職中期末試題和答案數(shù)學

一、單項選擇題(每題2分,共20分)1.集合\(A=\{1,2,3\}\),集合\(B=\{2,3,4\}\),則\(A\capB=(\)\)A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1\}\)D.\(\{4\}\)2.函數(shù)\(y=\sqrt{x-1}\)的定義域是()A.\(x\geq1\)B.\(x\gt1\)C.\(x\leq1\)D.\(x\lt1\)3.直線\(y=2x+1\)的斜率是()A.\(1\)B.\(2\)C.\(\frac{1}{2}\)D.\(-2\)4.已知\(\sin\alpha=\frac{1}{2}\),且\(\alpha\)是第一象限角,則\(\cos\alpha=(\)\)A.\(-\frac{\sqrt{3}}{2}\)B.\(\frac{\sqrt{3}}{2}\)C.\(-\frac{1}{2}\)D.\(\frac{1}{2}\)5.等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(d=2\),則\(a_5=(\)\)A.\(9\)B.\(10\)C.\(11\)D.\(12\)6.不等式\(x^2-3x+2\lt0\)的解集是()A.\(\{x|1\ltx\lt2\}\)B.\(\{x|x\lt1或x\gt2\}\)C.\(\{x|x\lt1\}\)D.\(\{x|x\gt2\}\)7.圓\((x-1)^2+(y+2)^2=4\)的圓心坐標是()A.\((1,-2)\)B.\((-1,2)\)C.\((1,2)\)D.\((-1,-2)\)8.已知向量\(\overrightarrow{a}=(1,2)\),\(\overrightarrow=(3,4)\),則\(\overrightarrow{a}+\overrightarrow=(\)\)A.\((4,6)\)B.\((2,2)\)C.\((-2,-2)\)D.\((-4,-6)\)9.函數(shù)\(y=\cosx\)的最小正周期是()A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)10.從\(5\)名同學中選\(2\)名參加比賽,不同選法有()A.\(10\)種B.\(20\)種C.\(25\)種D.\(30\)種二、多項選擇題(每題2分,共20分)1.以下屬于基本初等函數(shù)的有(）A.冪函數(shù)B.指數(shù)函數(shù)C.對數(shù)函數(shù)D.三角函數(shù)2.下列命題正確的有()A.若\(a\gtb\),則\(ac^2\gtbc^2\)B.若\(a\gtb\),\(c\gtd\),則\(a+c\gtb+d\)C.若\(a\gtb\),則\(a^2\gtb^2\)D.若\(a\gtb\gt0\),\(c\gtd\gt0\),則\(ac\gtbd\)3.直線\(Ax+By+C=0\)(\(A\)、\(B\)不同時為\(0\))的斜率存在的條件有()A.\(A=0\),\(B\neq0\)B.\(A\neq0\),\(B=0\)C.\(A\neq0\),\(B\neq0\)D.\(A=B=0\)4.下列是奇函數(shù)的函數(shù)有(）A.\(y=x^3\)B.\(y=\sinx\)C.\(y=\cosx\)D.\(y=x+1\)5.等差數(shù)列\(zhòng)(\{a_n\}\)的通項公式\(a_n=a_1+(n-1)d\)中,涉及的量有()A.\(a_n\)B.\(a_1\)C.\(n\)D.\(d\)6.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)(\(a\gtb\gt0\))的性質(zhì)有()A.焦點在\(x\)軸B.長軸長為\(2a\)C.短軸長為\(2b\)D.離心率\(e\lt1\)7.以下計算正確的有()A.\(\log_a(MN)=\log_aM+\log_aN\)B.\(\log_a\frac{M}{N}=\log_aM-\log_aN\)C.\(\log_aM^n=n\log_aM\)D.\(\log_aa=1\)8.對于二次函數(shù)\(y=ax^2+bx+c\)(\(a\neq0\)),下列說法正確的有()A.當\(a\gt0\)時,函數(shù)圖象開口向上B.對稱軸為\(x=-\frac{2a}\)C.頂點坐標為\((-\frac{2a},\frac{4ac-b^2}{4a})\)D.與\(y\)軸交點為\((0,c)\)9.已知\(\overrightarrow{a}=(x_1,y_1)\),\(\overrightarrow=(x_2,y_2)\),則\(\overrightarrow{a}\cdot\overrightarrow=(\)\)A.