9類試題及答案高一

單項選擇題(每題2分,共10題)1.集合\(\{1,2,3\}\)的子集個數是(）A.5B.6C.7D.82.函數\(y=\sqrt{x-1}\)的定義域是()A.\(x\geq0\)B.\(x\geq1\)C.\(x\gt1\)D.\(x\gt0\)3.若\(\sin\alpha=\frac{1}{2}\),且\(\alpha\)在第一象限,則\(\cos\alpha\)的值為()A.\(-\frac{\sqrt{3}}{2}\)B.\(\frac{\sqrt{3}}{2}\)C.\(\pm\frac{\sqrt{3}}{2}\)D.\(\frac{1}{2}\)4.直線\(y=2x+1\)的斜率是()A.1B.2C.\(\frac{1}{2}\)D.-25.等差數列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(a_3=5\),則公差\(d\)為()A.1B.2C.3D.46.已知向量\(\overrightarrow{a}=(1,2)\),\(\overrightarrow=(2,m)\),若\(\overrightarrow{a}\parallel\overrightarrow\),則\(m\)的值為()A.4B.-4C.1D.-17.函數\(y=\log_2x\)的圖象過點()A.\((0,1)\)B.\((1,0)\)C.\((1,1)\)D.\((2,0)\)8.圓\(x^2+y^2=4\)的半徑是()A.1B.2C.3D.49.若\(a\gtb\),則下列不等式成立的是()A.\(a^2\gtb^2\)B.\(\frac{1}{a}\lt\frac{1}\)C.\(a-1\gtb-1\)D.\(ac\gtbc\)10.正方體的棱長為\(2\),則其表面積是()A.12B.24C.36D.48多項選擇題(每題2分,共10題)1.以下哪些是偶函數(）A.\(y=x^2\)B.\(y=\cosx\)C.\(y=x^3\)D.\(y=\sinx\)2.下列屬于基本初等函數的有()A.冪函數B.指數函數C.對數函數D.三角函數3.等差數列的性質有()A.\(a_n=a_1+(n-1)d\)B.若\(m+n=p+q\),則\(a_m+a_n=a_p+a_q\)C.\(S_n=\frac{n(a_1+a_n)}{2}\)D.\(a_{n+1}-a_n=d\)4.關于直線\(Ax+By+C=0\)(\(A\)、\(B\)不同時為\(0\)),正確的是()A.斜率\(k=-\frac{A}{B}\)(\(B\neq0\))B.在\(x\)軸截距為\(-\frac{C}{A}\)(\(A\neq0\))C.在\(y\)軸截距為\(-\frac{C}{B}\)(\(B\neq0\))D.與直線\(A_1x+B_1y+C_1=0\)平行,則\(\frac{A}{A_1}=\frac{B}{B_1}\neq\frac{C}{C_1}\)5.以下哪些點在直線\(y=x+1\)上(）A.\((0,1)\)B.\((1,2)\)C.\((-1,0)\)D.\((2,3)\)6.已知集合\(A=\{1,2,3\}\),\(B=\{2,3,4\}\),則()A.\(A\capB=\{2,3\}\)B.\(A\cupB=\{1,2,3,4\}\)C.\(A\)的真子集個數為\(7\)D.\(B\)的子集個數為\(8\)7.對于函數\(y=\sin(x+\frac{\pi}{3})\),以下說法正確的是()A.周期是\(2\pi\)B.圖象關于點\((-\frac{\pi}{3},0)\)對稱C.圖象向左平移\(\frac{\pi}{3}\)個單位得到\(y=\sinx\)D.在\([0,\frac{\pi}{6}]\)上單調遞增8.下列不等式中,正確的是()A.\(x^2+1\geq2x\)B.\(a+b\geq2\sqrt{ab}\)(\(a\gt0\),\(b\gt0\))C.\(|x|+\frac{1}{|x|}\geq2\)(\(x\neq0\)）D.\(x^2+y^2\geq\frac{(x+y)^2}{2}\)9.一個三棱柱的底面是正三角形,側棱垂直于底面,下列說法正確的是()A.它有\(zhòng)(6\)個頂點B.它有\(zhòng)(9\)條棱C.它有\(zhòng)(5\)個面D.它的側面積等于底面周長乘側棱長10.已知向量\(\overrightarrow{a}=(1,-1)\),\(\overrightarrow=(-1,2)\),則()A.