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2022年3月14日 星期一

110年國中學力鑑定-數學詳解

110年度自學進修國民中小學畢業程度〈含身心障礙國民〉
學力鑑定 國中級-數學

一、選擇題:(每題3分,共90分)

解答:$$80+(-28)\div 4 =80+(-7)=73,故選\bbox[red,2pt]{(3)}$$
解答:$$\cases{24=2^3\times 3\\ 36= 2^2\times 3^2\\ 48=2^4\times 3} \Rightarrow \cases{(24,36,48) = 2^2\times 3 =12\\ [24,36,48] = 2^4\times 3^2 =144} \Rightarrow (24,36,48) +[24,36,48] =12+144 =156\\,故選\bbox[red,2pt]{(2)}$$
解答:$$0.000318 = {3.18 \over 10000} = {3.18 \over 10^{4}}=3.18 \times 10^{-4} \Rightarrow n=-4,故選\bbox[red,2pt]{(2)}$$
解答:$$\sqrt{3^2+4^2} =\sqrt{25} =5,故選\bbox[red,2pt]{(1)}$$
解答:$$(-{8\over 3})\times (-{1\over 4}) = {8\over 3}\times {1\over 4} ={2\over 3},故選\bbox[red,2pt]{(2)}$$
解答:$$x=-2 \Rightarrow 3x+5 =3\times (-2)+5 = -6+5=-1,故選\bbox[red,2pt]{(3)}$$
解答:$$a不小於6 \Rightarrow a \not \lt 6 \Rightarrow a\ge 6,故選\bbox[red,2pt]{(2)}$$
解答:$$x+4=0 \Rightarrow x=-4,只有(-4,0)的x座標為-4,其他點的x座標皆不是-4,故選\bbox[red,2pt]{(1)}$$
解答:$$f(x)是常數函數\Rightarrow f(x)=k \Rightarrow f(1)= f(2)=f(3) =f(4)= f(5)=k \\ \Rightarrow f(1)+ f(2)+f(3) +f(4)+f(5) =5k = 15 \Rightarrow k=3 \Rightarrow f(6)=k = 3,故選\bbox[red,2pt]{(4)}$$
解答:$$\cases{a:b= 5:2 =10:4\\ b:c=4:3} \Rightarrow a:b:c = 10:4:3,故選\bbox[red,2pt]{(4)}$$
解答:$$外角=30^\circ \Rightarrow 內角=180^\circ - 30^\circ =150^\circ \Rightarrow {(n-2)\times 180 \over n}=150 \Rightarrow {n-2\over n} ={150\over 180} \\ \Rightarrow 1-{2\over n}={5\over 6} \Rightarrow {2\over n}=1-{5\over 6}={1\over 6} \Rightarrow n=2\times 6=12,故選\bbox[red,2pt]{(3)}$$

解答:$$最大值是7、最小值是1,因此全距=7-1=6,故選\bbox[red,2pt]{(3)}$$
解答:$$假設正方形邊長為a,則a\times a=6 \Rightarrow a^2=6 \Rightarrow a=\sqrt 6,故選\bbox[red,2pt]{(3)}$$

解答:$$8^2 =(-8)^2 =64 \Rightarrow 64的平方根為\pm 8,故選\bbox[red,2pt]{(3)}$$
解答:$$\cases{A=(3x-4)(2x+3) \\ B=(x+5)(3x-4)} \Rightarrow 3x-4同時是A及B的因式,故選\bbox[red,2pt]{(1)}$$
解答:$$(3x+1) (x-5)=0 \Rightarrow \cases{3x+1=0 \Rightarrow x=-1/3\\ x-5=0 \Rightarrow x=5} \Rightarrow x=-{1\over 3}或5,故選\bbox[red,2pt]{(2)}$$
解答:$$\cases{8^2 =64\\ 9^2=81\\ 10^2=100\\ 11^2=121 \\ 12^2=144} \Rightarrow 81\lt 90\lt 100 \Rightarrow \sqrt{81} \lt \sqrt{90} \lt \sqrt{100} \Rightarrow 9\lt \sqrt{90}\lt 10,故選\bbox[red,2pt]{(2)}$$
解答:$$與30互質的數:1,7,11,13,共4個,因此機率為{4\over 15} ,故選\bbox[red,2pt]{(2)}$$
解答:$$(1)公差為2\\ (2)公差為1\\ (3)公差為0\\ (4) 1-0\ne 0-1 \Rightarrow 不是等差數列,故選\bbox[red,2pt]{(4)}$$
解答:$$\angle A=\angle D=90^\circ \Rightarrow RHS,故選\bbox[red,2pt]{(4)}$$
解答:$$2x^2+7x+3 =(2x+1)(x+3) = (2x+a)(x+b) \Rightarrow \cases{a=1 \\ b=3} \Rightarrow a+b= 4,故選\bbox[red,2pt]{(1)}$$
解答
$$垂直、水平各一條,如上圖,故選\bbox[red,2pt]{(2)}$$
解答:$$7出現五次,頻率最高,故選\bbox[red,2pt]{(1)}$$
解答:$$\triangle ADE \sim \triangle ABC \Rightarrow {\overline{AD} \over \overline{AE}} = {\overline{AB} \over \overline{AC}} \Rightarrow {8\over 6} ={12\over \overline{AC}} \Rightarrow \overline{AC} =6\times 12\div 8=9,故選\bbox[red,2pt]{(1)}$$
解答
$$見上圖,紅色三條、藍色三條,故選\bbox[red,2pt]{(3)}$$
解答:$$\overline{O_1O_2} =兩半徑之和=5+3=8,故選\bbox[red,2pt]{(4)}$$
解答:$$(12+ 3+ 25+35+ 38+58 +53+43+48)\div 9 = 315\div 9=35,故選\bbox[red,2pt]{(4)}$$

解答:$$G為重心\Rightarrow \triangle GFB= \triangle GCD ={1\over 6}\triangle ABC \Rightarrow 12={1\over 3}\triangle ABC \Rightarrow \triangle ABC=12\times 3=36,故選\bbox[red,2pt]{(3)}$$
解答:$$\angle A+\angle B=180^\circ \Rightarrow 2x+ 3x-80 =180 \Rightarrow 5x=260 \Rightarrow x=52 \Rightarrow \angle A=2x= 104,故選\bbox[red,2pt]{(2)}$$
解答:$$放大三倍,角度不變,\angle D'=\angle D=50^\circ,故選\bbox[red,2pt]{(1)}$$
解答:$$y=(x+5)^2-7 \Rightarrow x=-5時,y=-7 \Rightarrow 頂點座標為\bbox[red, 2pt]{(-5,-7)}$$
解答:$$該數列為等差,其中\cases{首項a_1=33\\ 公差d=43-33=10} \Rightarrow a_1+\cdots + a_{10} =(33+123)\times 10\div 2= \bbox[red,2pt]{780}$$


解答
(1)$$由上表可知: x=2y,因此重量越重、價錢越高,成正比,故:\bbox[red,2pt]{是}$$(2)$$ 150= 2y \Rightarrow y=75,即\bbox[red,2pt]{75}元$$
解答:$$\cases{a=甲\\ b=乙} \Rightarrow \cases{a+b=24\\ ab=108} \Rightarrow a(24-a)=108 \Rightarrow a^2-24a+108=0 \Rightarrow (a-18)(a-6)=0\\ \Rightarrow \cases{a=18 \Rightarrow b=6(不合,違反a\lt b)\\ a=6 \Rightarrow b=18} \Rightarrow 甲數=\bbox[red,2pt]{6}$$
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解題僅供參考,其他歷屆試題及詳解
 

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