國立中央大學附屬中壢高級中學105學年度第 1次教師甄選
一、填充題
解答:√an=2√an+1+√anan+1⇒√an√anan+1=2√an+1√anan+1+√anan+1√anan+1⇒1√an+1=21√an+1⇒bn+1=2bn+1,其中{bn=1√anb1=1√a1=1⇒bn=2bn−1+1=2(2bn−2+1)+1=22bn−2+2+1=⋯=2n−1b1+2n−2+2n−3+⋯+1=1+2+22+⋯+2n−1=2n−1⇒1√an=2n−1⇒an=1(2n−1)2=122n−2n+1+1解答:令α=x−[x],則0≤α<1;將[x]=x−α代回原式⇒4x2−20(x−α)+23=0⇒4x2−20x+23=−20α⇒4(x−52)2−2=−20α⇒−20<4(x−52)2−2≤0⇒{4(x−52)2≤2−18<4(x−52)2⇒0≤4(x−52)2≤2⇒52−1√2≤x≤52+1√2⇒[x]=1,2,3將[x]值代回原式⇒{[x]=1[x]=2[x]=3⇒{4x2+3=0(無解)4x2−17=04x2−37=0⇒x=√172,√372
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√x2+(mx−3m+2)2+√x2+(mx−3m+10)2=10⇒√x2+(mx−3m−(−2))2+√x2+(mx−3m−(−10))2=10上述方程式相當於求兩圖形{直線L:y=m(x−3)橢圓Γ:√x2+(y+2)2+√x2+(y+10)2=10的交點;而Γ:{焦點F1(0,−2)焦點F2(0,−10)2a=10⇒{中心點O(0,−6)a=5b=3c=4⇒Γ:x29+(y+6)225=1依題意有兩相異實根,即交點數為2;橢圓的切線方程式:y−k=m(x−h)±√b2m2+a2,其中(h,k)為中心點⇒y+6=mx±√9m2+25經過(3,0)⇒6=3m±√9m2+25⇒9m2+25=(3m−6)2⇒m=1136,另外一根m=∞,即x=−3⇒m>1136,兩圖形有兩交點;
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{A,B,C,D在同一圓上¯AB=¯BC=¯AD⇒ABCD為一等腰梯形⇒令{A(−15/2,0)B(15/2,0)圓心O(0,a),見上圖;作¯BP⊥¯CD及¯AQ⊥¯CD⇒¯CP=(¯CD−¯AB)÷2=5/2;在直角△BPC中,¯BP=√152−(5/2)2=52√35⇒C(10,52√35);圓半徑r=¯OB=¯OC⇒(152)2+a2=102+(52√35−a)2⇒a=32√35⇒r=√(152)2+a2=3√15⇒正方形A′B′C′D′面積=(2r)×(2r)=(6√15)2=540
解答:將↔AB:y=m(x−4)+1代入xy=1⇒mx(x−4)+x=1⇒mx2+(1−4m)x−1=0⇒兩根之和=4m−1m=4×2(∵兩根之平均=4)⇒m=−14⇒↔AB:x+4y=8因此xy=1上的點C(t,1/t)至↔AB的距離d=|t+4/t−8|√17≤|4−8|√17=4√1717(∵t+4t≥2√t⋅4t=4)
解答:令a=10^{100},則\left[\cfrac{10^{10000}}{10^{100}+1} \right] =\left[\cfrac{a^{100}}{a+1} \right] =\left[\cfrac{((a+1)-1)^{100}}{a+1} \right]\\ =\left[\sum_{k=0}^{100}C^{100}_k(a+1)^{k-1}(-1)^{100-k} \right] = \left[{1\over a+1} -C^{100}_1 + C^{100}_2(a+1)-\cdots +C^{100}_{100}(a+1)^{99}\right]\\ =\left[{1\over a+1} +很大的正整數 \right]= 很大的正整數=-C^{100}_1 + C^{100}_2(a+1)-\cdots +C^{100}_{100}(a+1)^{99}\cdots(1) \\ 由於C^{100}_m(a+1)^n= C^{100}_m(C^n_na^n +C^n_{n-1}a^{n-1}+\cdots+C^n_1a+1)=100k+C^{100}_m\\ 因此(1)=100K+ (C^{100}_{100}-C^{100}_{99}+\cdots +C^{100}_2-C^{100}_1+C^{100}_0)-C^{100}_0,m,n,k\in \mathbb{N}\\ =100K +(1+(-1))^{100}-1 =100k-1 \equiv 99 \mod 100,K\in \mathbb{N}\\ \Rightarrow \left[\cfrac{10^{10000}}{10^{100}+1} \right] \div 100的餘數為\bbox[red, 2pt]{99}
