江蘇數(shù)學(xué)高考試題答案

一、單項(xiàng)選擇題(每題2分,共10題)1.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.\(4\pi\)答案:A2.已知集合\(A=\{1,2,3\}\),\(B=\{2,4\}\),則\(A\cupB=\)()A.\(\{2\}\)B.\(\{1,2,3,4\}\)C.\(\{1,3\}\)D.\(\{2,4\}\)答案:B3.復(fù)數(shù)\(z=1+2i\)(\(i\)為虛數(shù)單位),則\(\vertz\vert=\)()A.\(\sqrt{5}\)B.\(5\)C.\(\sqrt{3}\)D.\(3\)答案:A4.直線\(l\)過點(diǎn)\((1,0)\)且與直線\(x-2y+3=0\)平行,則直線\(l\)的方程為()A.\(x-2y-1=0\)B.\(2x-y-2=0\)C.\(x-2y+1=0\)D.\(2x+y-2=0\)答案:A5.等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(a_3=5\),則\(a_5=\)()A.\(9\)B.\(10\)C.\(11\)D.\(12\)答案:A6.函數(shù)\(f(x)=\lnx-\frac{1}{x}\)的零點(diǎn)所在區(qū)間是()A.\((0,1)\)B.\((1,2)\)C.\((2,3)\)D.\((3,4)\)答案:B7.已知向量\(\overrightarrow{a}=(1,2)\),\(\overrightarrow=(-2,m)\),且\(\overrightarrow{a}\parallel\overrightarrow\),則\(m=\)()A.\(4\)B.\(-4\)C.\(1\)D.\(-1\)答案:B8.若\(\tan\alpha=3\),則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}=\)()A.\(2\)B.\(-2\)C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案:A9.圓\(x^2+y^2-4x+6y=0\)的圓心坐標(biāo)是()A.\((2,-3)\)B.\((-2,3)\)C.\((-2,-3)\)D.\((2,3)\)答案:A10.從\(1,2,3,4\)中任取兩個不同的數(shù),則取出的兩個數(shù)之和為\(5\)的概率是()A.\(\frac{1}{6}\)B.\(\frac{1}{4}\)C.\(\frac{1}{3}\)D.\(\frac{1}{2}\)答案:C二、多項(xiàng)選擇題(每題2分,共10題)1.下列函數(shù)中,是偶函數(shù)的有()A.\(y=x^2\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=|x|\)答案:ABD2.以下哪些是等比數(shù)列的性質(zhì)(）A.\(a_n^2=a_{n-1}a_{n+1}(n\gt1)\)B.\(S_n=\frac{a_1(1-q^n)}{1-q}(q\neq1)\)C.\(a_m+a_n=a_p+a_q(m+n=p+q)\)D.\(a_m\cdota_n=a_p\cdota_q(m+n=p+q)\)答案:ABD3.已知直線\(l_1:A_1x+B_1y+C_1=0\),\(l_2:A_2x+B_2y+C_2=0\),則\(l_1\perpl_2\)的條件可以是（）A.\(A_1A_2+B_1B_2=0\)B.\(A_1B_2-A_2B_1=0\)C.\(k_1k_2=-1\)(\(k_1,k_2\)分別為\(l_1,l_2\)斜率)D.\(\frac{A_1}{A_2}=-\frac{B_1}{B_2}\)答案:AC4.下列關(guān)于橢圓的說法正確的有(）A.橢圓的標(biāo)準(zhǔn)方程有兩種形式B.橢圓上的點(diǎn)到兩焦點(diǎn)距離之和為定值C.離心率\(e\)滿足\(0\lte\lt1\)D.長軸長一定大于短軸長答案:ABC5.以下哪些是基本初等函數(shù)()A.冪函數(shù)B.指數(shù)函數(shù)C.對數(shù)函數(shù)D.三角函數(shù)答案:ABCD6.已知\(\alpha\)為銳角,\(\sin\alpha=\frac{3}{5}\),則以下正確的是()A.\(\cos\alpha=\frac{4}{5}\)B.\(\tan\alpha=\frac{3}{4}\)C.\(\sin2\alpha=\frac{24}{25}\)D.\(\cos2\alpha=\frac{7}{25}\)答案:ABCD7.對于函數(shù)\(y=2\sin(2x+\frac{\pi}{6})\),下列說法正確的是（）A.周期為\(\pi\)B.圖象關(guān)于點(diǎn)\((-\frac{\pi}{12},0)\)對稱C.圖象關(guān)于直線\(x=\frac{\pi}{6}\)對稱D.在\([0,\frac{\pi}{6}]\)上單調(diào)遞增答案:ACD8.下列運(yùn)算正確的是()A.\((a^m)^n=a^{mn}\)B.\(a^m\cdota^n=a^{m+n}\)C.\(\log_a(MN)=\log_aM+\log_aN\)D.\(\log_a\frac{M}{N}=\log_aM-\log_aN\)答案:ABCD9.