114年身障生升四技二專-數學(A)詳解

114 學年度身心障礙學生升學大專校院甄試

甄試類(群)組別：四技二專組-數學(A)

單選題，共 20 題，每題 5 分

解答: $$-x^2+6x\le 5 \Rightarrow x^2-6x+5\ge 0 \Rightarrow (x-5)(x-1)\ge 0 \Rightarrow x\ge 5或x\le 1，故選\bbox[red, 2pt]{(D)}$$

解答: $$\cases{3x-5y\le 13\\ 2x+y\le 13\\ 7x-3y\ge 13}所圍三角形頂點坐標\cases{A(4,5)\\ B(6,1) \\C(1,-2)} \Rightarrow \cases{f(4,5)=12 +5a\\ f(6,1)=18+a\\ f(1,-2) =3-2a} \\ f(4,5)有最大值\Rightarrow \cases{12+5a\ge 18+a\\ 12+5a\ge 3-2a} \Rightarrow \cases{a\ge 3/2\\ a\ge -9/7} \Rightarrow a\ge {3\over 2}，故選\bbox[red, 2pt]{(A)}$$

解答: $$(A) \sqrt[3] 5 \lt \sqrt 5\Rightarrow \sqrt[3] 5-\sqrt 5\lt 0\\ (B) \log_76 ={\log_66 \over \log_6 7} ={1\over \log_6 7\gt 1} \lt 1 \Rightarrow \log_76 -1\lt 0\\ (C)\log_{\sqrt{123}}123 ={\log_{123}123 \over \log_{123}\sqrt{123}} ={1\over 1/2} =2 \Rightarrow \log_{\sqrt{123}}123 -\log_{\sqrt \pi} 1=2-1\gt 0\\ (D) \log_{1/3}5 ={\log 5\over -\log 3} \lt 0 \\ ，故選\bbox[red, 2pt]{(C)}$$

解答: $$2025=45^2 \Rightarrow \log 2025 = 4\log 3+2\log 5 \\ a\log_{2025} 3+ b\log_{2025} 5=2 \Rightarrow a {\log 3\over \log 2025}+ b{\log 5\over \log 2025}=2 \\\Rightarrow a\log 3+b \log 5=2 \log 2025 = 8 \log 3+4\log 5 \Rightarrow \cases{a=8\\ b=4} \Rightarrow a+b=12，故選\bbox[red, 2pt]{(B)}$$

解答: $$\cases{購物台排列數:4! \\ 綜藝台排列數:3!\\ 電玩台排列數:3!} ，再加上綜藝台與電玩台順序可對調\\\Rightarrow 共有4!\times 3! \times 3!\times 2分配方式，故選\bbox[red, 2pt]{(C)}$$

解答: $$C^3_1C^6_3C^2_1= 3\times 20\times 2=120，故選\bbox[red, 2pt]{(D)}$$

解答: $$\cases{機率總和為1 : 2x+4y=1\\ 共望值=3:x+2x+3y+4y+5y+6y= 3x+18y=3} \Rightarrow \cases{x =1/4\\ y=1/8}，故選\bbox[red, 2pt]{(A)}$$

解答: $$\cases{都是男生的機率:C^{15}_3/C^{25}_3 \\都是女生的機率:C^{10}_3/C^{25}_3} \Rightarrow 不為單一性別的機率=1-C^{15}_3/C^{25}_3 -C^{10}_3/C^{25}_3 \\ =1-{91\over 460}-{6\over 115}={3\over 4}，故選\bbox[red, 2pt]{(B)}$$

解答: $$\sqrt{(376-385)^2 + (385-385)^2+ (379-385)^2+(382-385)^2+ (394-385)^2+ (388-385)^2 +(391-385)^2\over 7} \\=\sqrt{9^2+ 0+6^2+ 3^2+9^2+ 3^2+6^2 \over 7} =\sqrt{252\over 7} =\sqrt{36}=6，故選\bbox[red, 2pt]{(C)}$$

解答: $$由小至大依序為61,63,67,69,70\\ (A)組距=70-61=9 \ne 8\\ (B) 平均=(61+63+ 67+69+70)/5=330/5=66 \ne 65\\ (C)中位數=67\ne 69 \\(D)5\times{3\over 4}=3.75 \Rightarrow Q_3=第4位= 69 \\ 應該都不正確, 公布的答案是\bbox[cyan,2pt]{(D)}$$

解答: $$(A) f(2)=6\\ (B)f(2)=9 \ne 6\\(C) f(2)=12 \ne 6\\ (D)f(2)=4+4+5=13\ne 6\\，故選\bbox[red, 2pt]{(A)}$$

解答: $$花團長為x, 寬為y \Rightarrow x+2y=32 \Rightarrow x=32-2y \\\Rightarrow 面積a =xy=(32-2y)y =-2y^2+32y =-2(y^2-16y+64)+128 =-2(y-8)^2+128\le 128\\ \Rightarrow a最大值為128，故選\bbox[red, 2pt]{(C)}$$

解答: $$與5x-7y+35=0垂直的直線為7x+5y=k，又通過(0,0) \Rightarrow k=0\\ \Rightarrow 7x+5y=0，故選\bbox[red, 2pt]{(C)}$$

解答: $$f(-1)=0 \Rightarrow -7+4+a=0 \Rightarrow a=3，故選\bbox[red, 2pt]{(D)}$$

解答: $$\alpha,\beta為x^2+3x-2=0的兩根\Rightarrow \cases{\alpha=-3\\ \alpha\beta=-2} \Rightarrow \alpha^2+\beta^2= (\alpha+ \beta)^2-2\alpha \beta=9+4=13\\，故選\bbox[red, 2pt]{(B)}$$

解答: $$\sin^2 30^\circ +\cos^2 60^\circ = ({1\over 2})^2+ ({1\over 2})^2 ={1\over 2}，故選\bbox[red, 2pt]{(A)}$$

解答: $$\tan \theta=2 \Rightarrow \cases{\cos \theta=1/\sqrt{10} \\ \sin \theta=3/\sqrt{10}} \Rightarrow {5\cos \theta+2 \sin \theta \over 4\cos \theta-\sin \theta} ={5+6\over 4-3}=11，故選\bbox[red, 2pt]{(C)}$$

解答: $$(x-2)^2+ (y+7)^2 =4^2，故選\bbox[red, 2pt]{(D)}$$

解答: $$\sum_{k=1}^{10} (5+4k) =\sum_{k=1}^{10} 5+ 4\sum_{k=1}^{10}k =5\cdot 10+4\cdot 55=270，故選\bbox[red, 2pt]{(B)}$$

解答: $$100000(1+2\%)^2 =100000\times 1.0404 = 104040，故選\bbox[red, 2pt]{(B)}$$

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解題僅供參考,其他歷年試題及詳解