2025年今年高二數(shù)學(xué)考試試題及答案

一、單項(xiàng)選擇題1.已知數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列，\(a_{3}=5\)，\(a_{7}=13\)，則\(a_{10}\)的值為（）A.18B.19C.20D.21答案：B2.若雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)（\(a\gt0\)，\(b\gt0\)）的漸近線方程為\(y=\pm\frac{3}{4}x\)，則其離心率為（）A.\(\frac{5}{4}\)B.\(\frac{5}{3}\)C.\(\frac{4}{3}\)D.\(\frac{\sqrt{7}}{4}\)答案：A3.在\(\triangleABC\)中，\(a=3\)，\(b=5\)，\(\sinA=\frac{1}{3}\)，則\(\sinB=（）A.\(\frac{1}{5}\)B.\(\frac{5}{9}\)C.\(\frac{\sqrt{5}}{3}\)D.1答案：B4.已知向量\(\vec{a}=(1,-2)\)，\(\vec=(m,4)\)，且\(\vec{a}\parallel\vec\)，則\(m\)的值為（）A.-2B.2C.-8D.8答案：A5.函數(shù)\(f(x)=x^{3}-3x^{2}+1\)的單調(diào)遞減區(qū)間是（）A.\((2,+\infty)\)B.\((-\infty,2)\)C.\((-\infty,0)\)D.\((0,2)\)答案：D6.已知直線\(l\)過點(diǎn)\((1,0)\)且垂直于\(x\)軸，若\(l\)被拋物線\(y^{2}=4ax\)截得的線段長為\(4\)，則拋物線的焦點(diǎn)坐標(biāo)為（）A.\((1,0)\)B.\((2,0)\)C.\((0,1)\)D.\((0,2)\)答案：A7.從\(5\)名男生和\(3\)名女生中選\(3\)人參加某項(xiàng)活動(dòng)，至少有\(zhòng)(1\)名女生參加的選法種數(shù)為（）A.24B.30C.46D.60答案：C8.已知\(\cos\alpha=-\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})\)的值為（）A.\(\frac{1}{7}\)B.7C.-\frac{1}{7}\)D.-7答案：A9.設(shè)\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geq2\\x-y\leq2\\y\leq2\end{cases}\)，則\(z=2x-y\)的最大值為（）A.4B.6C.8D.10答案：B10.已知球\(O\)的半徑為\(R\)，\(A\)，\(B\)，\(C\)三點(diǎn)在球\(O\)的球面上，\(\triangleABC\)是邊長為\(\sqrt{3}\)的正三角形，且球心\(O\)到平面\(ABC\)的距離為\(\frac{R}{2}\)，則球\(O\)的表面積為（）A.\(4\pi\)B.\(8\pi\)C.\(12\pi\)D.\(16\pi\)答案：D二、多項(xiàng)選擇題1.下列說法正確的是（）A.若直線\(l_{1}\)：\(A_{1}x+B_{1}y+C_{1}=0\)與直線\(l_{2}\)：\(A_{2}x+B_{2}y+C_{2}=0\)垂直，則\(A_{1}A_{2}+B_{1}B_{2}=0\)B.若\(a\)，\(b\)，\(c\)成等比數(shù)列，則\(b^{2}=ac\)C.命題“\(\forallx\inR\)，\(x^{2}-x+1\gt0\)”的否定是“\(\existsx_{0}\inR\)，\(x_{0}^{2}-x_{0}+1\leq0\)”D.若\(p\)，\(q\)都是真命題，則\(p\landq\)為真命題答案：ABC2.已知函數(shù)\(f(x)=\sin(2x+\varphi)\)（\(0\lt\varphi\lt\frac{\pi}{2}\)），則下列說法正確的是（）A.函數(shù)\(f(x)\)的最小正周期為\(\pi\)B.函數(shù)\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{6},0)\)對稱C.函數(shù)\(f(x)\)在\([0,\frac{\pi}{3}]\)上單調(diào)遞增D.函數(shù)\(f(x)\)的圖象向右平移\(\frac{\pi}{6}\)個(gè)單位長度后得到的函數(shù)\(g(x)=\sin(2x-\frac{\pi}{6})\)答案：AD3.已知\(a\)，\(b\)為正實(shí)數(shù)，且\(a+b=1\)，則（）A.\(ab\leq\frac{1}{4}\)B.\(\frac{1}{a}+\frac{1}\geq4\)C.\(a^{2}+b^{2}\geq\frac{1}{2}\)D.\(\sqrt{a}+\sqrt\leq\sqrt{2}\)答案：ABCD4.設(shè)等比數(shù)列\(zhòng)(\{a_{n}\}\)的公比為\(q\)，其前\(n\)項(xiàng)和為\(S_{n}\)，前\(n\)項(xiàng)積為\(T_{n}\)，并且滿足條件\(a_{1}\gt1\)，\(a_{7}a_{8}\gt1\)，\(\frac{a_{7}-1}{a_{8}-1}\lt0\)，則下列結(jié)論正確的是（）A.\(0\ltq\lt1\)B.\(a_{7}a_{9}\gt1\)C.\(S_{n}\)的最大值為\(S_{9}\)D.\(T_{n}\)的最大值為\(T_{7}\)答案：AD5.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\(F_{1}\)，\(F_{2}\)，點(diǎn)\(P\)在橢圓\(C\)上，且\(\angleF_{1}PF_{2}=60^{\circ}\)，若\(\triangleF_{1}PF_{2}\)的面積為\(\sqrt{3}\)，則下列結(jié)論正確的是（）A.\(b=1\)B.