已知\(a,b\gt0\),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.\(\sqrt{a}+\sqrt\leqslant\sqrt{2}\)答案:ABCD10.關(guān)于空間直線與平面,下列說法正確的是()A.若直線\(a\parallel\)平面\(\alpha\),直線\(b\subset\alpha\),則\(a\parallelb\)B.若直線\(a\perp\)平面\(\alpha\),直線\(b\parallel\alpha\),則\(a\perpb\)C.若平面\(\alpha\parallel\)平面\(\beta\),直線\(a\subset\alpha\),則\(a\parallel\beta\)D.若平面\(\alpha\perp\)平面\(\beta\),平面\(\alpha\cap\beta=l\),直線\(a\subset\alpha\)且\(a\perpl\),則\(a\perp\beta\)答案：BCD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)\(y=x^3\)是奇函數(shù)。(√)3.若\(a\gtb\),則\(a^2\gtb^2\)。（×)4.直線\(x=1\)的斜率不存在。(√)5.拋物線\(y^2=4x\)的焦點(diǎn)坐標(biāo)是\((1,0)\)。(√)6.若\(\sin\alpha=\sin\beta\),則\(\alpha=\beta\)。（×)7.數(shù)列\(zhòng)(1,1,1,1,\cdots\)是等差數(shù)列也是等比數(shù)列。(√)8.若向量\(\overrightarrow{a}\cdot\overrightarrow=0\),則\(\overrightarrow{a}\perp\overrightarrow\)。（×,需考慮零向量情況)9.函數(shù)\(y=\log_2x\)在定義域上是增函數(shù)。(√)10.正方體的體對角線與面對角線一定垂直。(×）四、簡答題(每題5分,共4題)1.求函數(shù)\(y=3x^2-2x+1\)的對稱軸和頂點(diǎn)坐標(biāo)。答案:對于二次函數(shù)\(y=ax^2+bx+c\),對稱軸\(x=-\frac{2a}\)。此函數(shù)\(a=3\),\(b=-2\),對稱軸\(x=\frac{1}{3}\)。把\(x=\frac{1}{3}\)代入得\(y=\frac{2}{3}\),頂點(diǎn)坐標(biāo)\((\frac{1}{3},\frac{2}{3})\)。2.已知\(\sin\alpha=\frac{1}{2}\),\(\alpha\in(0,\pi)\),求\(\cos\alpha\)的值。答案：因?yàn)閈(\sin^2\alpha+\cos^2\alpha=1\),\(\sin\alpha=\frac{1}{2}\),所以\(\cos^2\alpha=1-(\frac{1}{2})^2=\frac{3}{4}\)。又\(\alpha\in(0,\pi)\),當(dāng)\(\sin\alpha=\frac{1}{2}\)時,\(\alpha=\frac{\pi}{6}\)或\(\frac{5\pi}{6}\),\(\cos\alpha=\pm\frac{\sqrt{3}}{2}\)。3.求過點(diǎn)\((1,2)\)且與直線\(2x-y+1=0\)垂直的直線方程。答案:直線\(2x-y+1=0\)斜率\(k=2\),與其垂直直線斜率\(k'=-\frac{1}{2}\)。由點(diǎn)斜式\(y-y_0=k'(x-x_0)\),\((x_0=1,y_0=2)\),得\(y-2=-\frac{1}{2}(x-1)\),整理得\(x+2y-5=0\)。4.已知等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(d=2\),求\(S_{10}\)。答案：等差數(shù)列求和公式\(S_n=na_1+\frac{n(n-1)}{2}d\)。\(n=10\),\(a_1=1\),\(d=2\),則\(S_{10}=10\times1+\frac{10\times9}{2}\times2=10+90=100\)。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)\(y=\frac{1}{x}\)定義域?yàn)閈(x\neq0\)。在\((-\infty,0)\)和\((0,+\infty)\)上分別單調(diào)遞減。當(dāng)\(x_1\ltx_2\lt0\)或\(0\ltx_1\ltx_2\)時,\(f(x_1)-f(x_2)=\frac{1}{x_1}-\frac{1}{x_2}=\frac{x_2-x_1}{x_1x_2}\gt0\),即\(f(x_1)\gtf(x_2)\)。2.如何判斷直線與圓的位置關(guān)系?答案：可通過比較圓心到直線的距離\(d\)與圓半徑\(r\)的大小判斷。若\(d\gtr\),直線與圓相離；若\(d=r\),直線與圓相切；若\(d\ltr\),直線與圓相交。也可聯(lián)立直線與圓的方程,根據(jù)判別式判斷,\(\Delta\gt0\)相交,\(\Delta=0\)相切,\(\Delta\lt0\)相離。3.討論等比數(shù)列的公比\(q\)對數(shù)列性質(zhì)的影響。答案:當(dāng)\(q\gt1\)且\(a_1\