數(shù)學(xué)基礎(chǔ)自考試題及答案

一、單項(xiàng)選擇題（每題2分，共20分）1.若集合\(A=\{1,2,3\}\)，\(B=\{2,3,4\}\)，則\(A\capB=(\)\)A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1,4\}\)D.\(\varnothing\)2.函數(shù)\(y=x^2-2x+3\)的對(duì)稱(chēng)軸方程是\((\)\)A.\(x=1\)B.\(x=-1\)C.\(x=2\)D.\(x=-2\)3.已知等差數(shù)列\(zhòng)(\{a_n\}\)中，\(a_1=2\)，\(d=3\)，則\(a_5=(\)\)A.11B.12C.13D.144.函數(shù)\(y=\sinx\)的最小正周期是\((\)\)A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)5.若\(\log_2x=3\)，則\(x=(\)\)A.6B.8C.9D.126.直線\(2x+y-1=0\)的斜率是\((\)\)A.2B.-2C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)7.過(guò)點(diǎn)\((1,2)\)且平行于\(x\)軸的直線方程是\((\)\)A.\(x=1\)B.\(y=1\)C.\(x=2\)D.\(y=2\)8.已知向量\(\vec{a}=(1,2)\)，\(\vec=(3,m)\)，若\(\vec{a}\parallel\vec\)，則\(m=(\)\)A.3B.4C.5D.69.不等式\(x^2-4x+3\lt0\)的解集是\((\)\)A.\(\{x|1\ltx\lt3\}\)B.\(\{x|x\lt1或x\gt3\}\)C.\(\{x|-1\ltx\lt-3\}\)D.\(\{x|x\lt-1或x\gt-3\}\)10.若\(\cos\alpha=\frac{1}{2}\)，且\(\alpha\in(0,\frac{\pi}{2})\)，則\(\alpha=(\)\)A.\(\frac{\pi}{6}\)B.\(\frac{\pi}{3}\)C.\(\frac{\pi}{4}\)D.\(\frac{\pi}{2}\)二、多項(xiàng)選擇題（每題2分，共20分）1.下列函數(shù)中，是偶函數(shù)的有\(zhòng)((\)\)A.\(y=x^2\)B.\(y=x^3\)C.\(y=\cosx\)D.\(y=|x|\)2.下列數(shù)列中，是等差數(shù)列的有\(zhòng)((\)\)A.\(1,3,5,7\)B.\(2,4,8,16\)C.\(1,1,1,1\)D.\(0,-2,-4,-6\)3.下列直線中，與直線\(y=2x+1\)平行的有\(zhòng)((\)\)A.\(y=2x-1\)B.\(y=-2x+1\)C.\(4x-2y+3=0\)D.\(2x-y+5=0\)4.已知集合\(A=\{1,2,3\}\)，\(B=\{2,3,4\}\)，則下列關(guān)系正確的有\(zhòng)((\)\)A.\(A\capB=\{2,3\}\)B.\(A\cupB=\{1,2,3,4\}\)C.\(A\subseteqB\)D.\(B\)中有元素不在\(A\)中5.在\(\triangleABC\)中，已知\(A=30^{\circ}\)，\(B=60^{\circ}\)，則\(C\)的度數(shù)可能是\((\)\)A.\(90^{\circ}\)B.\(120^{\circ}\)C.\(30^{\circ}\)D.\(60^{\circ}\)6.下列函數(shù)中，在\((0,+\infty)\)上單調(diào)遞增的有\(zhòng)((\)\)A.\(y=x\)B.\(y=x^2\)C.\(y=\frac{1}{x}\)D.\(y=\log_2x\)7.若向量\(\vec{a}=(1,1)\)，\(\vec=(1,-1)\)，則下列運(yùn)算結(jié)果正確的有\(zhòng)((\)\)A.\(\vec{a}+\vec=(2,0)\)B.\(\vec{a}-\vec=(0,2)\)C.\(\vec{a}\cdot\vec=0\)D.\(|\vec{a}|=\sqrt{2}\)8.下列不等式中，解集為\(R\)的有\(zhòng)((\)\)A.\(x^2+2x+1\gt0\)B.\(x^2-2x+3\gt0\)C.\(2x^2+3x+4\gt0\)D.\(-x^2+x-1\lt0\)9.已知圓的方程為\((x-1)^2+(y-2)^2=4\)，則下列點(diǎn)在圓上的有\(zhòng)((\)\)A.\((1,0)\)B.\((3,2)\)C.\((1,4)\)D.\((-1,2)\)10.下列三角函數(shù)值正確的有\(zhòng)((\)\)A.\(\sin\frac{\pi}{6}=\frac{1}{2}\)B.