若\(P\)到\(x\)軸的距離為\(\frac{\sqrt{3}}{2}\)，則\(C\)的離心率為\(\frac{1}{2}\)C.若\(\triangleF_{1}PF_{2}\)為銳角三角形，則\(C\)的離心率的取值范圍是\((\frac{1}{2},1)\)D.若\(C\)的長軸長為\(4\)，則\(\vertPF_{1}\vert\cdot\vertPF_{2}\vert=4\)答案：ACD6.已知函數(shù)\(f(x)=x^{3}-3x\)，則下列說法正確的是（）A.函數(shù)\(f(x)\)有兩個(gè)極值點(diǎn)B.函數(shù)\(f(x)\)在區(qū)間\([-2,2]\)上的最大值為\(2\)C.函數(shù)\(f(x)\)的圖象關(guān)于原點(diǎn)對稱D.若關(guān)于\(x\)的方程\(f(x)-a=0\)有三個(gè)不同的實(shí)根，則\(-2\lta\lt2\)答案：ACD7.已知向量\(\vec{a}=(1,m)\)，\(\vec=(2,n)\)，則（）A.若\(\vec{a}\parallel\vec\)，則\(m=\frac{n}{2}\)B.若\(\vec{a}\perp\vec\)，則\(2+mn=0\)C.若\(\vert\vec{a}\vert=\vert\vec\vert\)，則\(m=n\)D.若\(\vec{a}\)與\(\vec\)的夾角為\(\frac{\pi}{4}\)，則\(m+n=2\)答案：AB8.已知圓\(C\)：\((x-1)^{2}+(y-2)^{2}=25\)，直線\(l\)：\((2m+1)x+(m+1)y-7m-4=0\)，則（）A.直線\(l\)恒過定點(diǎn)\((3,1)\)B.直線\(l\)與圓\(C\)可能相離C.直線\(l\)被圓\(C\)截得的弦長最短時(shí)，直線\(l\)的方程為\(2x-y-5=0\)D.直線\(l\)被圓\(C\)截得的弦長最長時(shí)，直線\(l\)的方程為\(x+y-4=0\)答案：AC9.已知\(x\gt0\)，\(y\gt0\)，且\(x+2y=1\)，則（）A.\(xy\leq\frac{1}{8}\)B.\(\frac{1}{x}+\frac{1}{y}\geq3+2\sqrt{2}\)C.\(2^{x}+4^{y}\geq2\sqrt{2}\)D.\(x^{2}+4y^{2}\geq\frac{1}{2}\)答案：ABCD10.已知函數(shù)\(f(x)=e^{x}-ax\)（\(a\inR\)），則下列說法正確的是（）A.當(dāng)\(a=1\)時(shí)，\(f(x)\)在\((0,+\infty)\)上單調(diào)遞增B.當(dāng)\(a=0\)時(shí)，\(f(x)\)的最小值為\(1\)C.當(dāng)\(a\gt0\)時(shí)，\(f(x)\)可能有兩個(gè)零點(diǎn)D.若\(f(x)\geq1\)對任意\(x\inR\)恒成立，則\(a=1\)答案：ACD三、判斷題1.若直線\(l\)的斜率\(k=\tan\alpha\)，則\(\alpha\)就是直線\(l\)的傾斜角。（×）2.若\(\vec{a}\cdot\vec=\vec{a}\cdot\vec{c}\)，則\(\vec=\vec{c}\)。（×）3.數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=n^{2}+1\)，則\(a_{n}=2n-1\)。（×）4.雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)（\(a\gt0\)，\(b\gt0\)）的漸近線方程為\(y=\pm\frac{a}x\)。（√）5.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有\(zhòng)(f^\prime(x)\gt0\)，則\(y=f(x)\)在區(qū)間\((a,b)\)上單調(diào)遞增。（√）6.已知\(a\)，\(b\)，\(c\)是\(\triangleABC\)的三邊，若\(a^{2}+b^{2}\gtc^{2}\)，則\(\triangleABC\)是銳角三角形。（×）7.若\(p\)：\(\forallx\inR\)，\(x^{2}+x+1\gt0\)，則\(\negp\)：\(\existsx_{0}\inR\)，\(x_{0}^{2}+x_{0}+1\leq0\)。（√）8.從\(5\)個(gè)不同元素中取出\(3\)個(gè)元素的組合數(shù)\(C_{5}^{3}\)與從\(5\)個(gè)不同元素中取出\(2\)個(gè)元素的組合數(shù)\(C_{5}^{2}\)相等。（√）9.已知球\(O\)的半徑為\(R\)，球的體積公式為\(V=\frac{4}{3}\piR^{3}\)。（√）10.若\(a\)，\(b\)為實(shí)數(shù)，且\(a+b=1\)，則\((a+\frac{1}{a})(b+\frac{1})\geq\frac{25}{4}\)。（√）四、簡答題1.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和為\(S_{n}\)，\(a_{3}=7\)，\(S_{4}=24\)。求\(\{a_{n}\}\)的通項(xiàng)公式。答案：設(shè)等差數(shù)列\(zhòng)(\{a_{n}\}\)的公差為\(d\)。由\(a_{3}=7\)可得\(a_{1}+2d=7\)；由\(S_{4}=24\)可得\(4a_{1}+\frac{4\times3}{2}d=24\)，即\(4a_{1}+6d=24\)。將\(a_{1}=7-2d\)代入\(4a_{1}+6d=24\)，得\(4(7-2d)+6d=24\)，\(28-8d+6d=24\)，\(-2d=-4\)，解得\(d=2\)，則\(a_{1}=7-2\times2=3\)。所以\(a_{n}=a_{1}+(n-1)d=3+2(n-1)=2n+1\)。2.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{2