


y=a\cos { \left( 3x \right)  } +b\sin { \left( 3x \right)  } +c\cos { \left( 4x \right)  } \\ \Rightarrow y'=-3a\sin { \left( 3x \right)  } +3b\cos { \left( 3x \right)  } -4c\sin { \left( 4x \right)  } \\ \Rightarrow \begin{cases} y(0)=0 \\ y'(0)=3 \end{cases}\Rightarrow \begin{cases} a+c=0 \\ 3b=3 \end{cases}\Rightarrow \begin{cases} a+c=0 \\ b=1 \end{cases}\Rightarrow a+b+c=1,故選\bbox[red,2pt]{(A)}


L\{t\sin{(at)} \}={2as\over (s^2+a^2)^2} \Rightarrow L\{{t\over 2\omega}\sin{(\omega t)} \}={1\over 2\omega}{2\omega s\over (s^2+\omega^2)^2} = { s\over (s^2+\omega^2)^2},故選\bbox[red,2pt]{(C)}



y''-{4\over x}y'+{4\over x^2}y=x^2+1,先求齊次解,即y''-{4\over x}y'+{4\over x^2}y=0\\ 令y=x^m\Rightarrow y'=mx^{m-1},y''=m(m-1)x^{m-2}代回齊次式\\ \Rightarrow m(m-1)x^{m-2}-4mx^{m-2}+4x^{m-2}=0\Rightarrow m(m-1)-4m+4=0 \\ \Rightarrow m^2-5m+4=0\Rightarrow (m-4)(m-1)=0\Rightarrow y_h=c_1x+c_2x^4\\ \Rightarrow 只剩下選項(C),(D)正確\\ (C) y_p=a_1+a_2x^2 \Rightarrow y_p'=2a_2x,y_p''=2a_2代入原式不合,故選\bbox[red,2pt]{(D)}。若要按部就班,可用參數變換法求解:y_h=c_1x+c_2x^4 \Rightarrow y_p=\phi_1 x+\phi_2 x^4 \Rightarrow \left[\begin{matrix}x&x^4\\{d\over dx}x &{d\over dx}x^4 \end{matrix}\right]\left[\begin{matrix}\phi_1'\\\phi_2' \end{matrix}\right]= \left[\begin{matrix}0\\x^2+1 \end{matrix}\right] \\ \Rightarrow \left[\begin{matrix}x&x^4\\1 &4x^3 \end{matrix}\right] \left[ \begin{matrix} \phi_1'\\\phi_2' \end{matrix}\right]= \left[\begin{matrix}0\\x^2+1 \end{matrix}\right] \Rightarrow \begin{cases} x\phi_1'+x^4\phi_2'=0\\\phi_1'+4x^3\phi_2'=x^2+1 \end{cases} \Rightarrow \begin{cases} \phi_1'=-x^2/3-1/3\\\phi_2'=1/3x+ 1/3x^3 \end{cases}\\ \Rightarrow \begin{cases} \phi_1=\int{(-x^2/3-1/3)\,dx}=-{1\over 9}x^3-{1\over 3}x\\\phi_2=\int{(1/3x+ 1/3x^3)\,dx} ={1\over 3}\ln{x}-{1\over 6x^2} \end{cases} \Rightarrow y_p=\phi_1 x+\phi_2 x^4\\ = -{1\over 9}x^4-{1\over 3}x^2+{1\over 3}x^4\ln{x}-{1\over 6}x^2= -{1\over 9}x^4-{1\over 2}x^2+{1\over 3}x^4\ln{x}\\ \Rightarrow y=y_h+y_p=c_1 x+(c_2-1/9)x^4-{1\over 2}x^2+{1\over 3}x^4\ln{x}\equiv C_1x+C_2x^4+A_1x^2+A_2x^4\ln{x}


y=e^{-2x}\cos{x} \Rightarrow \begin{cases}y'=-2e^{-2x}\cos{x}-e^{-2x}\sin{x}\\ y''=3e^{-2x}\cos{x}+4e^{-2x}\sin{x}\\ y'''=-2e^{-2x}\cos{x}-11e^{-2x}\sin{x} \end{cases}\Rightarrow Ay'''+By''+Cy'+Dy=0\\ \Rightarrow (-2A+3B-2C+D)e^{-2x}\cos{x}+ (-11A+4B-C)e^{-2x}\sin{x}=0\\ \Rightarrow \begin{cases}-2A+3B-2c+D=0\\ -11A+4B-C=0\end{cases} \\(A)\times:A=1,B=7,C=16,D=-10\Rightarrow -11A+4B-C=-11+28-16=1\ne 0\\ (B)\bigcirc:A=B=1,C=-7,D=15 \Rightarrow \begin{cases}-2A+3B-2c+D=0\\ -11A+4B-C=0\end{cases} \\(C)\times:A=0,B=1,C=8,D=17 \Rightarrow -11A+4B-C=0+4-8=-4\ne 0 \\ (D) \times: A=3,B=2,C =-8,D=-16\Rightarrow -11A+4B-C=-33+8+8=-17\ne 0 \\,故選\bbox[red,2pt]{(B)}