解答:\lim_{n\to \infty}\left( \sqrt[n]{1+3^{2n}} +\sqrt[n]{3^{2n}+5^{2n}} + \cdots +\sqrt[n]{(2m-1)^{2n}+(2m+1)^{2n}} \right) \\ = \lim_{n\to \infty}\left(3^2 \sqrt[n]{({1\over 3})^{2n}+1} +5^2\sqrt[n]{({3\over 5})^{2n}+1} + \cdots +(2m+1)^2\sqrt[n]{({2m-1\over 2m+1})^{2n}+1} \right) \\ =3^2+5^2+\cdots +(2m+1)^2 =\sum_{k=1}^m(2k+1)^2 =4\sum_{k=1}^m k^2 +4\sum_{k=1}^m k+\sum_{k=1}^m 1\\ ={2\over 3}m(m+1)(2m+1)+2m(m+1)+m\\ 因此原式\Rightarrow \lim_{ m\to\infty}{{2\over 3}m(m+1)(2m+1)+2m(m+1)+m\over m^3} =\bbox[red,2pt]{4\over 3}
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\left[\cfrac{x+1}{2}\right]=k,k\in \mathbb{Z} \Rightarrow k\le \cfrac{x+1}{2}\lt k+1 \Rightarrow x \in [2k-1,2k+1) \Rightarrow x介於兩個連續奇數之間; \\\log_2 \left\{ \log_2 \left( \left| x-\left[\cfrac{x+1}{2} \right]\right|\right)\right\} \lt 0 \Rightarrow 0\lt \log_2 \left( \left| x-\left[\cfrac{x+1}{2} \right]\right|\right) \lt 1 \\ \Rightarrow 1 \lt \left| x-\left[\cfrac{x+1}{2} \right]\right|\lt 2 \Rightarrow \cases{ 1\lt x-\left[\cfrac{x+1}{2}\right] \lt 2 \\ -2\lt x-\left[\cfrac{x+1}{2}\right] \lt -1} \Rightarrow \cases{ x+1\lt \left[\cfrac{x+1}{2}\right] \lt x+2 \\ x-2\lt \left[\cfrac{x+1}{2}\right] \lt x-1}\\ \Rightarrow 所圍區域為四條水平線(見上圖),即\bbox[red, 2pt]{-4\lt x\lt -2或 2\lt x\lt 4,但x\ne \pm 3\qquad}
解答:(a,b,c)代表3個箱子內的物品數,由於箱子相同,不計算a,b,c的排列\\(9,0,0)\to 分法=1\\(8,1,0)\to C^9_8=9,(7,2,0)\to C^9_7=36,(6,3,0)\to C^9_6=84, (5,4,0)\to C^9_5=126;\\(7,1,1)\to C^9_7 =36, (6,2,1)\to C^9_6C^3_2=252, (5,3,1)\to C^9_5C^4_3=504,(5,2,2)\to C^9_5C^4_2\div 2=378;\\(4,4,1)\to C^9_4C^5_1\div 2=315, (4,3,2)\to C^9_4C^5_3=1260; (3,3,3)\to C^9_3C^6_3\div 3!=280 \\ 共有1+9+36+84+126 +36+252+504+378+315 +1260 +280= \bbox[red, 2pt]{3281}種分法
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解答:(a,b,c)代表3個箱子內的物品數,由於箱子相同,不計算a,b,c的排列\\(9,0,0)\to 分法=1\\(8,1,0)\to C^9_8=9,(7,2,0)\to C^9_7=36,(6,3,0)\to C^9_6=84, (5,4,0)\to C^9_5=126;\\(7,1,1)\to C^9_7 =36, (6,2,1)\to C^9_6C^3_2=252, (5,3,1)\to C^9_5C^4_3=504,(5,2,2)\to C^9_5C^4_2\div 2=378;\\(4,4,1)\to C^9_4C^5_1\div 2=315, (4,3,2)\to C^9_4C^5_3=1260; (3,3,3)\to C^9_3C^6_3\div 3!=280 \\ 共有1+9+36+84+126 +36+252+504+378+315 +1260 +280= \bbox[red, 2pt]{3281}種分法
