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2021年2月21日 星期日

109年中正預校教甄-數學詳解

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=26k=0C26kxk=1+26k=1C26kxk=1+8191=213x=2eikπ131,k=0,1,,25Re(x)=2coskπ131Re(x)>0,coskπ13>12=cosπ40k323k25Re(x)<04k22,19



解答
{E¯BC¯EF¯OA¯AE=¯ACsin60=4×32=23¯EF=3¯AF2=¯AE2¯EF2=123=9¯AF=3OCE:¯OE2=¯OC2¯EC2=a222=a24OEF:¯OE2=(¯OA¯AF)2+¯EF2a24=(a3)2+34=6a+12a=166=83


解答{a1+a2+a3++a2020=20211a1+1a2+1a3++1a2020=2021{a2+a3++a2020=2021a11a2+1a3++1a2020=20211a1(a2+a3++a2020)(1a2+1a3++1a2020)(1+1++1)2=20192(2021a1)(20211a1)20192202122021(a1+1a1)+120192a1+1a12021220192+12021=80812021
解答{a+b+2c=xa+2b+c=ya+b+c=z{a=xy+3zb=yzc=xz2b2ca+b+2c+2a+4ca+2b+c+ba+b+c=2y2xx+2x2y+2zy+yzz=2yx2+2x+2zy2+yz1=(2yx+2xy)+(2zy+yz)522yx2xy+22zyyz5=81
解答{a2+b2=4,O1(0,0),r1=2(c7)2+(d24)2=36,O2(7,24),r2=6|abcd|M=a2+b2×c2+d2=r1×(¯O1O2+r2)=2(25+6)=62(ac)2+(bd)2m=(¯O1O2r1r2)2=(2526)2=172=289M+m=62+289=351註:學校公布的答案是79
解答1220:2x+3y=8(x,y)=(4,0),(1,2){22221233=34112:2x+3y=12(x,y)=(6,0),(3,2),(0,4){222222822233108{3322232322233223,7333311127+1=88×4=32
解答
O(0,0,0)E:3x+2y+23z=5d(O,E)=59+4+12=12x=21y2πdx=π21(4x2)dx=π[4x13x3]|21=π(8834+13)=53π

解答A=[221131122]=[121121121]+[100010001]=B+IB=[121121121]Bn=22n2B,nNM=[abcdefghi],g(M)=h,g(Bn)=22n2g(B)=22n22=22n1A2=(B+I)2=B2+2B+Ig(A2)=g(B2)+2g(B)+g(I)=8+22+0=12A3=(B+I)3=B3+3B2+3B+Ig(A3)=32+24+6+0=62A4=(B+I)4=B4+4B3+6B2+4B+Ig(A4)=128+128+48+8+0=312A5=(B+I)5=B5+5B4+10B3+10B2+5B+Ig(A5)=512+640+320+80+10=1562g(f(A))=15627×312+9×62+9×127×2+8×0=30
解答α,β,γx32x26x+5=0{α+β+γ=2αβ+βγ+γα=6αβγ=5(α+β)(β+γ)(γ+α)=(2β)(2γ)(2α)=84(α+β+γ)+2(αβ+βγ+γα)αβγ=7(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y262y6(y2)(y6)0y28y+120(1)y=ax2+bx+6x2+2(ay)x2+bx+62y=0b24(ay)(62y)08y2(8a+24)y+24ab20y2(a+3)y+3ab280(1){a+3=83ab28=12a=5
解答:Cn0+Cn2+Cn4n=C120+C122+C12412=1+66+49512=550註:公式來源:
解答
¯AE¯BD¯CF¯BDMEAMFC(AAA)¯AECF=¯AM¯MC¯AM¯MC=ABDCBD=12¯AB¯ADsinBAD12¯CB¯CDsinBCD=20sinBAD16sin(πBAD)=54


解答(x+2)2+(y+3)2=(x+5)2+(y+7)2=(x+11)2+(y+k)24x+6y+13=10x+14y+74=22x+2ky+121+k2{6x+8y+61=012x+(2k14)y+47+k2=0612=82k146147+k2612=82k14k=15{82k14=126147+k2=6127282k146147+k2k=15
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


令\overline{BD}=a \Rightarrow \cases{\cos \angle A={3+1-a^2 \over 2\sqrt 3} \\ \cos \angle C={1+1-a^2\over 2}} \Rightarrow \cases{a^2= 4-2\sqrt 3\cos \angle A \\ a^2=2-2\cos \angle C} \Rightarrow 4-2\sqrt 3\cos \angle A= 2-2\cos \angle C \\ \Rightarrow \cos \angle C= \sqrt 3\cos \angle A-1 \cdots(1)\\ 又\cases{S=\triangle ABD ={1\over 2}\cdot \overline{AB}\cdot \overline{AD} \sin \angle A ={\sqrt 3\over 2}\sin \angle A\\ T=\triangle BCD ={1\over 2}\cdot \overline{CB}\cdot \overline{CD} \sin \angle C ={1\over 2}\sin \angle C} \Rightarrow S^2+T^2 ={3\over 4}\sin^2\angle A+ {1\over 4}\sin^2\angle C \cdots(2)\\ 將(1)代入(2)\Rightarrow S^2+T^2 ={3\over 4}(1-\cos^2\angle A)+ {1\over 4}(1-\cos^2\angle C)\\ ={3\over 4}(1-\cos^2\angle A)+ {1\over 4}(1-(\sqrt 3\cos \angle A-1)^2)= -{3\over 2}\cos^2 \angle A+{\sqrt 3\over 2}\cos \angle A+{3\over 4}\\ \Rightarrow 當\cos \angle A={\sqrt 3\over 6}時,S^2+T^2 有極大值\bbox[red,2pt]{7\over 8}


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



\cases{z_1=-3-\sqrt 3 i\\ z_2=\sqrt 3+ i\\ z=\sqrt 3\sin \theta+ i(\sqrt 3\cos \theta+2)} \Rightarrow \cases{A(z_1)=(-3,-\sqrt 3)\\ B(z_2)=(\sqrt 3,1)\\ \Gamma(z)=x^2+(y-2)^2 = 3} \Rightarrow L=\overleftrightarrow{AB}:x=\sqrt 3 y\\ 圓心(0,2)至L距離={2\sqrt 3\over \sqrt{3+1}} =\sqrt 3=圓半徑 \Rightarrow L為圓之切線 \\\Rightarrow |z-z_1|+ |z-z_2|的最小值=\overline{AB}= \sqrt{(3+\sqrt 3)^2 +(\sqrt 3+1)^2} =  \sqrt{16+2\sqrt{48}} \\=\sqrt{12}+\sqrt 4 = \bbox[red,2pt]{2+2\sqrt 3}
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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