2025數(shù)學(xué)省聯(lián)考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)的定義域?yàn)閈([1,3]\)，則函數(shù)\(y=f(x+1)\)的定義域?yàn)椋ǎ〢.\([0,2]\)B.\([1,3]\)C.\([2,4]\)D.\([-1,1]\)答案：A2.已知向量\(\vec{a}=(1,2)\)，\(\vec=(2,k)\)，若\(\vec{a}\parallel\vec\)，則\(k=\)（）A.1B.2C.3D.4答案：D3.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，公比\(q=2\)，則\(a_{3}=\)（）A.1B.2C.4D.8答案：C4.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(\frac{2\pi}{3}\)答案：A5.直線\(l:y=kx+1\)與圓\(x^{2}+y^{2}=1\)相切，則\(k=\)（）A.0B.\(\pm1\)C.\(\pm\sqrt{3}\)D.\(\pm2\)答案：B6.從5個(gè)不同元素中取出3個(gè)元素的組合數(shù)\(C_{5}^3=\)（）A.10B.20C.30D.60答案：A7.已知\(x\in(0,\pi)\)，\(\sinx=\frac{1}{2}\)，則\(x=\)（）A.\(\frac{\pi}{6}\)B.\(\frac{5\pi}{6}\)C.\(\frac{\pi}{6}\)或\(\frac{5\pi}{6}\)D.\(\frac{\pi}{3}\)答案：C8.雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0,b>0)\)的漸近線方程為\(y=\pm\frac{3}{4}x\)，則離心率\(e=\)（）A.\(\frac{5}{4}\)B.\(\frac{5}{3}\)C.\(\frac{\sqrt{7}}{4}\)D.\(\frac{\sqrt{7}}{3}\)答案：A9.若\(f(x)=x^{2}+2(a-1)x+2\)在\((-\infty,4]\)上是減函數(shù)，則\(a\)的取值范圍是（）A.\(a\leqslant-3\)B.\(a\geqslant-3\)C.\(a\leqslant5\)D.\(a\geqslant5\)答案：A10.若\(\log_{a}2<\log_{a}3\)，則\(a\)的取值范圍是（）A.\((0,1)\)B.\((1,+\infty)\)C.\((0,+\infty)\)D.\((-\infty,0)\)答案：B二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)是奇函數(shù)的有（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=e^{x}-e^{-x}\)D.\(y=\frac{1}{x}\)答案：ABCD2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)，其公差\(d\neq0\)，\(a_{1}=1\)，且\(a_{2},a_{4},a_{8}\)成等比數(shù)列，則（）A.\(d=1\)B.\(a_{n}=n\)C.\(a_{10}=10\)D.數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=\frac{n(n+1)}{2}\)答案：ABCD3.若\(\alpha\in(0,\frac{\pi}{2})\)，\(\beta\in(0,\frac{\pi}{2})\)，\(\sin\alpha=\frac{3}{5}\)，\(\cos\beta=\frac{4}{5}\)，則（）A.\(\cos\alpha=\frac{4}{5}\)B.\(\sin\beta=\frac{3}{5}\)C.\(\sin(\alpha+\beta)=1\)D.\(\cos(\alpha-\beta)=\frac{7}{25}\)答案：AC4.直線\(l_{1}:y=k_{1}x+b_{1}\)與直線\(l_{2}:y=k_{2}x+b_{2}\)平行的條件是（）A.\(k_{1}=k_{2}\)且\(b_{1}\neqb_{2}\)B.\(k_{1}=k_{2}\)C.\(k_{1}\timesk_{2}=-1\)D.\(\frac{b_{1}}{b_{2}}=\frac{k_{1}}{k_{2}}\)答案：A5.已知函數(shù)\(y=f(x)\)在\(x=x_{0}\)處可導(dǎo)，則（）A.函數(shù)\(y=f(x)\)在\(x=x_{0}\)處連續(xù)B.\(\lim\limits_{h\rightarrow0}\frac{f(x_{0}+h)-f(x_{0})}{h}=f'(x_{0})\)C.曲線\(y=f(x)\)在點(diǎn)\((x_{0},f(x_{0}))\)處有切線D.\(\lim\limits_{x\rightarrowx_{0}}f(x)=f(x_{0})\)答案：ABCD6.下列不等式成立的是（）A.\(x^{2}+1>2x\)B.\(a+b\geqslant2\sqrt{ab}(a,b>0)\)C.\(\frac{1}{x}+\frac{1}{y}\geqslant\frac{4}{x+y}(x,y>0)\)D.\(\sinx+\cosx\geqslant-\sqrt{2}\)答案：BCD7.關(guān)于二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)，下列說(shuō)法正確的是（）A.當(dāng)\(a>0\)時(shí)，函數(shù)圖象開口向上B.