國立東石高中 112 學年度第一次教師甄選
一、填充題(每格 5 分,共 80 分)
解答:x=θ−sinθ⇒dx=(1−cosθ)dθ⇒面積=∫ydx=∫y(t)x′(t)dt=∫2π0(1−cosθ)(1−cosθ)dθ=∫2π01−2cosθ+cos2θdθ==∫2π032−2cosθ+12cos2θdθ=[32θ−2sinθ+14sin2θ]|2π0=3π
解答:a=cosπ3=12⇒|1aa2a332a+1a2+2a3a23a+22a+13a1111|=|11/21/41/8325/43/435/223/21111|−3r1+r2→r2,−3r1+r3→r3,−r1+r4→r4→|11/21/41/801/21/23/8015/49/801/23/47/8|−2r2+r3→r3,−r2+r4→r4→|11/21/41/801/21/23/8001/43/8001/41/2|−r3+r4→r4→|11/21/41/801/21/23/8001/43/80001/8|=1⋅12⋅14⋅18=164
解答:x+y+z=5,6,…,12⇒H35+H36+⋯+H312=21+28+36+45+55+66+78+91=420
解答:an=Cn2⋅3n−2⇒limn→∞(32a2+33a3+⋯+3nan)=limn→∞n∑k=23kak=limn→∞n∑k=23kCk23k−2=limn→∞n∑k=218k(k−1)=limn→∞(18n∑k=2(1k−1−1k))=18
解答:f(θ)=2sinθ+13+cosθ⇒f′(θ)=0⇒6cosθ+sinθ+2(3+cosθ)2=0⇒6cosθ+sinθ+2=0取x=tanθ2⇒{sinθ=2x1+x2cosθ=1−x21+x2⇒6⋅1−x21+x2+2x1+x2+2=0⇒2x2−x−4=0⇒{x=1+√334⇒{sinθ=(4+4√33)/(25+√33)cosθ=−(9+√33)/(25+√33)⇒f(θ)=3+√338x=1−√334⇒{sinθ=(4−4√33)/(25−√33)cosθ=(−9+√33)/(25−√33)⇒f(θ)=3−√338⇒3−√338≤t≤3+√338
8. 有一個袋子,裡面裝了 2 顆紅球,3 顆白球,4 顆黃球,5 顆黑球(球的材質、大小都相同),將袋中的球取出,一次取一顆,取後不放回,求白球先被取完之機率= _____。
解答:(a-2b+3c-4d)^{40}-(a+2b-3c-4d)^{20} =((a-4d)-((2b-3c))^{40}-((a-4d)+(2b-3c))^{40} \\=\sum_{k=0}^{40}C^{40}_k \left((a-4d)^k(-1)^{40-k}(2b-3c)^{40-k}-(a-4d)^k (2b-3c)^{40-k} \right)\\ 當k=偶數時,左右項次相消;因此僅剩下k=奇數的項次,也就是\\ (a-4d)^{39}(2b-3c)+ (a-4d)^{37}(2b-3c)^{3}+ \cdots +(a-4d)(2b-3c)^{39}\\ \Rightarrow 項數=40\cdot 2+38\cdot 4+ 36\cdot 6+\cdots +2\cdot 40 =\sum_{k=1}^{20}(42-2k)\cdot 2k \\=\sum_{k=1}^{20}(84k-4k^2) =17640-11480= \bbox[red, 2pt]{6160}
解答:取\cases{u=2x+3y+z\\ v=3x-2y+z\\ w=2x+3y+2z} \Rightarrow \begin{Vmatrix} u_x & u_y & u_z \\v_x & v_y& v_z \\w_x & w_y &w_z \end{Vmatrix} =\begin{Vmatrix}2 & 3 & 1 \\3 & -2 & 1 \\2 & 3 &2 \end{Vmatrix}=18\\ u^2+v^2+z^2=1的體積={4\over 3}\pi \Rightarrow (2x+3y+z)^2 +(3x-2y+z)^2 +(x+3y+2z)^2=1\\ 的體積={4\over 3}\pi \times{1\over 18} = \bbox[red, 2pt]{2\pi \over 27}
解答:
解答:{1\over x+1}+{1\over x+2y}=1 \Rightarrow x^2+2xy-x-1=0\\ 取\cases{f(x,y)=2x+y\\ g(x,y)=x^2+2xy-x-1} \Rightarrow \cases{f_x= \lambda g_x\\ f_y= \lambda g_y\\ g=0} \Rightarrow \cases{2=\lambda(2x+2y-1) \\ 1=\lambda (2x)} \Rightarrow 2={2x+2y-1\over 2x} \\ \Rightarrow x=y-{1\over 2} 代入g=0 \Rightarrow 12y^2-12y-1=0 \Rightarrow y={3+2\sqrt 3\over 6} (負值不合)\\ 2x+y =2y-1+y=3y-1={3+2\sqrt 3\over 2}-1 ={1+2\sqrt 3\over 2} =\bbox[red, 2pt]{{1\over 2}+\sqrt 3}
解答:\sin{2\pi \over 7}+\sin{4 \pi \over 7}-\sin{\pi \over 7} =2\sin{3\pi \over 7} \cos {\pi \over 7} -\sin{6\pi \over 7} =2\sin{3\pi \over 7} \cos {\pi \over 7} -2\sin{3\pi \over 7}\cos{3\pi \over 7} \\ =2\sin{3\pi \over 7} (\cos{\pi \over 7} -\cos {3\pi \over 7}) =2\sin{3\pi \over 7} (2\sin{2\pi \over 7} \sin{\pi \over 7}) = \color{blue}{4\sin{\pi \over 7}\sin{2\pi \over 7}\sin{3\pi \over 7}} \\ 令\cases{x=\sin{\pi \over 7}\sin{2\pi \over 7}\sin{3\pi \over 7} \\y=\cos{\pi \over 7}\cos{2\pi \over 7} \cos{3\pi \over 7}} \Rightarrow xy={1\over 8}\sin{ 2\pi \over 