深圳一模試題及答案理數(shù)

一、單項(xiàng)選擇題（每題2分，共10題）1.已知集合\(A=\{1,2,3\}\)，\(B=\{2,3,4\}\)，則\(A\capB=(\)\)A.\(\{1,2\}\)B.\(\{2,3\}\)C.\(\{3,4\}\)D.\(\{1,4\}\)2.復(fù)數(shù)\(z=\frac{1+i}{1-i}\)（\(i\)為虛數(shù)單位），則\(\vertz\vert=(\)\)A.\(0\)B.\(1\)C.\(\sqrt{2}\)D.\(2\)3.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(3,-2)\)，且\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow\)，則\(m=(\)\)A.\(-8\)B.\(-6\)C.\(6\)D.\(8\)4.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和為\(S_{n}\)，若\(a_{3}=5\)，\(S_{9}=81\)，則\(a_{7}=(\)\)A.\(18\)B.\(13\)C.\(9\)D.\(7\)5.執(zhí)行如圖所示的程序框圖，若輸入的\(x=2\)，則輸出的\(y\)值為（）A.\(2\)B.\(3\)C.\(4\)D.\(5\)6.已知函數(shù)\(f(x)=\sin(2x+\frac{\pi}{3})\)，則\(f(x)\)的最小正周期為（）A.\(4\pi\)B.\(2\pi\)C.\(\pi\)D.\(\frac{\pi}{2}\)7.設(shè)\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geqslant1\\x-y\leqslant1\\y\leqslant1\end{cases}\)，則\(z=2x+y\)的最大值為（）A.\(1\)B.\(3\)C.\(5\)D.\(9\)8.已知雙曲線\(C:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a\gt0,b\gt0)\)的離心率為\(\sqrt{2}\)，則\(C\)的漸近線方程為（）A.\(y=\pmx\)B.\(y=\pm\sqrt{2}x\)C.\(y=\pm2x\)D.\(y=\pm\frac{\sqrt{2}}{2}x\)9.已知\(a=\log_{2}0.2\)，\(b=2^{0.2}\)，\(c=0.2^{0.3}\)，則（）A.\(a\ltb\ltc\)B.\(a\ltc\ltb\)C.\(c\lta\ltb\)D.\(b\ltc\lta\)10.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\gt0\)時(shí)，\(f(x)=x^{2}-2x\)，則\(f(-1)=(\)\)A.\(-1\)B.\(1\)C.\(3\)D.\(-3\)二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^{\frac{1}{2}}\)B.\(y=2^{x}\)C.\(y=\log_{0.5}x\)D.\(y=\frac{1}{x}\)2.已知直線\(l\)過點(diǎn)\((0,1)\)，且傾斜角為\(\frac{\pi}{4}\)，則直線\(l\)的方程可以是（）A.\(x-y+1=0\)B.\(x+y-1=0\)C.\(y=x+1\)D.\(y=-x+1\)3.已知\(a\)，\(b\)為兩條不同的直線，\(\alpha\)，\(\beta\)為兩個(gè)不同的平面，則下列說法正確的是（）A.若\(a\parallel\alpha\)，\(b\parallel\alpha\)，則\(a\parallelb\)B.若\(a\parallel\alpha\)，\(a\parallel\beta\)，則\(\alpha\parallel\beta\)C.若\(a\perp\alpha\)，\(b\perp\alpha\)，則\(a\parallelb\)D.若\(\alpha\perp\beta\)，\(a\subset\alpha\)，\(a\perp\beta\)，則\(a\parallel\alpha\)4.以下關(guān)于正態(tài)分布\(N(\mu,\sigma^{2})\)的說法正確的是（）A.正態(tài)分布密度曲線關(guān)于直線\(x=\mu\)對(duì)稱B.\(\sigma\)越小，正態(tài)分布密度曲線越“瘦高”C.\(P(X\gt\mu)=0.5\)D.隨機(jī)變量\(X\)服從正態(tài)分布\(N(3,4)\)，則\(P(X\lt3)=0.5\)5.已知函數(shù)\(f(x)=\cos(2x+\varphi)(\vert\varphi\vert\lt\frac{\pi}{2})\)，若\(f(\frac{\pi}{6})=0\)，則（）A.\(\varphi=-\frac{\pi}{6}\)B.\(f(x)\)在\((0,\frac{\pi}{6})\)上單調(diào)遞減C.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{5\pi}{12},0)\)對(duì)稱D.\(f(x)\)在\((0,\frac{\pi}{2})\)上的最大值為\(1\)6.已知數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=n^{2}\)，則（）A.\(a_{1}=1\)B.\(a_{n}=2n-1\)C.\(a_{n}=2n\)D.數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列7.已知\(a\gt0\)，\(b\gt0\)，且\(a+b=1\)，則（）A.\(ab\leqslant\frac{1}{4}\)B.\(\frac{1}{a}+\frac{1}\geqslant4\)C.\(a^{2}+b^{2}\geqslant\frac{1}{2}\)D.\(\sqrt{a}+\sqrt\leqslant\sqrt{2}\)8.已知橢圓\(C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a\gtb\gt0)\)的左、右焦點(diǎn)分別為\(F_{1}\)，\(F_{2}\)，\(P\)為橢圓上一點(diǎn)，且\(\angleF_{1}PF_{2}=60^{\circ}\)，則（）A.