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2023年5月16日 星期二

112年嘉義高中教甄-數學詳解

國立嘉義高級中學 112 學年度第 1 學期第 1 次教師甄選

解答:m,n=m+(m+1)++(m+n1)=nm+n(n1)2=2nm+n(n1)2=2023n(2m+n1)=40464046=2×7×172{4046=1×4046{n=1m=2023,n24046=2×2023{n=2m=10114046=7×578,14×289,17×238,34×1195
解答:

|1z1|1|z||1z1||z||z1||z|(x1)2+y2x2+y22x+10x12Ω1={x12}|z1|1Ω2={(x1)2+y21};Ω1Ω2,x=12(x1)2+y2=1{A(1/2,3/2)B(1/2,3/2)¯AB=3cosAOB=1+132=12AOB=120{OAB=3/4OAB=π/3Ω1Ω2=π334=4π3312

解答:{A(2,4,5)C(4,4,7)G(3,5,6)F(a,b,c){AC=(2,0,2)AG=(1,1,1)H=¯AC(3,4,6)n=AC×AG=(2,0,2)AnEx+z=3¯AC=22=2{ACHFAGHF{(2,0,2)(a3,b4,c6)=0(1,1,1)(a3,b4,c6)=0{a+c=9b=4¯HF=d(H,E)=2|a+c3|2=2a+c=51{a=2,c=7a=4,c=5{F=(2,4,7)F=(4,4,5){FH=(1,0,1)=2nFH=(1,0,1)=2nF=(2,4,7)
解答:n5=aCn5+bCn4+cCn3+dCn2+eCn1n=11=0+0+0+0+ee=1n=232=d+2ed=30n=3243=c+3d+3ec=150n=41024=b+4c+6d+4eb=240n=53125=a+5b+10c+10d+5ea=120a+b+c+d+e=120+240+150+30+1=541
解答:f(n)=4f(n1)+4n2=4(4f(n2)+4n3)+4n2=42f(n2)+24n2==4n2f(2)+(n2)4n2=4n2(n1)(f(2)=1)f(9)=478=21717118
解答:n2abcd11,1,1,11224,1,1,142,2,1,16323,3,1,16424,4,1,164,2,2,1122,2,2,21525,5,1,16626,6,1,166,3,2,1243,3,4,1123,3,2,26824,4,2,264,4,4,14923,3,3,311025,5,2,265,5,4,1121226,6,4,1126,6,2,266,4,3,2244,4,3,361525,5,3,361624,4,4,411826,6,3,362025,5,4,462426,6,4,462525,5,5,513026,6,5,563626,6,6,61200p=20064=251621p=16225
解答:

{3x5f(x)=x35tdt=12x2+5x212x5g(x)=535tdt+x5t5dt=12x25x+292Ly=2x2023k,L{y=f(x)y=g(x){2x2023k=f(x),0k=4046/12337.22x2023k=g(x),=0k=4046/20=202.3k=203337,337203+1=135
解答:

{A(1,0,1)B(0,1,1)C(1,1,0)O(0,0,0)OABC=2{D=¯AO=(1/2,0,1/2)E=(2A+B)/3=(2/3,1/3,1)F=(2B+A)/3=(1/3,2/3,1)G=¯BC=(1/2,1,1/2)CDEOFG,{E2=CDEE3=OFG{CD=(1/2,1,1/2)CE=(1/3,2/3,1)n=CD×CE=(2/3,1/3,0)CDE12|n|=56AB=(1,1,0)cosθ=ABn|AB||n|=310d(E1,E4)=352=352/310=15CDE12(15)2=560
解答:
解答:利用公式:{2\over 3}R^3\tan \theta ={2\over 3}\cdot 6^3\tan 30^\circ =\bbox[red,2pt]{48\sqrt 3}\\ \href{https://chu246.blogspot.com/2020/12/blog-post.html}{公式來源}
解答:欲求之原式=\sum_{k=1}^{16} k(k+2)C^{16}_k x^ky^{16-k},其中x={3\over 4},y={1\over 4}\\ 令f(x,y)=(x+y)^{16} =\sum_{k=0}^{16}C^{16}_kx^ky^{16-k} \Rightarrow f_x= \sum_{k=1}^{16}k C^{16}_kx^{k-1} y^{16-k} \\ \Rightarrow x^3f_x =\sum_{k=1}^{16}k C^{16}_kx^{k+2} y^{16-k} \Rightarrow {\partial \over \partial x}\left(x^3f_x \right) =3x^2 f_x+ x^3f_{xx}=\sum_{k=1}^{16}k (k+2)C^{16}_kx^{k+1} y^{16-k} \\ \Rightarrow {1\over x}\left( 3x^2 f_x+ x^3f_{xx}\right) =3xf_x+ x^2f_{xx} =\sum_{k=1}^{16}k (k+2)C^{16}_kx^{k} y^{16-k} \triangleq 原式\\ f=(x+y)^{16} \Rightarrow f_x=16(x+y)^{15} \Rightarrow \cases{f_{xx}= 240(x+y)^{14} \\ 3xf_x= 48x(x+y)^{15}} \\ \Rightarrow g(x,y)=3xf_x+x^2f_{xx}= 48x(x+y)^{15} +240x^2(x+y)^{14} \\\Rightarrow g({3\over 4},{1\over 4}) =36+135= \bbox[red, 2pt]{171}
解答:\begin{array}{} 前4球&前4球排列&變色數&後4球&後4球排列數\\\hline 1B3W & BWWW & 1 & 4B & 1\\ & WBWW & 2\\ & WWBW& 2\\ & WWWB& 1&&\\\hdashline 2B2W & BBWW & 1& 3B1W& 4\\ & BWBW & 3\\ & BWWB & 2\\ & WBBW & 2\\ & WBWB& 3\\ & WWBB& 1&& \\\hdashline 3B1W & BBBW & 1& 2B2W & 6\\ & BBWB & 2\\ & BWBB& 2\\ & WBBB& 1\\\hline\end{array} \\ 總變色數=1\cdot (1+2+2+1)+ 4\cdot(1+3+2+2+3+1)+ 6\cdot(1+2+2+1) =90 \\ \Rightarrow 期望值={90\over C^8_3}={90\over 56}=\bbox[red, 2pt]{45\over 28}
解答:P(1,a) \Rightarrow a\sin({\pi\over 3}+\theta) =a \Rightarrow {\pi\over 3}+\theta={\pi \over 2} \Rightarrow \theta={\pi\over 6} \Rightarrow f(x)=a\sin({\pi\over 3}x+ {\pi\over 6}) \\又Q在最低點\Rightarrow {\pi\over 3}x+ {\pi\over 6}={3\over 2}\pi \Rightarrow x=4 \Rightarrow Q(4,-a)\\ 因此我們有\cases{P(1,a)\\ R(1,0)\\ Q(4,-a)} \Rightarrow \cases{\overrightarrow{RP} =(0,a) \\ \overrightarrow{RQ}=(3,-a)} \Rightarrow \cos \angle PRQ ={\overrightarrow{RP} \cdot \overrightarrow{RQ} \over |\overrightarrow{RP}||\overrightarrow{RQ}} \\ \Rightarrow {-a^2\over a\cdot \sqrt{9+a^2}} =-{1\over 2} \Rightarrow a^2=3 \Rightarrow a=\bbox[red, 2pt] {\sqrt 3}

解答:假設\overline{AC}=a \Rightarrow \overline{AB}=10-a \Rightarrow 三角柱體積=f(a)={1\over 2}a(10-a)\sin 60^\circ\cdot 2a\\ \Rightarrow f(a)={\sqrt 3\over 2}(10a^2-a^3) \Rightarrow f'(a)={\sqrt 3\over 2}(20a-3a^2) \Rightarrow f''(a)={\sqrt 3\over 2}(20-6a)\\ f'(a)=0 \Rightarrow \cases{a=0\\ a=20/3} \Rightarrow \cases{f''(0)\gt 0\\ f''(20/3)\lt 0} \Rightarrow f({20\over 3})= {\sqrt 3\over 2}\cdot {400\over 9}\cdot(10-{20\over 3}) =\bbox[red,2pt]{2000\sqrt 3\over 27}
解答:80(1+3.53\%)^n \ge 100(1+1.5\%)^n \Rightarrow \left({ 1.0353\over 1.015}\right)^n \ge {100\over 80} \Rightarrow \left({102\over 100}\right) ^n \ge {10\over 8}\\ \Rightarrow n\log 1.02 \ge 1-3\log 2=1-3\cdot 0.301=0.097 \Rightarrow n\ge {0.097\over 0.0086} \approx 11.3\\ \Rightarrow 最少\bbox[red,2pt]{12}年後