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\triangle OAB:正弦定理 \Rightarrow {|\vec z|\over \sin \alpha} ={|\vec w|\over \sin (\pi-\theta)} ={|\vec w|\over \sin \theta} \Rightarrow \sin\theta = {|\vec w|\over |\vec z|}\sin \alpha ={1\over 10}\sin \alpha \\ \Rightarrow \sin ^2\theta={\sin^2\alpha \over 100} \le {1\over 100} \Rightarrow \tan^2\theta ={\sin^2\theta \over \cos^2\theta} ={\sin^2\theta \over 1-\sin^2\theta}= {1 \over 1-\sin^2\theta} -1\le {1\over 1-1/100}-1\\= \bbox[red,2pt]{1\over 99}
二、計算題
解答:(1)x^2f(x)={3\over 5}x^5+{1\over 2}ax^4 -{1\over 3}x^3+ 2\int_0^xtf(t)\;dt \\兩邊微分\Rightarrow 2xf(x)+x^2f'(x) =3x^4+2ax^3-x^2 +2xf(x)\\ \Rightarrow f'(x)=3x^2+2ax-1 \Rightarrow f(x)=x^3+ax^2-x+C,C為常數\\ f(0)=0 \Rightarrow C=0 \Rightarrow \bbox[red,2pt]{f(x)=x^3+ax^2-x}(2)f(x)=x^3+ax^2-x=0 \Rightarrow x(x^2+ax-1)=0 \Rightarrow x=0,\alpha,\beta,其中\cases{\alpha={-a-\sqrt{a^2+4}\over 2}\\ \beta={-a+\sqrt{a^2+4}\over 2}} \\ \Rightarrow \cases{\alpha^2+\beta^2= (\alpha+\beta)^2-2\alpha\beta=a^2+2 \\ \alpha^3+\beta^3 =(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)= -a^3-3a\\ \alpha^4+\beta^4= (\alpha^2+ \beta^2)^2 -2\alpha^2\beta^2=(a^2+2)^2-2 =a^4+4a^2+2} \\ \Rightarrow S(a)= \int_\alpha^0 f(x)\;dx -\int_0^\beta f(x)\;dx =\left .\left[ {1\over 4}x^4+{1\over 3}ax^3-{1\over 2}x^2 \right]\right|_\alpha^0 -\left .\left[ {1\over 4}x^4+{1\over 3}ax^3-{1\over 2}x^2 \right]\right|_0^\beta \\ =-{1\over 4}\alpha^4-{1\over 3}a\alpha^3+{1\over 2}\alpha^2 -{1\over 4}\beta^4-{1\over 3}a\beta^3+{1\over 2}\beta^2 =-{1\over 4}(\alpha^4+\beta^4)-{1\over 3}a(\alpha^3+\beta^3)+{1\over 2}(\alpha^2+\beta^2) \\ =-{1\over 4}(a^4+4a^2+2)-{1\over 3}a(-a^3-3a)+{1\over 2}(a^2+2) = \bbox[red, 2pt]{{1\over 12}a^4+{1\over 2}a^2+{1\over 2}}3S'(a)=0 \Rightarrow {1\over 3}a^3+ a=0 \Rightarrow a(a^2+3)=0 \Rightarrow 當a=0時,S(a)有最小值S(0)=\bbox[red,2pt]{1\over 2}