當(dāng)\(a=2b\)時(shí)，橢圓的離心率\(e=\frac{\sqrt{3}}{3}\)B.當(dāng)\(a=2b\)時(shí)，\(\vertPF_{1}\vert\cdot\vertPF_{2}\vert=\frac{4}{3}b^{2}\)C.若\(\triangleF_{1}PF_{2}\)的面積為\(\sqrt{3}c^{2}\)，則橢圓的離心率\(e=\frac{1}{2}\)D.若\(\overrightarrow{PF_{1}}\cdot\overrightarrow{PF_{2}}=0\)，則\(\triangleF_{1}PF_{2}\)的面積為\(b^{2}\)9.已知函數(shù)\(f(x)\)的導(dǎo)函數(shù)為\(f^\prime(x)\)，且滿足\(f(x)=x^{3}+f^\prime(\frac{2}{3})x^{2}-x\)，則（）A.\(f^\prime(\frac{2}{3})=-1\)B.\(f(x)\)在\(x=\frac{2}{3}\)處取得極大值C.\(f(x)\)在\(x=\frac{2}{3}\)處取得極小值D.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{2}{3},f(\frac{2}{3}))\)對(duì)稱10.已知函數(shù)\(y=f(x)\)的圖象關(guān)于點(diǎn)\((a,b)\)對(duì)稱，則有\(zhòng)(f(a+x)+f(a-x)=2b\)。若函數(shù)\(f(x)=\frac{1}{x-1}+1\)，則（）A.\(f(x)\)的圖象關(guān)于點(diǎn)\((1,1)\)對(duì)稱B.\(f(x)+f(2-x)=2\)C.\(f(\frac{1}{2023})+f(\frac{2}{2023})+\cdots+f(\frac{4045}{2023})=4045\)D.\(f(x)\)在\((1,+\infty)\)上單調(diào)遞減三、判斷題（每題2分，共10題）1.若\(a\gtb\)，則\(ac^{2}\gtbc^{2}\)。（）2.直線\(x=1\)的傾斜角為\(0^{\circ}\)。（）3.若向量\(\overrightarrow{a}\)，\(\overrightarrow\)滿足\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}\perp\overrightarrow\)。（）4.函數(shù)\(y=\sinx\)在\([0,2\pi]\)上的圖象關(guān)于直線\(x=\pi\)對(duì)稱。（）5.若\(a\)，\(b\)，\(c\)成等比數(shù)列，則\(b^{2}=ac\)。（）6.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有\(zhòng)(f^\prime(x)\gt0\)，則\(y=f(x)\)在\((a,b)\)上單調(diào)遞增。（）7.拋物線\(y^{2}=4x\)的焦點(diǎn)坐標(biāo)為\((1,0)\)。（）8.若\(\alpha\)，\(\beta\)是兩個(gè)不同的平面，直線\(m\subset\alpha\)，且\(m\parallel\beta\)，則\(\alpha\parallel\beta\)。（）9.已知函數(shù)\(f(x)\)是奇函數(shù)，當(dāng)\(x\gt0\)時(shí)，\(f(x)=x^{2}-1\)，則\(f(-2)=-3\)。（）10.若\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geqslant0\\x-y\leqslant0\\y\leqslant1\end{cases}\)，則\(z=2x+y\)的最大值為\(3\)。（）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=\sin^{2}x+\sqrt{3}\sinx\cosx\)的最小正周期及單調(diào)遞增區(qū)間。答案：化簡(jiǎn)\(y=\frac{1-\cos2x}{2}+\frac{\sqrt{3}}{2}\sin2x=\sin(2x-\frac{\pi}{6})+\frac{1}{2}\)。最小正周期\(T=\pi\)。由\(2k\pi-\frac{\pi}{2}\leqslant2x-\frac{\pi}{6}\leqslant2k\pi+\frac{\pi}{2}\)，\(k\inZ\)，得單調(diào)遞增區(qū)間為\([k\pi-\frac{\pi}{6},k\pi+\frac{\pi}{3}]\)，\(k\inZ\)。2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和為\(S_{n}\)，\(a_{3}=5\)，\(S_{5}=25\)，求\(a_{n}\)的通項(xiàng)公式。答案：設(shè)等差數(shù)列\(zhòng)(\{a_{n}\}\)公差為\(d\)，由\(S_{5}=\frac{5(a_{1}+a_{5})}{2}=5a_{3}=25\)，得\(a_{3}=5\)。又\(a_{3}=a_{1}+2d=5\)，\(a_{1}+2d=5\)且\(a_{3}=5\)，解得\(a_{1}=1\)，\(d=2\)，所以\(a_{n}=a_{1}+(n-1)d=2n-1\)。3.已知橢圓\(C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a\gtb\gt0)\)的離心率為\(\frac{\sqrt{3}}{2}\)，且過點(diǎn)\((\sqrt{3},\frac{1}{2})\)，求橢圓\(C\)的方程。答案：離心率\(e=\frac{c}{a}=\frac{\sqrt{3}}{2}\)，即\(c=\frac{\sqrt{3}}{2}a\)，又\(b^{2}=a^{2}-c^{2}=\frac{1}{4}a^{2}\)。橢圓過點(diǎn)\((\sqrt{3},\frac{1}{2})\)，代入橢圓方程\(\frac{3}{a^{2}}+\frac{1}{4b^{2}}=1\)，將\(b^{2}=\frac{1}{4}a^{2}\)代入得\(a^{2}=4\)，\(b^{2}=1\)，橢圓方程為\(\frac{x^{2}}{4}