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2022年2月8日 星期二

105年身心障礙學生四技二專甄試-數學(B)-詳解

105 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:$$(A)斜率=1/2\\(B)斜率=-5\\(C)斜率=3/2\\(D)斜率=3\\ 斜率最大為3,故選\bbox[red,2pt]{(D)}$$
解答:$$\theta為第二象限角\Rightarrow \cases{\sin \theta\gt 0\\ \cos \theta \lt 0} \Rightarrow \cases{\sin(\pi/2-\theta) =\cos \theta \lt 0\\ \tan(\pi+\theta) =\tan \theta \lt 0} \Rightarrow 第三象限,故選\bbox[red,2pt]{(C)}$$
解答:$$后羿坐標(0,0,0) \Rightarrow 野雁坐標(50,0,120) \Rightarrow 老鷹坐標(70,0,240)\\ \Rightarrow 后羿與老鷹距離\sqrt{70^2+240^2} =250,故選\bbox[red,2pt]{(C)}$$
解答:$$D為\overline{BC}中點\Rightarrow D =(B+C)/2 = ((2,3)+(-4,1))/2 =(-1,2) \Rightarrow \cases{\overrightarrow{AB} =(2,3) \\\overrightarrow{AD} =(-1,2)} \\ \Rightarrow \overrightarrow{AB}\cdot \overrightarrow{AD} =-2+6=4,故選\bbox[red,2pt]{(B)}$$
解答:$$(A) 10^{\log_{10}8} =8\\(B) 2^{1/2} =\sqrt 2\\ (C){\log_{10}8 \over \log_{10} 2} =\log_2 8=3\\ (D) (1/2)^2=1/4 \\ \Rightarrow (A)最大,故選\bbox[red,2pt]{(A)}$$
解答:$$\log_{10} x^2y = 2\log_{10}x +\log_{10}y = 2\cdot {1\over 2}-4 =-3,故選\bbox[red,2pt]{(A)}$$
解答:$$ \cases{2,5,8為等差數列,公差d=3\\ 1,2,4為等比數列,公比r=2} \Rightarrow r+d= 2+3=5,故選\bbox[red,2pt]{(B)}$$
解答:$$f(x)\times g(x)+2h(x) = (x+1)(2x+1)+2(3x+5) =2x^2+3x+1+6x+10 =2x^2+9x+11 \\,故選\bbox[red,2pt]{(C)}$$
解答:$$ \cases{2x+y=5\\ 3x-y=5} \Rightarrow 5x=10 \Rightarrow x=2 \Rightarrow y=5-4=1 \Rightarrow x+y=2+1=3,故選\bbox[red,2pt]{(A)}$$
解答:$$\begin{vmatrix}x & 1\\ 6 & 2 \end{vmatrix} =\begin{vmatrix} 5 & x\\ 4 & 6 \end{vmatrix} \Rightarrow 2x-6=30-4x \Rightarrow x=6,故選\bbox[red,2pt]{(B)}$$
解答:$$(A) \times: (1,1)\Rightarrow x-y=0 \not \gt 0\\(B)\times:(1,2) \Rightarrow x-y=-1\not \gt 0\\ (C)\bigcirc: \\(D)\times: (5,4) \Rightarrow y-3=1 \not \lt 0\\,故選\bbox[red,2pt]{(C)}$$
解答:$$3!=6,故選\bbox[red,2pt]{(B)}$$
解答:$$\cases{P(A)抽到獎的機率=20/50\\ P(B)抽到超級大獎的機率=1/50} \Rightarrow P(B\mid A)= {1/50\over 20/50} = {1\over 20},故選\bbox[red,2pt]{(D)}$$    
解答:$$x,y均為整數\Rightarrow \cases{x=4,5\\ y=6,7,8} ,又\cases{算術平均數=(2+2+\cdots +9)/10=(39+x+y)/10\\ 中位數=(x+5)/2} \\\Rightarrow {39+x+y\over 10} ={x+5\over 2}  \Rightarrow 14+y=4x \Rightarrow \cases{x=4 \Rightarrow y=4不合\\ x=5\Rightarrow y=6} \Rightarrow (x,y)=(5,6) \\ \Rightarrow 算術平均數=中位數=(x+5)/2=10/2=5,故選\bbox[red,2pt]{(C)}$$
解答:$$\cases{抽到甲:100\times {100\over 10000}=1\\ 抽到乙:100\times {40\over 5000} ={4\over 5}} \Rightarrow 0.5\times 抽到甲+0.5\times 抽到乙= 0.9,故選\bbox[red,2pt]{(B)}$$
解答:$$直角\triangle ABC外接圓直徑2R=斜邊長 =5 \Rightarrow 正弦定理{3\over \sin \theta} =2R=5 \Rightarrow \sin \theta ={3\over 5}\\ \Rightarrow \sin 2\theta = 2\sin \theta \cos \theta = 2\cdot {3\over 5}\cdot {4\over 5} ={24\over 25},故選\bbox[red,2pt]{(D)}$$
解答:$$\cases{半徑=1\\ 圓心(0,1)} \Rightarrow 圓方程式:x^2+(y-1)^2 =1 \Rightarrow (0,2)在圓上,故選\bbox[red,2pt]{(A)}$$
解答:$$25x^2+9y^2=225 \Rightarrow {x^2\over 9}+ {y^2\over 25}=1 \Rightarrow \cases{a=5\\ b=3} \Rightarrow 2a+2b=16,故選\bbox[red,2pt]{(D)}$$
解答:$$\cases{f(x)=x^2+1 \\ g(x)=3x+2} \Rightarrow \cases{f'(x)=2x\\ g'(x)=3} \Rightarrow {d\over dx}(f(x)\times g(x)) = f'(x)g(x) + f(x)g'(x)\\ =2x(3x+2)+3(x^2+1)= 9x^2+4x+3,故選\bbox[red,2pt]{(D)}$$
解答:$$(\int_0^3f(x)\;dx) -(\int_0^3g(x)\;dx) =\int_0^3(f(x)-g(x))\;dx =\int_0^3 2\;dx=6,故選\bbox[red,2pt]{(A)}$$
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