\(\cos\frac{\pi}{3}=\frac{1}{2}\)C.\(\tan\frac{\pi}{4}=1\)D.\(\sin\frac{\pi}{2}=0\)三、判斷題（每題2分，共20分）1.空集是任何集合的子集。\((\)\)2.函數(shù)\(y=\frac{1}{x}\)是奇函數(shù)。\((\)\)3.若\(a\gtb\)，則\(ac^2\gtbc^2\)。\((\)\)4.直線\(x+y-1=0\)的傾斜角是\(45^{\circ}\)。\((\)\)5.等比數(shù)列\(zhòng)(\{a_n\}\)中，\(a_1=1\)，\(q=2\)，則\(a_3=4\)。\((\)\)6.函數(shù)\(y=\cosx\)在\([0,\pi]\)上單調(diào)遞增。\((\)\)7.若向量\(\vec{a}=(1,2)\)，\(\vec=(2,4)\)，則\(\vec{a}\)與\(\vec\)共線。\((\)\)8.不等式\(x^2-2x+1\leq0\)的解集是\(\{1\}\)。\((\)\)9.圓\((x-2)^2+(y+3)^2=4\)的圓心坐標(biāo)是\((2,-3)\)，半徑是\(2\)。\((\)\)10.\(\log_28=3\)。\((\)\)四、簡(jiǎn)答題（每題5分，共20分）1.求函數(shù)\(y=x^2-4x+5\)的最小值。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)中，\(a_3=5\)，\(a_7=13\)，求\(a_1\)和\(d\)。3.求過(guò)點(diǎn)\((2,-1)\)且斜率為\(3\)的直線方程。4.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，求\(\cos\alpha\)的值。五、討論題（每題5分，共20分）1.討論函數(shù)\(y=x^2-2x-3\)的單調(diào)性。2.討論直線\(y=kx+1\)與圓\(x^2+y^2=1\)的位置關(guān)系。3.討論等比數(shù)列\(zhòng)(\{a_n\}\)中，公比\(q\)對(duì)數(shù)列單調(diào)性的影響。4.討論不等式\(ax^2+bx+c\gt0\)（\(a\neq0\)）的解集情況。答案一、單項(xiàng)選擇題1.B2.A3.C4.C5.B6.B7.D8.D9.A10.B二、多項(xiàng)選擇題1.ACD2.ACD3.ACD4.ABD5.A6.ABD7.ABCD8.BCD9.ABC10.ABC三、判斷題1.√2.√3.×4.×5.√6.×7.√8.√9.√10.√四、簡(jiǎn)答題1.對(duì)\(y=x^2-4x+5\)配方得\(y=(x-2)^2+1\)，因?yàn)閈((x-2)^2\geq0\)，所以\(y\geq1\)，最小值為\(1\)。2.由等差數(shù)列通項(xiàng)公式\(a_n=a_1+(n-1)d\)，可得\(\begin{cases}a_1+2d=5\\a_1+6d=13\end{cases}\)，兩式相減得\(4d=8\)，\(d=2\)，代入\(a_1+2d=5\)得\(a_1=1\)。3.由點(diǎn)斜式\(y-y_0=k(x-x_0)\)，過(guò)點(diǎn)\((2,-1)\)，斜率\(k=3\)，則直線方程為\(y+1=3(x-2)\)，即\(3x-y-7=0\)。4.因?yàn)閈(\sin^2\alpha+\cos^2\alpha=1\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，\(\cos\alpha\lt0\)，所以\(\cos\alpha=-\sqrt{1-\sin^2\alpha}=-\sqrt{1-(\frac{3}{5})^2}=-\frac{4}{5}\)。五、討論題1.對(duì)\(y=x^2-2x-3\)配方得\(y=(x-1)^2-4\)，對(duì)稱(chēng)軸為\(x=1\)。在\((-\infty,1)\)上，\(y\)隨\(x\)增大而減小；在\((1,+\infty)\)上，\(y\)隨\(x\)增大而增大。2.圓\(x^2+y^2=1\)圓心\((0,0)\)，半徑\(r=1\)，圓心到直線\(y=kx+1\)即\(kx-y+1=0\)的距離\(d=\frac{1}{\sqrt{k^2+1}}\)。當(dāng)\(d\gt1\)即\(k=0\)時(shí)，相離；當(dāng)\(d=1\)即\(k\)不存在時(shí)，相切；當(dāng)\(d\lt1\)即\(k\neq0\)時(shí)，相交。3.當(dāng)\(a_1\gt0\)，\(q\gt1\)或\(a_1\lt0\)，\(0\ltq\lt1\)時(shí)，數(shù)列遞增；當(dāng)\(a_1\gt0\)，\(0\ltq\lt1\)或\(a_1\lt0\)，\(q\gt1\)時(shí)，數(shù)列遞減；當(dāng)\(q=1\)時(shí)，