中正國防幹部預備學校109年教師甄選測驗試題
解答:
(x+1)26=26∑k=0C26kxk=1+26∑k=1C26kxk=1+8191=213⇒x=√2eikπ13−1,k=0,1,…,25⇒Re(x)=√2coskπ13−1若Re(x)>0,則coskπ13>1√2=cosπ4⇒0≤k≤3或23≤k≤25⇒Re(x)<0⇒4≤k≤22,共19個
令{E為¯BC的中點¯EF⊥¯OA,如上圖⇒¯AE=¯ACsin60∘=4×√32=2√3依題意¯EF=√3⇒¯AF2=¯AE2−¯EF2=12−3=9⇒¯AF=3直角△OCE:¯OE2=¯OC2−¯EC2=a2−22=a2−4直角△OEF:¯OE2=(¯OA−¯AF)2+¯EF2⇒a2−4=(a−3)2+3⇒−4=−6a+12⇒a=166=83
解答:{a1+a2+a3+⋯+a2020=20211a1+1a2+1a3+⋯+1a2020=2021⇒{a2+a3+⋯+a2020=2021−a11a2+1a3+⋯+1a2020=2021−1a1由於(a2+a3+⋯+a2020)(1a2+1a3+⋯+1a2020)≥(1+1+⋯+1)2=20192⇒(2021−a1)(2021−1a1)≥20192⇒20212−2021(a1+1a1)+1≥20192⇒a1+1a1≤20212−20192+12021=80812021
解答:假設{a+b+2c=xa+2b+c=ya+b+c=z⇒{a=−x−y+3zb=y−zc=x−z⇒2b−2ca+b+2c+2a+4ca+2b+c+ba+b+c=2y−2xx+2x−2y+2zy+y−zz=2yx−2+2x+2zy−2+yz−1=(2yx+2xy)+(2zy+yz)−5≥2√2yx⋅2xy+2√2zy⋅yz−5=√8−1
解答:{a2+b2=4可以視為一圓,其圓心O1(0,0),半徑r1=2(c−7)2+(d−24)2=36可以視為一圓,其圓心O2(7,24),半徑r2=6|abcd|的最大值M=√a2+b2×√c2+d2=r1×(¯O1O2+r2)=2(25+6)=62(a−c)2+(b−d)2最小值m=(¯O1O2−r1−r2)2=(25−2−6)2=172=289因此M+m=62+289=351註:學校公布的答案是79
解答:第12階至第20階的走法:2x+3y=8⇒(x,y)=(4,0),(1,2)⇒{2222排列數1233排列數=3⇒共4種第1階至第12階的走法:2x+3y=12⇒(x,y)=(6,0),(3,2),(0,4)⇒{222222經過第8階22233排列數10,需扣除經過第8階的{3322232322233223種走法,剩下7種3333排列數1⇒第1階至第12階有7+1=8種走法;因此共有8×4=32種走法。
解答:

球心O(0,0,0)至平面E:3x+2y+2√3z=5的距離d(O,E)=5√9+4+12=1所求體積相當於半徑為2的圓形弓形部分繞x軸旋轉所得體積,見上圖斜線區域;體積=∫21y2πdx=π∫21(4−x2)dx=π[4x−13x3]|21=π(8−83−4+13)=53π
解答:A=[221131122]=[121121121]+[100010001]=B+IB=[121121121]⇒Bn=22n−2B,n∈N若矩陣M=[abcdefghi],定義g(M)=h,則g(Bn)=22n−2g(B)=22n−2⋅2=22n−1A2=(B+I)2=B2+2B+I⇒g(A2)=g(B2)+2g(B)+g(I)=8+2⋅2+0=12A3=(B+I)3=B3+3B2+3B+I⇒g(A3)=32+24+6+0=62A4=(B+I)4=B4+4B3+6B2+4B+I⇒g(A4)=128+128+48+8+0=312A5=(B+I)5=B5+5B4+10B3+10B2+5B+I⇒g(A5)=512+640+320+80+10=1562⇒g(f(A))=1562−7×312+9×62+9×12−7×2+8×0=30
解答:α,β,γ為x3−2x2−6x+5=0之三根⇒{α+β+γ=2αβ+βγ+γα=−6αβγ=−5⇒(α+β)(β+γ)(γ+α)=(2−β)(2−γ)(2−α)=8−4(α+β+γ)+2(αβ+βγ+γα)−αβγ=−7因此(α+β+γ)5−(α5+β5+γ5)=5(α+β)(β+γ)(γ+α)((α+β+γ)2−(αβ+βγ+γα)⇒25−(α5+β5+γ5)=5⋅(−7)(4+6)⇒α5+β5+γ5=32+350=382
解答:y的最小值2,最大值6⇒2≤y≤6⇒(y−2)(y−6)≤0⇒y2−8y+12≤0⋯(1)由於y=ax2+bx+6x2+2⇒(a−y)x2+bx+6−2y=0此函數有解⇒b2−4(a−y)(6−2y)≥0⇒8y2−(8a+24)y+24a−b2≤0⇒y2−(a+3)y+3a−b28≤0與式(1)相同⇒{a+3=83a−b28=12⇒a=5
解答:公式:Cn0+Cn2+Cn4−n=C120+C122+C124−12=1+66+495−12=550註:公式來源:按這裡
解答:
作¯AE⊥¯BD及¯CF⊥¯BD,如上圖;則△MEA∼△MFC(AAA)⇒¯AECF=¯AM¯MC⇒¯AM¯MC=△ABD△CBD=12⋅¯AB⋅¯AD⋅sin∠BAD12⋅¯CB⋅¯CD⋅sin∠BCD=20sin∠BAD16sin(π−∠BAD)=54解答:(x+2)2+(y+3)2=(x+5)2+(y+7)2=(x+11)2+(y+k)2⇒4x+6y+13=10x+14y+74=22x+2ky+121+k2⇒{6x+8y+61=012x+(2k−14)y+47+k2=0無解⇒612=82k−14≠6147+k2612=82k−14⇒k=15⇒{82k−14=126147+k2=61272⇒82k−14≠6147+k2;因此,k=15無解
解答:a=1223334444\cdots 999\\ \Rightarrow \cases{奇位數字和= 9 \times 5+8\times 4 + (7 +6 + 5)\times 3+ 4\times 2 + 3+ 2+ 1 =145\\ 偶位數字和= (9+8+7)\times 4 + 6\times 3+ (5+4+3)\times 2+2 = 140}\\ \Rightarrow 奇位數字和-偶位數字和=145-140=5 \Rightarrow a-5是11的倍數 \Rightarrow a=11k +5,k\in N \\ \Rightarrow a^2= (11k+5)^2 = 11^2k^2 + 110k + 25(= 11\times 2+3) =11s+ 3, s\in N\\ \Rightarrow a \equiv \bbox[red,2pt]{3} \mod 11
解答:每個座位的鄰居數量如下表:\\\begin{array}{|c|c|c|c|c|}\hline 3& 5& 5& 5&3\\\hline 5& 8& 8& 8&5\\\hline5& 8& 8& 8&5\\\hline5& 8& 8& 8&5\\\hline 3& 5 & 5 & 5 &3\\\hline \end{array} \Rightarrow 鄰居總數=3\times 4+5\times 12+8\times 9=144\\ 因此機率為{144\over 25\times 24} ={6\over 25} ={n\over m}\Rightarrow m+n=\bbox[red,2pt]{31}
解答:
\cases{s=\sqrt{29}=\sqrt{2^2+(2+3)^2} \\ t= \sqrt{37} =\sqrt{1^2+(1+2+3)^2} \\u=\sqrt{52} =\sqrt{(1+3)^2 +(1+2+3)^2}}\\將s,t,u以不同的直角三角形的斜邊繪製,如上圖;則此三角形面積=正方形扣除三個直角三角形\\ =6\times 6-{1\over 2}(5\times 2+1\times 6+ 4\times 6)= 36-20=\bbox[red,2pt]{16}解答:
f(f(x))=-5 \Rightarrow \cases{f(x)=2 \Rightarrow 兩圖形\cases{y=f(x)\\ y=2}有4個交點(上圖藍色圈圈)\\ f(x)=-3 \Rightarrow 兩圖形\cases{y=f(x)\\ y=-3}有5個交點(上圖綠色圈圈)\\ f(x)=k (上圖紅點) \Rightarrow 兩圖形\cases{y=f(x)\\ y=k(k>5)}有1個交點(上圖棕色圈圈)}\\ \Rightarrow f(f(x))=-5 有4+5+1= 10個交點,即有\bbox[red,2pt]{10}個相異實根註:學校公布的答案是\bbox[blue,2pt]{9}
解答:{1\over 7} ={a_1\over 9}+ {a_2\over 9^2}+ {a_3\over 9^3}+ \cdots +{a_n\over 9^n}+\cdots \Rightarrow {9\over 7}=1+{2\over 7} =a_1+ {a_2\over 9}+ {a_3\over 9^2}+ \cdots +{a_n\over 9^{n-1}}+\cdots\\ 由於a_1 \in \{0,1,...,6\},因此a_1=1 \Rightarrow {2\over 7} ={a_2\over 9}+ {a_3\over 9^2}+ \cdots +{a_n\over 9^{n-1}}+\cdots \\ \Rightarrow {18\over 7}=2+{4\over 7}= a_2+ {a_3\over 9} +{a_4\over 9^2}+ \cdots \Rightarrow a_2=2 \Rightarrow {4\over 7}={a_3\over 9} +{a_4\over 9^2}+ \cdots \\\Rightarrow {36\over 7}=5+{1\over 7}=a_3+{a_4\over 9}+\cdots\\ \Rightarrow a_3=5 \Rightarrow \langle a_n \rangle = 1,2,5,1,2,5,\cdots \Rightarrow a_{300}= \bbox[red,2pt]{5} (\because 300\equiv 0 \mod 3)
解答:
解答:\cases{a(4-b) = 4\cdots(1)\\ b(4-c)=4\cdots(2) \\ c(4-a)=4\cdots (3)},由(1) \Rightarrow a={4\over 4-b} 代入(3) \Rightarrow c(4-{4\over 4-b})=4 \Rightarrow c=1+{1\over 3-b}代入(2)\\ \Rightarrow b(3-{1\over 3-b})=4 \Rightarrow b={12-4b \over 8-3b} \Rightarrow b^2-4b+4=0 \Rightarrow (b-2)^2=0 \Rightarrow b=2 \\將 \cases{b=2代入(1) \Rightarrow a=2 \\ b=2代入(2) \Rightarrow c=2} \Rightarrow a+b+c=2+2+2 = \bbox[red,2pt]{6}

解答:
令\cases{\overline{CD}=w \\ \overline{AD}=h \\\overline{AE}=a\\ \overline{CF}=b} \Rightarrow \cases{\overline{BE}=w-a\\ \overline{BF}= h-b},依題意\cases{\triangle ADE=2 \\\triangle BEF=3 \\ \triangle CDF=4} \Rightarrow \cases{ah=4 \cdots(1)\\ (w-a)(h-b)=6 \cdots(2)\\ bw=8 \cdots(3)} \\ 由(2) \Rightarrow wh-wb-ah+ab=6 \Rightarrow wh-8-4+ab=6 \Rightarrow wh+ab=18\cdots(3)\\ \triangle ADE \times \triangle CDF = 2\times 4 \Rightarrow {1\over 2}ah \times {1\over 2}bw = 8 \Rightarrow abhw=32 \cdots(4)\\ (3) \Rightarrow ab=18-wh 代入(4) \Rightarrow (18-wh)(wh)=32 \Rightarrow (wh)^2-18(wh)+32=0 \\ \Rightarrow (wh-2)(wh-16)=0 \Rightarrow wh=16 (2不合,\because 矩形ABCD \gt \triangle \triangle CDF)\\ \Rightarrow \triangle DEF = wh-\triangle ADE-\triangle BEF-\triangle CDF =16-2-3-4= \bbox[red,2pt]{7}解答:
解答:a,b,c為x^3-3x^2+6x-1=0的三根\Rightarrow \cases{a+b+c=3 \\ ab+bc+ca = 6 \\ abc=1}\\ \Rightarrow a^2+b^2+c^2 =(a+b+c)^2-2(ab+ bc+ca)= 3^2-2\times 6= -3;\\ \begin{vmatrix} b^2+c^2 & ab & ac \\ ab & c^2+a^2 & bc\\ ac & bc & a^2+b^2 \end{vmatrix} = abc\begin{vmatrix} {b^2+c^2 \over a} & b & c \\ a & {c^2+a^2 \over b}& c\\ a & b & {a^2+b^2 \over c} \end{vmatrix} =\begin{vmatrix} b^2+c^2 & b^2 & c^2 \\ a^2 & c^2+a^2 & c^2\\ a^2 & b^2 & a^2+b^2 \end{vmatrix} \\ \underrightarrow{c_2+c_1,c_3+c_2}\begin{vmatrix} a^2+b^2+c^2 & a^2+b^2+c^2 & 2c^2 \\ 2a^2 & a^2+b^2+c^2 & a^2+b^2+c^2\\ a^2 & b^2 & a^2+b^2 \end{vmatrix} =\begin{vmatrix} -3 & -3 & 2c^2 \\ 2a^2 & -3 & -3\\ a^2 & b^2 & -3-c^2 \end{vmatrix} \\ = 9(-3-c^2)+4a^2b^2c^2 +9a^2 +6a^2c^2+6a^2(-3-c^2)-9b^2\\ =-23-9(a^2+b^2+c^2) =-23+27 = \bbox[red,2pt]{4}
解答:
111111+4444444444-66666=4444488889,再手算開根號,如上圖;因此 x=66667,各位數字和=6+6+6+6+7= \bbox[red,2pt]{31}
解答:
920 \div 2= 460 = 120+210+130 \\\Rightarrow 一個月的電費=120\times 1.63+210\times 2.38+130\times 3.52 = 1153元\\ \Rightarrow 兩個月的電費=1153\times 2= \bbox[red,2pt]{2306}元
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