7}\sin{4 \pi \over 7}\sin{6\pi \over 7} ={1\over 8}\sin{ 2\pi \over 7}\sin{3 \pi \over 7}\sin{\pi \over 7} \\ \Rightarrow xy={1\over 8}x \Rightarrow x(y-{1\over 8})=0 \Rightarrow y={1\over 8}\; (x\ne 0)\\ 又8x^2 =(2\sin^2{\pi \over 7})(2 \sin^2{2\pi \over 7})(2 \sin^2{3\pi \over 7} ) =(1-\cos {2\pi \over 7}) (1-\cos {4\pi \over 7}) (1-\cos {6\pi \over 7}) \\= (1-\cos {2\pi \over 7}) (1+\cos {3\pi \over 7}) (1+\cos {\pi \over 7})=\cdots =1-{1\over 8}+0 ={7\over 8} \Rightarrow x={\sqrt 7\over 8} \Rightarrow 4x= \bbox[red, 2pt]{\sqrt 7\over 2}\\ \href{https://www.doubtnut.com/question-answer/find-the-value-of-sinpi-7sin2pi-7sin4pi-7-1847631}{完整計算過程}
16.將 2 顆綠色珠子,4 顆紅色珠子,3 顆藍色珠子串成一個項圈,試求共有幾種不同方法(假設珠子大小一樣) =____。
解答:
解答:考慮x={x+3 \over 2x-4} \Rightarrow 2x^2-5x-3=0 \Rightarrow (2x+1)(x-3)=0 \Rightarrow 不動點x=-{1\over 2},3 \\ 原式a_{n+1}={a_n+3\over 2a_n-4} \Rightarrow {a_{n+1}+1/2 \over a_{n+1}-3} ={ {a_n+3\over 2a_n-4} +1/2 \over {a_n+3\over 2a_n-4} -3} =-{2\over 5}\cdot {a_n+1/2\over a_n-3}\\ \Rightarrow \left\langle {a_n+1/2\over a_n-3} \right\rangle 為一等比數列,公比為-{2\over 5} \Rightarrow {a_n+1/2\over a_n-3}={a_1+1/2\over a_1-3}\cdot \left( -{2\over 5}\right)^{n-1} =\left( -{2\over 5}\right)^{n}\\ \Rightarrow a_n={ -3\cdot 2^n-{1\over 2}\cdot (-5)^n \over (-5)^n-2^n} =\bbox[red, 2pt]{3\cdot 2^{n+1}+(-5)^n \over 2^{n+1}-2 \cdot (-5)^n}\\ \href{https://web.math.sinica.edu.tw/math_media/d281/28109.pdf}{參考資料}
二、計算題(需有計算過程) (每題 10 分,共 20 分)
解答:I=\int_{-\infty}^\infty x^4 e^{-x^2/2}\,dx =\int_{-\infty}^\infty y^4 e^{-y^2/2}\,dy \Rightarrow I^2 =\int_{-\infty}^\infty\int_{-\infty}^\infty x^4y^4e^{-(x^2+y^2)/2} \,dydx\\ 令\cases{x=r\cos \theta\\ y=r\sin \theta } \Rightarrow I^2= \int_0^{2\pi} \int_0^{\infty} \cos^4 \theta \sin^4\theta \cdot r^9 e^{-r^2/2}\,drd\theta\\ =\int_0^{2\pi} \cos^4 \theta \sin^4\theta\left. \left[-e^{-r^2/2}(r^8+ 8r^6+ 48r^4+ 192r^2 +384) \right] \right|_0^\infty\, d\theta \\ =\int_0^{2\pi} 384 \cos^4 \theta \sin^4\theta \,d\theta =\left. \left[ 9x-3\sin(4x)+{3\over 8} \sin(8x)) \right] \right|_0^{2\pi} =18\pi \Rightarrow I=\sqrt{18\pi} =\bbox[red, 2pt]{3\sqrt{2\pi}}解答:考慮x={x+3 \over 2x-4} \Rightarrow 2x^2-5x-3=0 \Rightarrow (2x+1)(x-3)=0 \Rightarrow 不動點x=-{1\over 2},3 \\ 原式a_{n+1}={a_n+3\over 2a_n-4} \Rightarrow {a_{n+1}+1/2 \over a_{n+1}-3} ={ {a_n+3\over 2a_n-4} +1/2 \over {a_n+3\over 2a_n-4} -3} =-{2\over 5}\cdot {a_n+1/2\over a_n-3}\\ \Rightarrow \left\langle {a_n+1/2\over a_n-3} \right\rangle 為一等比數列,公比為-{2\over 5} \Rightarrow {a_n+1/2\over a_n-3}={a_1+1/2\over a_1-3}\cdot \left( -{2\over 5}\right)^{n-1} =\left( -{2\over 5}\right)^{n}\\ \Rightarrow a_n={ -3\cdot 2^n-{1\over 2}\cdot (-5)^n \over (-5)^n-2^n} =\bbox[red, 2pt]{3\cdot 2^{n+1}+(-5)^n \over 2^{n+1}-2 \cdot (-5)^n}\\ \href{https://web.math.sinica.edu.tw/math_media/d281/28109.pdf}{參考資料}
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