對(duì)稱軸為\(x=-\frac{2a}\)C.\(\Delta=b^{2}-4ac\)決定函數(shù)圖象與\(x\)軸的交點(diǎn)個(gè)數(shù)D.當(dāng)\(c=0\)時(shí)，函數(shù)圖象過原點(diǎn)答案：ABCD8.一個(gè)正方體的棱長(zhǎng)為\(a\)，則（）A.正方體的表面積為\(6a^{2}\)B.正方體的體積為\(a^{3}\)C.正方體的體對(duì)角線長(zhǎng)為\(\sqrt{3}a\)D.正方體的面對(duì)角線長(zhǎng)為\(\sqrt{2}a\)答案：ABCD9.在\(\triangleABC\)中，\(A,B,C\)所對(duì)的邊分別為\(a,b,c\)，已知\(a=3\)，\(b=4\)，\(c=5\)，則（）A.\(\sinA=\frac{3}{5}\)B.\(\cosA=\frac{4}{5}\)C.\(\triangleABC\)是直角三角形D.面積\(S=\frac{1}{2}ab=6\)答案：ABCD10.若\(z_{1}=1+i\)，\(z_{2}=2-i\)，則（）A.\(z_{1}+z_{2}=3\)B.\(z_{1}-z_{2}=-1+2i\)C.\(z_{1}\timesz_{2}=3+i\)D.\(\frac{z_{1}}{z_{2}}=\frac{3}{5}+\frac{1}{5}i\)答案：ABC三、判斷題（每題2分，共10題）1.空集是任何集合的子集。（）答案：正確2.若\(a>b\)，則\(ac^{2}>bc^{2}\)。（）答案：錯(cuò)誤3.函數(shù)\(y=\sqrt{x}\)在其定義域上是增函數(shù)。（）答案：正確4.平行四邊形的對(duì)角線互相平分。（）答案：正確5.若\(\lim\limits_{n\rightarrow\infty}a_{n}=A\)，\(\lim\limits_{n\rightarrow\infty}b_{n}=B\)，則\(\lim\limits_{n\rightarrow\infty}(a_{n}+b_{n})=A+B\)。（）答案：正確6.對(duì)于二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)，當(dāng)\(a<0\)時(shí)，函數(shù)圖象開口向下。（）答案：正確7.同弧所對(duì)的圓周角是圓心角的一半。（）答案：正確8.若\(z=a+bi(a,b\inR)\)，則\(\vertz\vert=\sqrt{a^{2}+b^{2}}\)。（）答案：正確9.兩個(gè)向量的夾角范圍是\([0,\pi]\)。（）答案：正確10.若函數(shù)\(y=f(x)\)在\(x=x_{0}\)處有極值，則\(f'(x_{0})=0\)。（）答案：錯(cuò)誤四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^{2}-2x-3\)在\([0,3]\)上的最值。答案：\(y=x^{2}-2x-3=(x-1)^{2}-4\)，對(duì)稱軸為\(x=1\)。當(dāng)\(x=1\)時(shí)，\(y_{min}=-4\)；當(dāng)\(x=3\)時(shí)，\(y=(3-1)^{2}-4=0\)，\(y_{max}=0\)。2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=n^{2}+n\)，求\(a_{n}\)。答案：當(dāng)\(n=1\)時(shí)，\(a_{1}=S_{1}=1^{2}+1=2\)；當(dāng)\(n\geqslant2\)時(shí)，\(a_{n}=S_{n}-S_{n-1}=n^{2}+n-[(n-1)^{2}+(n-1)]=2n\)，\(n=1\)時(shí)也滿足，所以\(a_{n}=2n\)。3.求過點(diǎn)\(A(1,2)\)且與直線\(y=2x+1\)平行的直線方程。答案：已知直線\(y=2x+1\)的斜率\(k=2\)，所求直線平行于它，斜率也為\(2\)。由點(diǎn)斜式\(y-y_{1}=k(x-x_{1})\)，過點(diǎn)\(A(1,2)\)，則直線方程為\(y-2=2(x-1)\)，即\(y=2x\)。4.若\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\frac{\pi}{2})\)，求\(\sin2\alpha\)。答案：因?yàn)閈(\sin^{2}\alpha+\cos^{2}\alpha=1\)，\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\frac{\pi}{2})\)，所以\(\sin\alpha=\sqrt{1-\cos^{2}\alpha}=\frac{2\sqrt{2}}{3}\)，\(\sin2\alpha=2\sin\alpha\cos\alpha=2\times\frac{2\sqrt{2}}{3}\times\frac{1}{3}=\frac{4\sqrt{2}}{9}\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x}\)的單調(diào)性。答案：函數(shù)\(y=\frac{1}{x}\)在\((-\infty,0)\)和\((0,+\infty)\)上分別單調(diào)遞減。2.討論等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}\)的最值情況。答案：當(dāng)\(d>0\)時(shí)，\(S_{n}\)有最小值；當(dāng)\(d<0\)時(shí)，\(S_{n}\)有最大值，可通過\(a_{n}\)的正負(fù)性變化確定最值。3.討論直線\(y=kx+b\)與圓\(x^{2}+y^{