解答:\omega = \cos{2\pi\over 7}+i\sin{2\pi \over 7} \Rightarrow \omega^7=1 \\ (2-\omega)(2-\omega^3)(2-\omega^5) =(4-2\omega-2\omega^3 +\omega^4)(2-\omega^5)= 8-2\omega-\omega^2-4\omega^3+2\omega^4-4\omega^5+2\omega^6\\ \Rightarrow 實部=8-2\cos{2\pi\over 7}-\cos{4\pi\over 7}-4\cos{6\pi\over 7} +2\cos{8\pi\over 7} -4\cos{10\pi\over 7} +2\cos{12\pi\over 7}\\ = 8-2\cos{2\pi\over 7}-\cos{4\pi\over 7}+4\cos{\pi\over 7} -2\cos{\pi\over 7} -4\cos{4\pi\over 7} +2\cos{2\pi\over 7}=8+2\cos{\pi\over 7}-5\cos{4\pi \over 7}\\ \Rightarrow 虛部=-2\sin{2\pi\over 7}-\sin{4\pi\over 7}-4\sin{6\pi\over 7} +2\sin{8\pi\over 7} -4\sin{10\pi\over 7} +2\sin{12\pi\over 7} \\= -2\sin{2\pi\over 7}-\sin{4\pi\over 7}-4\sin{\pi\over 7} -2\sin{ \pi\over 7} +4\sin{4\pi\over 7} -2\sin{2\pi\over 7} \\ =-6\sin{\pi\over 7}-4\sin{2\pi\over 7}+3\sin{4\pi\over 7} \\因此欲求之解為\bbox[red,2pt]{8+2\cos{\pi\over 7}-5\cos{4\pi \over 7}+i(-6\sin{\pi\over 7}-4\sin{2\pi\over 7}+3\sin{4\pi\over 7})}
解答:\omega = \cos{2\pi\over 7}+i\sin{2\pi \over 7} \Rightarrow \omega^7=1 \\ (2-\omega)(2-\omega^3)(2-\omega^5) =(4-2\omega-2\omega^3 +\omega^4)(2-\omega^5)= 8-2\omega-\omega^2-4\omega^3+2\omega^4-4\omega^5+2\omega^6\\ \Rightarrow 實部=8-2\cos{2\pi\over 7}-\cos{4\pi\over 7}-4\cos{6\pi\over 7} +2\cos{8\pi\over 7} -4\cos{10\pi\over 7} +2\cos{12\pi\over 7}\\ = 8-2\cos{2\pi\over 7}-\cos{4\pi\over 7}+4\cos{\pi\over 7} -2\cos{\pi\over 7} -4\cos{4\pi\over 7} +2\cos{2\pi\over 7}=8+2\cos{\pi\over 7}-5\cos{4\pi \over 7}\\ \Rightarrow 虛部=-2\sin{2\pi\over 7}-\sin{4\pi\over 7}-4\sin{6\pi\over 7} +2\sin{8\pi\over 7} -4\sin{10\pi\over 7} +2\sin{12\pi\over 7} \\= -2\sin{2\pi\over 7}-\sin{4\pi\over 7}-4\sin{\pi\over 7} -2\sin{ \pi\over 7} +4\sin{4\pi\over 7} -2\sin{2\pi\over 7} \\ =-6\sin{\pi\over 7}-4\sin{2\pi\over 7}+3\sin{4\pi\over 7} \\因此欲求之解為\bbox[red,2pt]{8+2\cos{\pi\over 7}-5\cos{4\pi \over 7}+i(-6\sin{\pi\over 7}-4\sin{2\pi\over 7}+3\sin{4\pi\over 7})}
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解題僅供參考,其他教甄試題及詳解
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