(A)y=x\Rightarrow y''+Ay'+By=0+A+BX=0 \Rightarrow A=0且B=0\\(B)y=x^2 \Rightarrow y''+Ay'+By=2+2Ax+Bx^2=0,無法找出A,B滿足該式\\ (C)y=e^{x+1}\Rightarrow y''+Ay'+By= e^{x+1}+Ae^{x+1}+Be^{x+1}=(A+B+1)e^{x+1}=0\\ \Rightarrow A+B=-1\\ (D)y=e^x\cos{(2x+3)} \Rightarrow y''+Ay'+By= (A+B-3)e^x\cos{(2x+3)}-(2A+4)e^x\sin{(2x+3)} =0\\ \Rightarrow \begin{cases}A+B-3=0\\2A+4=0\end{cases}\\,故選\bbox[red,2pt]{(B)}


(A)\bigcirc :u(t,x)=\sin{2t}\sin{x}\Rightarrow \begin{cases}\frac{\partial u}{\partial x}=\sin{2t}\cos{x}\\ \frac{\partial u}{\partial t}=2\cos{2t}\sin{x}\end{cases} \Rightarrow \begin{cases}\frac{\partial^2 u}{\partial x^2}=-\sin{2t}\sin{x}\\ \frac{\partial^2 u}{\partial t^2}=-4\sin{2t}\sin{x}\end{cases} \Rightarrow \frac{\partial^2 u}{\partial t^2}= 2^2\frac{\partial^2 u}{\partial x^2}\\(B)\bigcirc :u(t,x)=e^{-4t}\cos{3x} \Rightarrow \begin{cases}\frac{\partial u}{\partial x}=-3e^{-4t}\sin{3x} \Rightarrow \frac{\partial^2 u}{\partial x^2}=-9e^{-t}\cos{3x}\\ \frac{\partial u}{\partial t}= -4e^{-4t}\cos{3x} \end{cases} \Rightarrow \frac{\partial u}{\partial t}= (3/2)^2\frac{\partial^2 u}{\partial x^2}\\ (C)\times :u(t,x)=e^t\sin{3x} \Rightarrow \begin{cases}\frac{\partial u}{\partial x}=3e^t\cos{3t}\\ \frac{\partial u}{\partial t}= e^t\sin{3x} \end{cases} \Rightarrow \begin{cases}\frac{\partial^2 u}{\partial x^2}=-9e^t\sin{3x}\\ \frac{\partial^2 u}{\partial t^2}=e^t\sin{3x}\end{cases} \Rightarrow \frac{\partial^2 u}{\partial t^2}=- (1/3)^2\frac{\partial^2 u}{\partial x^2}\\ (D)\bigcirc :u(t,x)=e^{-t}\sin{3x} \Rightarrow \begin{cases}\frac{\partial u}{\partial x}=3e^{-t}\cos{3x} \Rightarrow \frac{\partial^2 u}{\partial x^2}=-9e^{-t}\sin{3x}\\ \frac{\partial u}{\partial t}= -e^{-t}\sin{3x} \end{cases} \Rightarrow \frac{\partial u}{\partial t}= 3^2\frac{\partial^2 u}{\partial x^2}\\,故選\bbox[red,2pt]{(C)}


L\left\{t*e^{2t} \right\}=L\{t\}\cdot L\{e^{2t}\}=\frac{1}{s^2}\cdot\frac{1}{s-2}=\frac{1}{s^2(s-2)},故選\bbox[red,2pt]{(C)}


b_n=\frac{1}{5} \int_{-5}^5{F(x)\sin{\left(\frac{n\pi}{5}x\right)}\,dx} = \frac{1}{5} \int_{0}^5 {3 \sin{\left(\frac{n\pi}{5}x\right)}\,dx} = \frac{3}{5}\left.\left[-\frac{5}{n\pi}\cos{\left(\frac{n\pi}{5}x\right)}\right]\right|_0^5\\ =-\frac{3}{n\pi}(\cos{(n\pi)}-1)=\frac{3(1-\cos{(n\pi))}}{n\pi} ,故選\bbox[red,2pt]{(C)}


P(0< X\le 1)=\int_0^1{\frac{x^2}{3}\,dx} = \left.\left[\frac{1}{9}x^3\right]\right|_0^1=\frac{1}{9},故選\bbox[red,2pt]{(A)}


P[4< X \le 12,0 < Y < \infty]=\int_0^\infty{\int_4^{12}{\frac{1}{12}e^{-(x/4)-(y/3)}\,dx}\,dy}= \int_0^\infty{\left. \left[-\frac{1}{3}e^{-(x/4)-(y/3)}\right]\right|_4^{12}\,dy}\\ = -\frac{1}{3}\left(\int_0^\infty{e^{-3-y/3}-e^{-1-y/3}\,dy}\right)=-\frac{1}{3}\left.\left[-3e^{-3-y/3}+3e^{-1-y/3}\right]\right|_0^\infty= -\frac{1}{3}\left(0-(-3e^{-3}+3e^{-1}) \right)\\ =-\frac{1}{3}\left(3e^{-3}-3e^{-1}\right)=e^{-1}-e^{-3},故選\bbox[red,2pt]{(B)}


E[X]=x\int{f(x)\,dx}=\int_0^1{2x^2\,dx}=\left.\left[\frac{2}{3}x^3\right]\right|_0^1={2\over 3},故選\bbox[red,2pt]{(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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    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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    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?

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