\(\overrightarrow{a}+\overrightarrow=(0,1)\)B.\(\overrightarrow{a}-\overrightarrow=(2,-3)\)C.\(\overrightarrow{a}\cdot\overrightarrow=-3\)D.\(|\overrightarrow{a}|=\sqrt{2}\)判斷題(每題2分,共10題)1.空集是任何集合的真子集。()2.函數\(y=x^2\)在\((-\infty,0)\)上單調遞增。()3.若\(\tan\alpha=1\),則\(\alpha=\frac{\pi}{4}\)。()4.兩條直線斜率相等,則這兩條直線平行。(）5.等比數列\(zhòng)(\{a_n\}\)中,\(a_1=1\),公比\(q=2\),則\(a_3=4\)。（)6.向量\(\overrightarrow{a}=(1,2)\)與\(\overrightarrow=(2,4)\)共線。()7.對數函數\(y=\log_ax\)(\(a\gt0\)且\(a\neq1\))的定義域是\((0,+\infty)\)。()8.圓\((x-1)^2+(y+2)^2=9\)的圓心坐標是\((1,-2)\)。()9.若\(a\gtb\gt0\),\(c\gtd\gt0\),則\(ac\gtbd\)。（)10.長方體的對角線長\(l=\sqrt{a^2+b^2+c^2}\)(\(a\)、\(b\)、\(c\)為長方體棱長）。(）簡答題(每題5分,共4題)1.求函數\(y=\frac{1}{\sqrt{x-2}}\)的定義域。答案:要使函數有意義,則\(x-2\gt0\),解得\(x\gt2\),所以定義域為\((2,+\infty)\)。2.已知等差數列\(zhòng)(\{a_n\}\)中,\(a_1=3\),\(d=2\),求\(a_5\)。答案：根據等差數列通項公式\(a_n=a_1+(n-1)d\),當\(n=5\),\(a_1=3\),\(d=2\)時,\(a_5=3+(5-1)×2=3+8=11\)。3.求\(\sin150^{\circ}\)的值。答案:根據誘導公式\(\sin(180^{\circ}-\alpha)=\sin\alpha\),則\(\sin150^{\circ}=\sin(180^{\circ}-30^{\circ})=\sin30^{\circ}=\frac{1}{2}\)。4.已知直線\(l\)過點\((1,2)\)且斜率為\(3\),求直線\(l\)的方程。答案：由直線的點斜式方程\(y-y_1=k(x-x_1)\)（其中\(zhòng)((x_1,y_1)\)為直線上一點,\(k\)為斜率）,可得直線\(l\)方程為\(y-2=3(x-1)\),整理得\(3x-y-1=0\)。討論題(每題5分,共4題)1.討論函數\(y=\cosx\)在\([0,2\pi]\)上的單調性。答案:在\([0,\pi]\)上,\(y=\cosx\)單調遞減;在\([\pi,2\pi]\)上,\(y=\cosx\)單調遞增。因為\(\cosx\)的導數\(-\sinx\)在\([0,\pi]\)上小于等于\(0\),在\([\pi,2\pi]\)上大于等于\(0\)。2.已知集合\(A=\{x|x^2-3x+2=0\}\),\(B=\{x|x^2-ax+a-1=0\}\),討論\(A\)與\(B\)的關系。答案：解方程\(x^2-3x+2=0\)得\(x=1\)或\(x=2\),所以\(A=\{1,2\}\)。解方程\(x^2-ax+a-1=0\)得\((x-1)[x-(a-1)]=0\),\(x=1\)或\(x=a-1\)。若\(a-1=1\)即\(a=2\),\(B=\{1\}\),\(B\subsetA\)；若\(a-1=2\)即\(a=3\),\(B=\{1,2\}\),\(A=B\)；若\(a-1\neq1\)且\(a-1\neq2\),\(B\subsetA\)。3.討論直線\(y=kx+1\)與圓\(x^2+y^2=1\)的位置關系。答案:圓\(x^2+y^2=1\)圓心\((0,0)\),半徑\(r=1\)。根據點到直線距離公式,圓心到直線\(y=kx+1\)(即\(kx-y+1=0\))的距離\(d=\frac{1}{\sqrt{k^2+1}}\)。當\(d=\frac{1}{\sqrt{k^2+1}}\lt1\)即\(k\neq0\)時,直線與圓相交;當\(d=\frac{1}{\sqrt{k^2+1}}=1