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)
解:
(一)C包住z=0⇒∫Ccoszz(z2+8)dz≡∫Cf(z)zdz=2πi×f(0)=2πi×cos002+8=2πi×18=πi4
(二)C包住z=0⇒∫Ccoshzz4dz≡∫Cf(z)z4dz=2πi3!×f(3)(0)=2πi3!×sinh0=0
解:2sin(y2)dx+xycos(y2)dy=0⇒{M(x,y)=2sin(y2)N(x,y)=xycos(y2)⇒{∂M∂y=4ycos(y2)∂N∂x=ycos(y2)⇒∂M∂y≠∂N∂x⇒非正合假設F=F(x)為積分因子⇒∂(MF)∂y=∂(NF)∂x⇒4ycos(y2)F=ycos(y2)F+xycos(y2)dFdx⇒3ycos(y2)F=xycos(y2)dFdx⇒1FdFdx=3x⇒∫1FdFdxdx=∫3xdx⇒lnF=3lnx⇒F=x3⇒{∂u∂x=MF=2x3sin(y2)∂u∂yNF=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)=0為其通解,即12x4sin(y2)+C=0為其通解,再由y(2)=√π2⇒8sin(π/2)+C=0⇒C=−8⇒其解為x4sin(y2)=16。
解:(一)A=[00−2121103]⇒det(A−λI)=0⇒|−λ0−212−λ1103−λ|=0⇒−λ(λ−2)(λ−3)−2(λ−2)=0⇒(λ−2)(−λ2+3λ−2)=0⇒(λ−2)2(λ−1)=0⇒λ=1,2,即A的特徵值為1及2(二)λ=1⇒A−λI=[−10−2111102]⇒[−10−2111102][x1x2x3]=0⇒{−x1−2x3=0x1+x2+x3=0x1+2x3=0⇒[x1x2x3]=a[−211]λ=2⇒A−λI=[−20−2101101]⇒[−20−2101101][x1x2x3]=0⇒x1+x3=0⇒[x1x2x3]=b[10−1]+c[010]⇒特徵向量為[−211],[10−1],[010]
解:(A)少一個條件T(ax)=aT(x)(B)T不只是函數,更重要的是線性(D)不一定找得到線性的函數T,故選(C)
解:(D){T(2,2)=(0,0,0)T(1,1)=(0,0,0)⇒T(1,1)=T(2,2)⇒不是一對一,故選(D)
解:ATB=[12121212][13572468]=[5111723511172351117235111723]−r1+r2,−r1+r3,−r1+r4,→[5111723000000000000]⇒rank(ATB)=1,故選(A)
解:[√32−1212√32]=[cosπ6−sinπ6sinπ6cosπ6]為旋轉π6矩陣⇒[√32−1212√32]100=[cos100π6−sin100π6sin100π6cos100π6]=[cos2π3−sin2π3sin2π3cos2π3]=[−12−√32√32−12],故選(D)
解:L(x,y,z)=(2x+3y+z,3x+3y+z,2x+4y+z)=[231331241][xyz]=AX⇒A−1=[−110−1016−2−3]⇒L−1(x,y,z)=(−x+y,−x+z,6x−2y−3z),故選(A)
解:A=[5321701220−27](A)det(A−5I)=|032170720−22|=0(B)det(A−8I)=|−332170420−2−1|=|−332170000−2−1|=0(C)det(A−11I)=|−632170120−2−4|=|−63217012000|=0(D)det(A−14I)=|−932170−220−2−7|≠0,故選(D)當然也可以按部就班的計算det(A−λI)=0⇒λ3−24λ2+183λ−440=0⇒(λ−5)(λ−8)(λ−11)=0⇒特徵值λ=5,8,11
解:z=1+i=√2(1√2+i1√2)=√2(cos9π4+isin9π4)=√2ei9π4⇒f(z)=r13cosθ+2π3+ir13sinθ+2π3=√213cos9π4+2π3+i√213sin9π4+2π3=6√2(cos17π12+isin17π12)=6√2ei17π12,故選(C)
解:C={t+t2i∣t:0→2}⇒{x=ty=t2⇒{dx=dtdy=2tdt⇒∫Cz2dz=∫C(x+yi)2d(x+yi)=∫C(x2−y2)dx−2xydy+i∫C(x2−y2)dy+2xydx=∫20(t2−t4)dt−2t3⋅2tdt+i∫202(t2−t4)tdt+2t3dt=∫20(t2−5t4)dt+i∫20(4t3−2t5)dt=[13t3−t5]|20+i[t4−13t6]|20=(83−16−32)+i(16−643)=−883−163i,故選(A)
解:nnn!(z+i)2n=nnn!(z−(−i))2n⇒令cn=√nnn!⇒R=lim
解: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)}
解: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)}
解:
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}
解: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)}
解:(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)}
解:(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)}
解: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)}
解: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)}
解: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)}
解: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)}
解: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)}
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