解答:

兩圖形皆對稱y軸,取\cases{A(a,-a^2+4)\\ B(-a,-a^2+4)\\ C(-a,2a^2-8)\\ D(a,2a^2-8)} \Rightarrow \cases{\overline{AB}=2a\\ \overline{AD}= -a^2+4+8-2a^2=-3a^2+12} \\ \Rightarrow 矩形面積= f(a)=2a(-3a^2+12) =-6a^3+24a \Rightarrow f'(x)=-18a^2+24 =0 \\ \Rightarrow a={2\over \sqrt 3} \Rightarrow f({2\over \sqrt 3})=-{16\over \sqrt 3}+{48\over \sqrt 3}=\bbox[red, 2pt]{32\sqrt 3\over 3}
解答:I為內心,X為任一點 \Rightarrow \overrightarrow{XI}={a\over a+b+c}\overrightarrow{XA} +{b\over a+b+c}\overrightarrow{XB} +{c\over a+b+c}\overrightarrow{XC} \href{https://web.math.sinica.edu.tw/math_media/d341/34103.pdf}{公式來源} \\ 因此\overrightarrow{AI} = {3\over 2+3+4}\overrightarrow{AB} +{4\over 2+3+4}\overrightarrow{AC} ={1\over 3}\overrightarrow{AB} +{4\over 9} \overrightarrow{AC} \\ \Rightarrow \overrightarrow{AO} \cdot \overrightarrow{AI} ={1\over 3}\overrightarrow{AO} \cdot \overrightarrow{AB} +{4\over 9}\overrightarrow{AO}\cdot \overrightarrow{AC}  ={1\over 3} \cdot {1\over 2}\overline{AB}^2 +{4\over 9} \cdot {1\over 2}  \overline{AC}^2 \\={1\over 6}\cdot 16+{2\over 9}\cdot 9 =\bbox[red, 2pt]{14\over 3}
解答:\cases{E_1:2x+y+2z+3=0\\ E_2:3x+4y-5=0} \Rightarrow \cases{E_1法向量\vec u=(2,1,2)\\ E_2法向量\vec v=(3,4,0)} \Rightarrow \cases{\vec n_1=\vec u/|\vec u| =(2/3,1/3,2/3)\\ \vec n_2=\vec v/|\vec v|=(3/5,4/5,0)} \\ \Rightarrow \cos \theta ={\vec n_1\cdot \vec n_2\over |\vec n_1||\vec n_2|} \gt 0 \Rightarrow \vec n_1與\vec n_2的夾角為銳角\\ 因此取 \vec s= -\vec n_1+\vec n_2 =(-1/15,7/15,-10/15)為角平分面之法向量,\\並任找一點P(-17/5,19/5,0)\in E_1\cap E_2 \\\Rightarrow 角平分面方程式:-{1\over 15}(x+{17\over 5})+{7\over 15}(y-{19\over 5})-{10\over 15}z=0 \Rightarrow \bbox[red, 2pt]{x-7y+10z+30=0}
解答:橢圓{x^2\over m}+y^2=1 \Rightarrow \cases{a=\sqrt m\\ b=1} \Rightarrow c=\sqrt{m-1} \Rightarrow 焦點\cases{F_1(-\sqrt{m-1},0) \\ F_1(\sqrt{m-1},0)} \\ 雙曲線{x^2\over n}-{y^2\over 3}=1 \Rightarrow \cases{a=\sqrt n\\ b=\sqrt 3} \Rightarrow c=\sqrt{n+3} \Rightarrow 焦點\cases{F_1(-\sqrt{n+3},0) \\ F_2(\sqrt{n+3},0)}\\ 兩焦點相同\Rightarrow m-1=n+3 \Rightarrow m-n=4 \Rightarrow 取\cases{n=1\\ m=5} \Rightarrow \cases{F_1(-2,0) \\ F_2(2,0)}\\ 兩圖形交點:{x^2\over 5}+y^2=x^2-{y^2\over 3}\Rightarrow x^2={5\over 3}y^2 \Rightarrow y^2={3\over 4} \Rightarrow 取y={\sqrt 3\over 2} \\ \Rightarrow x={\sqrt 5\over 2} \Rightarrow P(\sqrt 5/2,\sqrt 3/2) \Rightarrow \cases{\overline{PF_1}=\sqrt 5+1 \\ \overline{PF_2}=\sqrt 5-1} \Rightarrow \cos \angle F_1PF_2={(\sqrt 5+1)^2+ (\sqrt 5-1)^2-4^2 \over 2(\sqrt 5+1)(\sqrt 5-1)}\\ =-{1\over 2} \Rightarrow \angle F_1PF_2=120^\circ \Rightarrow \tan \angle F_1PF_2=\bbox[red, 2pt]{-\sqrt 3}
解答:

