112 年特種考試地方政府公務人員考試試題
等 別:三等考試
類 科:電力工程、電子工程
科 目:工程數學
甲、申論題部分:(50分)
解答:先求齊次解:y″+4y=0⇒λ2+4=0⇒λ=±2i⇒yh=c1cos(2x)+c2sin(2x)令yp=acosx+bsinx+cx+d⇒y′p=−asinx+bcosx+c⇒y″p=−acosx−bsinx⇒y″p+4yp=3acosx+3bsinx+4cx+4d=1+x+sinx⇒{a=0b=1/3c=1/4d=1/4⇒yp=13sinx+14x+14⇒y=yh+yp=c1cos(2x)+c2sin(2x)+13sinx+14x+14⇒y′=−2c1sin(2x)+2c2cos(2x)+13cosx+14⇒{y(0)=c1+14=0y′(0)=2c2+13=0⇒{c1=−1/4c2=−1/3⇒y=−14cos(2x)−13sin(2x)+13sinx+14x+14
解答:(一)A=[0.90.10.20.8]⇒det(A−λI)=0⇒(λ−0.7)(λ−1)=0⇒特徵值為λ=0.7及1λ1=0.7⇒(A−λ1I)v=0⇒[0.20.10.20.1][x1x2]=0⇒v=[−k/2k],取v1=[−1/21]λ2=1⇒(A−λ2I)v=0⇒[−0.10.10.2−0.2][x1x2]=0⇒v=[kk],取v2=[11]⇒相對應的特徵向量為[−1/21]及[11](二)Av=λv⇒A(Av)=A(λv)=λAv=λ2v⇒A2的特徵值為λ2,即0.49,1,相對應的特徵向量與A相同,即[−1/21]及[11] (三)A=PDP−1=[−1/2111][0.7001][−2/32/32/31/3]⇒An=PDnP−1⇒limn→∞An=[−1/2111][0001][−2/32/32/31/3]=[2/31/32/31/3]
解答:f(z)={1\over (1+z^2)^2}= {1\over (z+i)^2 (z-i)^2} \text{ has two poles of order 2 at }z= i,-i\\只考慮上半部\Rightarrow Res(f,i)= {d\over dz} \left. {1\over (z+i)^2}\right|_{z=i} =\left. -{2\over (z+i)^3}\right|_{z=i} ={1\over 4i}\\ 又f(-x)=f(x)\Rightarrow f\text{ is even function } \Rightarrow \int_0^\infty {1\over 1+x^2}\,dx ={1\over 2}\int_{-\infty}^\infty {1\over 1+x^2}\,dx \\={1\over 2}\oint_c f(z)\,dz ={1\over 2}\cdot 2\pi i\times Res(f,i) =\bbox[red, 2pt]{\pi \over 4}
解答:f_Z(z)=\begin{cases}\lambda^2 e^{-\lambda z} , & 0\le z\lt \infty\\ 0,& z\lt 0\end{cases} \Rightarrow \int f_z(z)\,dz =1 \Rightarrow \int_0^\infty \lambda^2 e^{-\lambda z}\,dz=1 \\\Rightarrow \left. \left[ -\lambda e^{-\lambda z}\right] \right|_0^\infty =\lambda =1 \Rightarrow f_Z(z) =\begin{cases} e^{- z} , & 0\le z\lt \infty\\ 0,& z\lt 0\end{cases} \\ \mathbf{(一)}\; E(Z)= \int_0^\infty ze^{-z}\,dz = \left. \left[ -ze^{-z}-e^{-z} \right] \right|_0^\infty =\bbox[red, 2pt]1 \\\mathbf{(一)}\; F(Z) =f(Z\lt z)= \int_0^z e^{-x}\,dx =\left. \left[ -e^{-x}\right] \right|_0^z =\bbox[red, 2pt]{1-e^{-z}}
乙、測驗題部分:(50分)
解答:\phi=2x^2y-yz^2 \Rightarrow \nabla\phi =(\phi_x,\phi_y, \phi_z)= (4xy,2x^2-z^2,-2yz) \Rightarrow \nabla \phi(1,-1,2)=(-4,-2,4)\\ \bbox[red, 2pt]{無解},公布的答案是\bbox[cyan, 2pt]{(B)}解答:(2,-1,0)\cdot ((1,3,-1)\times (3,0,-1)) =(2,-1,0) \cdot (-3,-2,-9) = -4, 故選\bbox[red, 2pt]{(A)}
