115年身障升四技二專-數學B詳解

115 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(B)

解答:$$假設公差d \Rightarrow \cases{a_0=-1\\ a_1=a_0+d=-1+d=a\\a_2=a_0+2d=-1+2d=b\\ a_3=a_0+3d=-1+3d=8} \Rightarrow \cases{a+b=-2+3d\\ a_0+a_3=-2+3d=-1+8= 7} \\ \Rightarrow a+b= 7，故選\bbox[red, 2pt]{(B)}$$

解答:$$(a,b+2)在第三象限\Rightarrow \cases{a\lt 0\\ b+2\lt 0 \Rightarrow b\lt -2} \Rightarrow \cases{-a\gt 0\\ b+1\lt -1} \Rightarrow (-a,b+1)在第四象限\\，故選\bbox[red, 2pt]{(D)}$$

解答:$$(A)\times: f(x)=-x^2+2x+3 =-(x^2-2x+1)+4=-(x-1)^2+4 \Rightarrow 頂點(1,4) \\(B)\times:二次項係數為負值，圖形凹向下\\ (C)\bigcirc: f(x)=-(x-3)(x+1)=0 \Rightarrow x=3,-1有兩個交點 \\ (D)\times: 交點坐標為(3,0)或(-1,0)\\，故選\bbox[red, 2pt]{(C)}$$

解答:$$\cases{x+y\le 4的解包含第一、二、三、四象限\\ x-y\ge 2的解不包含第二象限}，故選\bbox[red, 2pt]{(B)}$$

解答:$$\left( x-{3\over x} \right)^2=2^2 \Rightarrow x^2-6+{{9\over x^2}} =4 \Rightarrow x^2+6+{9\over x^2}=4+12 \Rightarrow \left( x+{3\over x} \right)^2=16 \\ \Rightarrow x+{3\over x}=-4 (因為x\lt 0)，故選\bbox[red, 2pt]{(D)}$$

解答:$$|2x-1|\gt 3 \Rightarrow \cases{2x-1\gt 3\\ 2x-1\lt -3} \Rightarrow \cases{x\gt 2\\ x\lt -1} \Rightarrow x\gt 2或x\lt -1，故選\bbox[red, 2pt]{(D)}$$

解答:$$\log_2 3+ \log_2 16-\log_2 6+\log_5 25 =\log_2 3+ \log_2 2^4-\log_2 (2\times 3)+\log_5 5^2 \\ =\log_2 3+ 4-\log_2 2-\log_2 3+2 = \log_2 3+4-1-\log_2 3+2= 5，故選\bbox[red, 2pt]{(B)}$$

解答:$$\cases{x+y=2\\ x-2y=-1} \Rightarrow \cases{x=1\\ y=1} 代入\cases{2ax+3y=5\\ 3x-2by=7} \Rightarrow \cases{2a+3=5\\3-2b=7} \Rightarrow \cases{a=1\\ b=-2} \\ \Rightarrow a+2b=1-4=-3，故選\bbox[red, 2pt]{(D)}$$

解答:$$x=0代入三條直線\Rightarrow \cases{L_1: 3y-6=0 \Rightarrow a=y=2 \\ L_2:-2y-6=0 \Rightarrow b=y=-3\\ L_3:-y-4=0 \Rightarrow c=y=-4} \Rightarrow \Rightarrow a\gt b\gt c，故選\bbox[red, 2pt]{(B)}$$

解答:$${({1\over 3})^{3/2} 3^{5/2} \over \sqrt[3]{27}} ={3^{-3/2} \cdot 3^{5/2} \over 3} ={3^{2/2}\over 3} =1，故選\bbox[red, 2pt]{(B)}$$

解答:$$\sin B={5\over 13} \Rightarrow \cos B= \sqrt{1-({5\over 13})^2} ={12\over 13} = {\overline{BC}   \over \overline{AB}} ={\overline{BC} \over 39} \Rightarrow \overline{BC}=36，故選\bbox[red, 2pt]{(D)}$$

解答:$$\vec u\bot \vec v \Rightarrow \vec u\cdot \vec v=0 \Rightarrow (2,3) \cdot(k+1,k-1) =2k+2+3k-3=5k-1=0 \Rightarrow k={1\over 5}，故選\bbox[red, 2pt]{(C)}$$

解答:$$x^2+y^2+4x-6y+12=0 \Rightarrow (x^2+4x+4)+(y^2-6y+9)+12=4+9 \\ \Rightarrow (x+2)^2+(y-3)^2=1 \Rightarrow 圓心(-2,3)，故選\bbox[red, 2pt]{(C)}$$

解答:$$100(1+1.745\%)^3 =100\times 1.05327=105.327萬元=1053270，故選\bbox[red, 2pt]{(D)}$$

解答:$$\cases{兩個字 :第1個字有5種選擇，第2個字也有5種選擇 \Rightarrow 5^2=25 \\ 三個字:三個字都有5種選擇\Rightarrow 5^3=125} \Rightarrow 25+125=150，故選\bbox[red, 2pt]{(D)}$$

解答:$$C^5_2\times C^7_3=10 \times 35=350，故選\bbox[red, 2pt]{(C)}$$

解答:$$P(A\cup B) =P(A)+P(B)-P(A\cap B) =0.6+0.35-0.6\times 0.35= 0.74，故選\bbox[red, 2pt]{(B)}$$

解答:$$1000\times 0.2+ 4000\times 0.5+ 3000\times 0.3=3100萬元，故選\bbox[red, 2pt]{(B)}$$

解答:$$95\% \Rightarrow 平均值\pm 2個標準差\Rightarrow [75-2\times 4, 75+2\times 4] =[67,83] \Rightarrow 67分到83分，故選\bbox[red, 2pt]{(A)}$$

解答:$$抽樣誤差\pm 3.2\%  \Rightarrow 84.2\% \pm 3.2\% \Rightarrow[0.842-0.032,0.842+0.032] =[0.81,0.874]，故選\bbox[red, 2pt]{(B)}$$

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解題僅供參考，身障升大學歷年試題及詳解