假設A(R\cos \alpha,R\sin \alpha)對稱軸為直線L:y=\tan \theta x,則對稱點A'(R\cos(2\theta-\alpha), R\sin (2\theta-\alpha))\\ 也就是\cases{x=R\cos \alpha\\ y=R\sin \alpha} 對應到\cases{x'=R\cos(2\theta-\alpha)= R\cos 2\theta \cos \alpha +R\sin 2\theta \sin \alpha=x\cos 2\theta+y\sin 2\theta\\ y'=R\sin(2\theta-\alpha) =R(\sin 2\theta \cos \alpha-\sin\alpha \cos 2\theta)= x\sin 2\theta -y\cos 2\theta} \\ 即\begin{bmatrix} \cos 2\theta & \sin 2\theta\\ \sin 2\theta &-\cos 2\theta\end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix} =\begin{bmatrix} x'\\ y'\end{bmatrix} \Rightarrow 對直線L的鏡射矩陣為\bbox[red,2pt]{\begin{bmatrix} \cos 2\theta & \sin 2\theta\\ \sin 2\theta &-\cos 2\theta\end{bmatrix}}
解答:假設兩鏡射矩陣分別為\cases{A=\begin{bmatrix}\cos 2\alpha & \sin 2\alpha\\ \sin 2\alpha & -\cos 2\alpha \end{bmatrix} \\[1ex] B=\begin{bmatrix}\cos 2\beta & \sin 2\beta\\ \sin 2\beta & -\cos 2\beta \end{bmatrix}} \\ \Rightarrow AB=\begin{bmatrix}\cos 2\alpha \cos 2\beta+ \sin 2\alpha \sin 2\beta& \cos 2\alpha \sin 2\beta-\sin 2\alpha \cos 2\beta\\ \sin 2\alpha \cos 2\beta-\cos 2\beta \sin 2\alpha & \sin 2\alpha \sin 2\beta+ \cos 2\alpha \cos 2\beta \end{bmatrix} \\ =\begin{bmatrix} \cos(2\alpha-2\beta) & -\sin(2\alpha-2\beta) \\ \sin (2\alpha-2\beta) & \cos (2\alpha-2\beta)\end{bmatrix}為一\bbox[red,2pt]{旋轉}矩陣,旋轉角度為兩對稱軸角度差的2倍
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解題僅供參考,其他教甄試題及詳解

1 則留言:

  1. 朱老師你好!請問第16題,我算出AI的長度與A B的夾角,AO同樣算出其長度與AB的夾角,而AO與AB的夾角餘弦值和A l與AB的夾角餘弦值相等,即AI和AO夾角為0,兩向量內積,即其長度相乘,但答案是16 /3,老師我那裏弄錯了,能否解迷?謝謝你

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