解答:\oint_c y^2dx+ (xy+x^2)dx \Rightarrow \cases{P=y^2\\ Q=xy+x^2} \Rightarrow u=Q_x-P_y=y+2x-2y=2x-y\\ 所圍面積 = \int_0^1 \int_{x^2} ^x (2x-y)\,dydx ={1\over 10}, 故\bbox[red, 2pt]{無解},公布的答案是\bbox[cyan,2pt]{(C)}
解答:A=\begin{bmatrix}1& 4& 7\\ 2& 5& 8\\ 3& 6 & 9 \end{bmatrix} \Rightarrow rref(A)= \begin{bmatrix}1& 0& -1\\ 0& 1& 2\\ 0& 0 & 0 \end{bmatrix} \Rightarrow 列空間=\{a(1,0,-1)+ b(0,1,2) \mid a,b \in \mathbb R\} \\ \cases{(B) (2,2,2)=2(1,0,-1)+ 2(0,1,2) \\(C) (6,15,24)= 6(1,0,-1)+ 15(0,1,2)\\ (D) (1,2,3)=(1,0,-1)+ 2(0,1,2)} \Rightarrow 只有(A)不在列空間中,故選\bbox[red, 2pt]{(A)}
解答:由題意可知特徵值為2,-1\\ \lambda_1=2 \Rightarrow (A-\lambda_1 I)v=0 \Rightarrow \begin{bmatrix} -6& -6\\3 & 3\end{bmatrix} \begin{bmatrix} x_1\\ x_2\end{bmatrix}=0 \Rightarrow v=\begin{bmatrix} -k\\ k\end{bmatrix}, 取v_1= \begin{bmatrix} 2\\ -2\end{bmatrix} \\ \lambda_2=-1 \Rightarrow (A-\lambda_2 I) v=0 \Rightarrow \begin{bmatrix} -3& -6\\3 & 6\end{bmatrix} \begin{bmatrix} x_1\\ x_2\end{bmatrix}=0 \Rightarrow v=\begin{bmatrix} -2k \\ k\end{bmatrix}, 取v_2= \begin{bmatrix} -2\\ 1\end{bmatrix} \\ \Rightarrow P=[v_1\; v_2]= \begin{bmatrix} 2 &-2\\ -2 & 1 \end{bmatrix}故選\bbox[red, 2pt]{(C)}
解答:\det(A)=3 \Rightarrow \cases{\det(A^{-1})=1/3\\ \det(A^T)=3} \Rightarrow \det(3A^T)=3^3\times 3=81\\ \Rightarrow \det(3A^T)+ \det(A^{-1}) = 81+{1\over 3}= {244\over 3} \Rightarrow c=244, 故選\bbox[red, 2pt]{(C)}
解答:A=\begin{bmatrix} 1& 0 & -1\\ 0 & 1& 2\\ -1& 2& 5\end{bmatrix} =PDP^T ,其中P={1\over \sqrt{30}} \begin{bmatrix} \sqrt 5& 2\sqrt 6 & -1\\-2 \sqrt 5 & \sqrt 6& 2\\ \sqrt 5& 0& 5\end{bmatrix},D= \begin{bmatrix} 0& 0 & 0\\ 0 & 1& 0\\ 0& 0& 6\end{bmatrix} \\ \Rightarrow A可正交對角化,但\det(A)=0 ,故選\bbox[red, 2pt]{(D)}
解答:(B) \times: (0,1,1) \in S 但2(0,1,1)=(0,2,2) \not \in S\\ (C)\times: (2,1,0)\in S,但2(2,1,0)=(4,2,0)\not \in S\\ (D)\times: (0,0,0)\not \in S\\ 故選\bbox[red, 2pt]{(A)}
解答:(D) \cases{u(x,y)=e^x\cos y\\ v(x,y)=e^x\sin y} \Rightarrow \cases{u_x=e^x\cos y=v_y\\ u_y=-e^x\sin y=-v_x} \Rightarrow f 可解析\\ 故選\bbox[red, 2pt]{(D)}
