Loading [MathJax]/jax/element/mml/optable/BasicLatin.js

2024年11月22日 星期五

113年專技高考電機工程技師-工程數學詳解

113年專門職業及技術人員高等考試

等 別:高等考試
類 科:電機工程技師
科 目:工程數學(包括線性代數、微分方程、複變函數與機率)

解答:y4y+5y=0λ24λ+5=0λ=2±iyh=e2x(c1cos(x)+c2sin(x))Using variations of parameters, let {y1=cos(x)e2xy2=sin(x)e2x, then W=|y1y2y1y2|=e4xyp=cos(x)e2xsin(x)e2xe2xcscxe4xdx+sin(x)e2xcos(x)e2xe2xcscxe4xdx=cos(x)e2x1dx+sin(x)e2xcot(x)dx=xcos(x)e2x+sin(x)e2xln(sinx)y=yh+ypy=e2x(c1cosx+c2sinxxcosx+sinxln(sinx))


解答:L{f(t)}=12β3L{sin(βt)}12β2L{tcos(βt)}F(s)=12β3βs2+β212β2s2β2(s2+β2)2F(s)=1(s2+β2)2

解答:
Let f(z)=21+16z4=2/16z4116=216(z12eπi/4)(z12e3πi/4)(z12e5πi/4)(z12e7πi/4),and  let C be the closed contour consisting of the interval [R,R] on x-axis and the semicircle CR of radius R>1.Cf(z)dz=RR21+16x4dx+CRf(z)dz=I1+I2=2πi(Res(f,12eπi/4)+Res(f,12e2πi/4)),=2πi(14+4i+14+4i)=2πi(14i)=π2lim

解答:\textbf{(一) }P為對稱矩陣 \Rightarrow P= \begin{bmatrix} a & c\\ c& b\end{bmatrix} \Rightarrow PA+A^TP =\begin{bmatrix} a & c\\ c& b\end{bmatrix} \begin{bmatrix} 0 & 1\\ -5& -6\end{bmatrix} +\begin{bmatrix} 0 & -5\\ 1& -6\end{bmatrix} \begin{bmatrix} a & c\\ c& b\end{bmatrix} =-I_2 \\ \Rightarrow \begin{bmatrix} -10c & a-5b-6c\\ a-5b-6c& 2c-12b \end{bmatrix} =\begin{bmatrix} -1 & 0\\ 0& -1\end{bmatrix} \Rightarrow \cases{c=1/10\\ a-5b=6c\\ 2c+1=12b} \Rightarrow \cases{a=11/10\\ b=c=1/10} \\\quad \Rightarrow \bbox[red, 2pt]{P=\begin{bmatrix}11/ 10& 1/10\\ 1/10& 1/10 \end{bmatrix}} \\\textbf{(二) }P=\begin{bmatrix}11/ 10& 1/10\\ 1/10& 1/10 \end{bmatrix} \Rightarrow \det(P-\lambda I) =\lambda^2-{6\over 5}\lambda +{1\over 10} =0 \\ \quad \Rightarrow 特徵值\lambda =\bbox[red, 2pt]{{3\over 5}\pm {\sqrt{26} \over 10}}


解答:\textbf{(一) } r(t) =[3t,4t^2,8t^4] \Rightarrow 切線向量r'(t)= \bbox[red, 2pt]{[3,8t,32t^3]}\\ \Rightarrow 單位切線向量{r'(t) \over ||r'(t)||} ={1 \over \sqrt{3^2+(8t)^2+ (32t^3)^2}}[3,8t,32t^3]   = \bbox[red, 2pt]{{1\over \sqrt{1024t^6 +64t^2+9}}[ 3,8t,32t^3]} \\\textbf{(二) }\text{By Gauss Divergence Theorem, } \iint_S F\cdot ndA = \iiint \text{div }F\,dV =\iiint (7+3-1)\,dV \\\qquad =9\times {4\over 3}\pi\cdot 3^3= \bbox[red, 2pt]{324\pi}

 

解答:\textbf{(一) } \iint P\,dxdy=1 \Rightarrow \int_0^\infty \int_0^\infty ke^{-x-y/2}\,dxdy =\int_0^\infty ke^{-y/2}\, dy= \left. \left[ -2ke^{-y/2} \right] \right|_0^\infty =2k=1\\ \qquad \Rightarrow \bbox[red, 2pt]{k={1\over 2}} \\\textbf{(二) } p(y) =\int_0^\infty {1\over 2}e^{-x-y/2}\,dx= {1\over 2}e^{-y/2} \Rightarrow E\{Y\} =\int_0^\infty y\cdot {1\over 2}e^{-y/2}\,dy = \left. \left[ -(y+2)e^{-y/2}\right] \right|_0^\infty =\bbox[red, 2pt]2 \\ \textbf{(三) }E\{X^2Y^2\} =\int_0^\infty \int_0^\infty {1\over 2}x^3y^2e^{-x-y/2} \,dxdy = \int_0^\infty {1\over 2}y^2 \left. \left[ -(x^3+3x^2+6x+6)e^{-x-y/2} \right] \right|_0^\infty\, dy \\\qquad = \int_0^\infty3y^2e^{-y/2}\, dy = 3\left. \left[ -2(y^2+4y+8)e^{-y/2} \right] \right|_0^\infty =3\times 16= \bbox[red, 2pt]{48}



======================= END ======================

解題僅供參考,高普考歷年試題及詳解

沒有留言:

張貼留言