大?？荚嚁?shù)學(xué)題庫及答案

一、單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=\frac{1}{x-1}\)的定義域是(）A.\(x\neq0\)B.\(x\neq1\)C.\(x\gt1\)D.\(x\lt1\)2.下列函數(shù)中,既是奇函數(shù)又是增函數(shù)的是()A.\(y=x+1\)B.\(y=-x^3\)C.\(y=\frac{1}{x}\)D.\(y=x^3\)3.已知\(\sin\alpha=\frac{3}{5}\),且\(\alpha\)是第二象限角,則\(\cos\alpha=\)()A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)4.等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=2\),\(a_3=6\),則公差\(d=\)()A.1B.2C.3D.45.拋物線\(y=2x^2\)的焦點(diǎn)坐標(biāo)是()A.\((0,\frac{1}{2})\)B.\((0,\frac{1}{8})\)C.\((\frac{1}{8},0)\)D.\((\frac{1}{2},0)\)6.已知向量\(\vec{a}=(1,2)\),\(\vec=(2,3)\),則\(\vec{a}-\vec=\)()A.\((-1,-1)\)B.\((1,1)\)C.\((-3,-1)\)D.\((3,1)\)7.函數(shù)\(y=\log_2(x+1)\)的反函數(shù)是()A.\(y=2^x-1\)B.\(y=2^x+1\)C.\(y=\log_2(x-1)\)D.\(y=\log_2(x+1)\)8.不等式\(x^2-2x-3\lt0\)的解集是()A.\((-1,3)\)B.\((-3,1)\)C.\((-\infty,-1)\cup(3,+\infty)\)D.\((-\infty,-3)\cup(1,+\infty)\)9.若\(x\gt0\),\(y\gt0\),且\(x+y=1\),則\(\frac{1}{x}+\frac{1}{y}\)的最小值是()A.2B.3C.4D.510.曲線\(y=x^3\)在點(diǎn)\((1,1)\)處的切線方程是()A.\(y=3x-2\)B.\(y=3x+2\)C.\(y=-3x-2\)D.\(y=-3x+2\)答案:1.B2.D3.B4.B5.B6.A7.A8.A9.C10.A二、多項(xiàng)選擇題(每題2分,共10題)1.下列函數(shù)中,是冪函數(shù)的有（）A.\(y=x^2\)B.\(y=2x\)C.\(y=x^{\frac{1}{2}}\)D.\(y=\frac{1}{x^2}\)2.下列三角函數(shù)值為正數(shù)的是()A.\(\sin120^{\circ}\)B.\(\cos120^{\circ}\)C.\(\sin(-30^{\circ})\)D.\(\cos(-30^{\circ})\)3.已知數(shù)列\(zhòng)(\{a_n\}\)是等比數(shù)列,公比\(q=2\),\(a_1=1\),則()A.\(a_2=2\)B.\(a_3=4\)C.\(a_4=8\)D.\(a_5=16\)4.下列函數(shù)中,在\((0,+\infty)\)上單調(diào)遞增的是()A.\(y=x^2\)B.\(y=\lnx\)C.\(y=2^x\)D.\(y=\frac{1}{x}\)5.已知向量\(\vec{a}=(1,0)\),\(\vec=(0,1)\),則()A.\(\vec{a}\cdot\vec=0\)B.\(|\vec{a}|=1\)C.\(|\vec|=1\)D.\(\vec{a}\)與\(\vec\)垂直6.下列說法正確的是()A.一個(gè)命題的逆命題為真,則它的否命題為假B.一個(gè)命題的逆命題為真,則它的逆否命題為真C.一個(gè)命題的逆否命題為真,則它的否命題為真D.一個(gè)命題的否命題為真,則它的逆命題為真7.已知函數(shù)\(y=f(x)\)的定義域?yàn)閈([-1,1]\),則函數(shù)\(y=f(2x-1)\)的定義域?yàn)椋ǎ〢.\([0,1]\)B.\([-1,0]\)C.\([0,2]\)D.\([-1,2]\)8.下列曲線中,關(guān)于原點(diǎn)對(duì)稱的有()A.\(y=x^3\)B.\(y=\sinx\)C.\(y=\lnx\)D.\(y=\frac{1}{x}\)9.已知\(a\gtb\gt0\),則()A.\(a^2\gtb^2\)B.\(\frac{1}{a}\lt\frac{1}\)C.\(a^3\gtb^3\)D.\(\sqrt{a}\gt\sqrt\)10.若直線\(l_1:y=k_1x+b_1\),\(l_2:y=k_2x+b_2\)平行,則()A.\(k_1=k_2\)B.\(b_1=b_2\)C.\(k_1k_2=-1\)D.\(k_1\neqk_2\)答案:1.ACD2.AD3.ABCD4.ABC5.ABCD6.D7.A8.ABD9.ABCD10.A三、判斷題(每題2分,共10題)1.空集是任何集合的子集。()2.函數(shù)\(y=\sinx\)的最小正周期是\(2\pi\)。()3.若\(a\gtb\),則\(ac^2\gtbc^2\)。(）4.向量\(\vec{a}=(x_1,y_1)\),\(\vec=(x_2,y_2)\),則\(\vec{a}\cdot\vec=x_1x_2+y_1y_2\)。（)5.函數(shù)\(y=\sqrt{x-1}\)的定義域是\(x\geq1\)。()6.等比數(shù)列的公比不能為0。()7.若\(f(x)\)是奇函數(shù),則\(f(0)=0\)。（)8.直線\(x=1\)的斜率不存在。()9.橢圓的離心率\(e\)的范圍是\((0,1)\)。()10.若\(A\capB=A\),則\(A\subseteqB\)。（）答案:1.√2.√3.×4.√5.√6.√7.×8.√9.√10.√四、簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=2x^2-4x+3\)的最小值。配方得\(y=2(x-1)^2+1\),所以當(dāng)\(x=1\)時(shí),\(y\)取最小值\(1\)。2.已知\(\tan\alpha=2\),求\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值。分子分母同除以\(\cos\alpha\),得\(\frac{\tan\alpha+1}{\tan\alpha-1}=\frac{2+1}{2-1}=3\)。3.求過點(diǎn)\((1,2)\)且與直線\(y=3x+1\)平行的直線方程。設(shè)所求直線方程為\(y=3x+b\),代入點(diǎn)\((1,2)\)得\(b=-1\),方程為\(y=3x-1\)。4.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=n^2+2n\),求\(a_n\)。\(n\geq2\)時(shí),\(a_n=S_n-S_{n-1}=2n+1\)；\(n=1\)時(shí),\(a_1=S_1=3\)也滿足,所以\(a_n=2n+1\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=x^3\)的單調(diào)性。\(y^\prime=3x^2\geq0\),僅當(dāng)\(x=0\)時(shí)等號(hào)成立,所以在\(R\)上單調(diào)遞增。2.如何判斷兩個(gè)向量是否平行?若存在實(shí)數(shù)\(\lambda\),使\(\vec{a}=\lambda\vec\),則兩向量平行；或?qū)?yīng)坐標(biāo)成比例。3.討論等比數(shù)列的通項(xiàng)公