解答:z^2-z+1+i=0 \Rightarrow z={1\pm \sqrt{-3-4i}\over 2} ={1\pm (-1+2i)\over 2}=i,1-i\\ |z_1| \gt |z_2| \Rightarrow \cases{z_1=1-i\\ z_2=i} \Rightarrow \cases{(A)\times:z_1+z_2=1 \ne 1+2i\\ (B)\bigcirc: z_1-z_2=1-2i \\ (C)\times: |z_1|=\sqrt 2\ne 1\\ (D)\times:|z_2|=1 \ne \sqrt 2},故選\bbox[red, 2pt]{(B)}
解答:f(z)={z(z-5)(z-i)\over (z+i)^3 (z+1)(z-4)^2} \Rightarrow \cases{零點: z=0,5,i\\ 極點:z=-i,-i,-i,-1,4,4} \Rightarrow \cases{N=2(z=0,i)\\ P=4(z=-i,-i,-i,-1)} \\ \Rightarrow \oint_c {f'(z)\over f(z)} =2\pi i \times(N-P) =2\pi i \times (2-4)=-4\pi i,故選 \bbox[red, 2pt]{(B)}
解答:f(z)={1\over (1+x^2)^3} ={1\over (z+i)^3 (z-i)^3} \Rightarrow Res(f,i)={1\over 2}\cdot \left.{d^2\over dz^2} {1\over (z+i)^3} \right|_{z=i} \\=\left. {6\over (z+i)^5}\right|_{z=i} ={6\over 32i} \Rightarrow \int_{-\infty}^\infty f(x)\,dx = 2\pi i\times Res(f,i)=2\pi i\times {6\over 32 i}={3\over 8}\pi,故選 \bbox[red, 2pt]{(D)}
解答:\cases{P=Ax^2y^2+ By^3\\ Q=2x^3y+ 12xy^2+5} \Rightarrow \cases{P_y=2Ax^2y +3By^2\\ Q_x=6x^2y+12y^2}\\ 正合\Rightarrow P_y=Q_x \Rightarrow \cases{2A=6\\ 3B=12} \Rightarrow \cases{A=3\\ B=4}, 故選\bbox[red, 2pt]{(C)}
解答:(4-y^2)y'=x^2 \ge 0 \Rightarrow y^2\gt 4可能不會有唯一解,即y\lt -2, y\gt 2, 故選\bbox[red, 2pt]{(B)}
解答:a_n= {1\over 3}\int_{-3}^3 f(x)\cos{n\pi x\over 3}\,dx ={2\over 3}\int_{0}^3 x\cos{n\pi x\over 3}\,dx ={6\over n^2\pi^2}((-1)^n-1)\\ \Rightarrow \cases{a_1=-{12\over \pi^2} \\ a_3=-{12\over 9\pi^2} \\ a_5=-{12\over 25\pi^2} \\ a_7=-{12\over 49\pi^2}},故選\bbox[red, 2pt]{(D)}
解答:{1\over \sqrt{2\pi}}\int_{-1}^1 e^{-i\omega t}\,dt ={1\over \sqrt{2\pi}}\cdot {1\over -i\omega}(e^{-i\omega}-e^{i\omega}) = {1\over \sqrt{2\pi}}\cdot {1\over -i\omega}\cdot (-2i)\cdot {\sin \omega } \\ ={2\over \sqrt{2\pi}} {\sin \omega \over \omega} =\sqrt{2\over \pi}{\sin \omega \over \omega},故選 \bbox[red, 2pt]{(B)}(題目應該是筆誤)
解答:\int_0^\infty {1\over t}e^{-st}\,dt = \infty, 故選\bbox[red, 2pt]{(D)}
解答:E(X^2)=(-2)^2\cdot 0.4+ 1^2\cdot 0.5+ 3^2\cdot 0.1=3 \Rightarrow E(2X^2+1)= 2E(X^2)+1=7,故選\bbox[red, 2pt]{(C)}
解答:0.4\times 0.03+0.25\times 0.02+ 0.35\times 0.04 ={31\over 1000},故選 \bbox[red, 2pt]{(D)}
解答:
============= 先這樣, 再想想!!!======
你好 請問選擇20怎麼算 謝謝
回覆刪除我算的結果是1/90, 不是1/40, 所以.......
刪除了解 謝謝😊
刪除