\(x_1x_2+y_1y_2\)B.\(|\overrightarrow{a}||\overrightarrow|\cos\theta\)(\(\theta\)為\(\overrightarrow{a}\)與\(\overrightarrow\)夾角）C.\(x_1y_2+x_2y_1\)D.\(|\overrightarrow{a}||\overrightarrow|\sin\theta\)10.以下屬于概率的性質(zhì)的有()A.\(0\leqP(A)\leq1\)B.\(P(\Omega)=1\)(\(\Omega\)為樣本空間)C.\(P(\varnothing)=0\)D.若\(A\)、\(B\)互斥,則\(P(A\cupB)=P(A)+P(B)\)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)\(y=x^2\)在\((-\infty,0)\)上單調(diào)遞增。()3.直線\(x=1\)的斜率不存在。()4.\(\sin^2\alpha+\cos^2\alpha=1\)。()5.等比數(shù)列的公比\(q\)可以為\(0\)。()6.不等式\(x^2+1\gt0\)的解集是\(R\)。()7.圓\(x^2+y^2=r^2\)的圓心在原點,半徑為\(r\)。(）8.若向量\(\overrightarrow{a}\)與\(\overrightarrow\)平行,則\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為\(0^{\circ}\)。（)9.函數(shù)\(y=\tanx\)的定義域是\(x\neqk\pi+\frac{\pi}{2},k\inZ\)。()10.從\(n\)個不同元素中取出\(m\)(\(m\leqn\)）個元素的排列數(shù)\(A_{n}^m=\frac{n!}{(n-m)!}\)。(）四、簡答題(每題5分,共20分)1.求函數(shù)\(y=\frac{1}{\sqrt{x-3}}\)的定義域。答案:要使函數(shù)有意義,則\(x-3\gt0\),即\(x\gt3\),所以定義域為\((3,+\infty)\)。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=3\),\(d=2\),求\(a_8\)。答案：根據(jù)等差數(shù)列通項公式\(a_n=a_1+(n-1)d\),\(n=8\)時,\(a_8=3+(8-1)\times2=3+14=17\)。3.求直線\(2x+y-5=0\)的斜率和在\(y\)軸上的截距。答案:將直線方程化為斜截式\(y=-2x+5\),斜率\(k=-2\),在\(y\)軸上的截距為\(5\)。4.計算\(\log_28+\log_3\frac{1}{9}\)的值。答案:\(\log_28=\log_22^3=3\),\(\log_3\frac{1}{9}=\log_33^{-2}=-2\),所以\(\log_28+\log_3\frac{1}{9}=3-2=1\)。五、討論題(每題5分,共20分)1.討論一次函數(shù)\(y=kx+b\)(\(k\neq0\))在\(k\gt0\)和\(k\lt0\)時的單調(diào)性。答案：當\(k\gt0\)時,\(y\)隨\(x\)的增大而增大,函數(shù)在\(R\)上單調(diào)遞增；當\(k\lt0\)時,\(y\)隨\(x\)的增大而減小,函數(shù)在\(R\)上單調(diào)遞減。2.討論橢圓和雙曲線在定義和性質(zhì)上的異同。答案:相同點:都是圓錐曲線。不同點:定義上,橢圓是到兩定點距離之和為定值,雙曲線是到兩定點距離之差的絕對值為定值;性質(zhì)上,橢圓離心率\(0\lte\lt1\),雙曲線\(e\gt1\),且雙曲線有漸近線,橢圓沒有。3.舉例說明如何運用等差數(shù)列的求和公式解決實際問題。答案:比如計算一堆堆放成梯形的鋼管總數(shù),最上層有\(zhòng)(a_1\)根,最下層有\(zhòng)(a_n\)根,層數(shù)為\(n\)。鋼管總數(shù)就是等差數(shù)列\(zhòng)(a_1,a_2,\cdots,a_n\)的和,可用\(S_n=\frac{n(a_1+a_n)}{2}\)計算。4.討論函數(shù)的奇偶性在實際解題中的作用。答案:利用函數(shù)奇偶性可簡化計算。若函數(shù)是奇函數(shù),\(f(-x)=-f(x)\),關(guān)于原點對稱;偶函數(shù)\(f(-x)=f(x)\),關(guān)于\(y\)軸對稱。比如求函數(shù)值,已知一半?yún)^(qū)間的情況可推出另一半,也有助于分析函