浙江省寧波市鎮(zhèn)海中學(xué)2022-2023學(xué)年高二上學(xué)期期中數(shù)學(xué)試題（原卷版）

鎮(zhèn)海中學(xué)2022學(xué)年第一學(xué)期期中考試高二年級數(shù)學(xué)試卷一、選擇題：本題共8小題，每小題5分，共40分.在每小題給出的四個(gè)選項(xiàng)中，只有一項(xiàng)是符合題目要求的.1.拋物線SKIPIF1<0焦點(diǎn)到準(zhǔn)線的距離為A.4 B.2 C.1 D.SKIPIF1<02.已知函數(shù)SKIPIF1<0可導(dǎo)，且滿足SKIPIF1<0，則函數(shù)SKIPIF1<0在SKIPIF1<0處的導(dǎo)數(shù)為()A.SKIPIF1<0 B.SKIPIF1<0 C.1 D.23.與雙曲線SKIPIF1<0有相同漸近線，且與橢圓SKIPIF1<0有共同焦點(diǎn)的雙曲線方程是()ASKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<04.等差數(shù)列SKIPIF1<0中，已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則n為()A.58 B.59 C.60 D.615.函數(shù)SKIPIF1<0的圖象大致是()A. B.C. D.6.設(shè)SKIPIF1<0，SKIPIF1<0是雙曲線E：SKIPIF1<0的兩個(gè)焦點(diǎn)，雙曲線E與以O(shè)為圓心SKIPIF1<0為半徑的圓在第一象限的交點(diǎn)為SKIPIF1<0，且SKIPIF1<0，則該雙曲線的離心率為()A.SKIPIF1<0 B.SKIPIF1<0 C.13 D.SKIPIF1<07.已知數(shù)列SKIPIF1<0滿足：對于任意的m，SKIPIF1<0，都有SKIPIF1<0恒成立，且SKIPIF1<0，則SKIPIF1<0的值為()A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<08.已知橢圓SKIPIF1<0，兩條直線SKIPIF1<0：SKIPIF1<0；SKIPIF1<0：SKIPIF1<0，過橢圓上一點(diǎn)P作SKIPIF1<0，SKIPIF1<0的平行線，分別交SKIPIF1<0，SKIPIF1<0于M，N，若SKIPIF1<0為定值，則SKIPIF1<0()A.9 B.4 C.3 D.2二、選擇題：本題共4小題，每小題5分，共20分.在每小題給出的選項(xiàng)中，有多項(xiàng)符合題目要求.全部選對的得5分，部分選對的得2分，有選錯(cuò)的得0分.9.已知等差數(shù)列SKIPIF1<0的公差SKIPIF1<0，當(dāng)且僅當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的前SKIPIF1<0項(xiàng)和SKIPIF1<0最大，則()A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<010.如圖，已知直線SKIPIF1<0與曲線SKIPIF1<0相切于A、B兩點(diǎn)，設(shè)A，B兩點(diǎn)的橫坐標(biāo)分別為a，b，函數(shù)SKIPIF1<0，下列說法正確的有()A.SKIPIF1<0有極大值，也有極小值B.SKIPIF1<0是SKIPIF1<0的極小值點(diǎn)C.SKIPIF1<0是SKIPIF1<0的極大值點(diǎn)D.SKIPIF1<0是SKIPIF1<0的極大值點(diǎn)11.已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0過焦點(diǎn)SKIPIF1<0分別交拋物線SKIPIF1<0于點(diǎn)SKIPIF1<0、SKIPIF1<0、SKIPIF1<0、SKIPIF1<0，其中SKIPIF1<0位于SKIPIF1<0軸同側(cè)，且SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0，記SKIPIF1<0，SKIPIF1<0的斜率分別為SKIPIF1<0，SKIPIF1<0，則下列正確的有()A.SKIPIF1<0 B.SKIPIF1<0過定點(diǎn) C.SKIPIF1<0 D.SKIPIF1<0的最小值為SKIPIF1<012.已知數(shù)列SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0；則()A.SKIPIF1<0或5 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0三、填空題：本大題共4小題，每小題5分，共20分.13.已知函數(shù)SKIPIF1<0，則SKIPIF1<0________.14.已知雙曲線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，過左焦點(diǎn)SKIPIF1<0交雙曲線左支于A、B兩點(diǎn)，若SKIPIF1<0則SKIPIF1<0等于________.15.把自然數(shù)按如下規(guī)律排列：0，1，1，2，2，2，3，3，3，3，……，則第2022個(gè)數(shù)是________.16.對于首項(xiàng)和公比均為q的等比數(shù)列SKIPIF1<0滿足：對于任意正整數(shù)n都有SKIPIF1<0成立，求正實(shí)數(shù)q的取值范圍為________.四、解答題：本題共6小題，共70分.解答應(yīng)寫出文字說明、證明過程或演算步驟.17.已知數(shù)列SKIPIF1<0的前n項(xiàng)和SKIPIF1<0，SKIPIF1<0是首項(xiàng)為1的等比數(shù)列，且SKIPIF1<0.(1)求數(shù)列SKIPIF1<0，SKIPIF1<0通項(xiàng)公式；(2)設(shè)數(shù)列SKIPIF1<0滿足SKIPIF1<0，求SKIPIF1<0的前12項(xiàng)的和SKIPIF1<0.18.已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0在SKIPIF1<0上恒成立,求實(shí)數(shù)SKIPIF1<0的取值范圍;(2)若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增,求實(shí)數(shù)a的取值范圍.19.已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，過點(diǎn)SKIPIF1<0的直線垂直x軸于Q，SKIPIF1<0為等腰直角三角形.(1)求拋物線C的方程；(2)若直線l交拋物線C于A，B兩點(diǎn)，且F恰為SKIPIF1<0的重心，求直線l的方程.20已知函數(shù)SKIPIF1<0，其中SKIPIF1<0.(1)討論函數(shù)SKIPIF1<0的單調(diào)性；(2)若對于任意的SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，都有SKIPIF1<0成立，求a的取值范圍.21.已知曲線C:SKIPIF1<0上一點(diǎn)SKIPIF1<0,過SKIPIF1<0作曲線C的切線交x軸于SKIPIF1<0點(diǎn),SKIPIF1<0垂直于x軸且交曲線于SKIPIF1<0﹔再過SKIPIF1<0作曲線C的切線交x軸于SKIPIF1<0….,依次過SKIPIF1<0作曲線C的切線x軸于SKIPIF1<0﹐SKIPIF1<0垂直于x軸,得到一系列的點(diǎn)SKIPIF1<0,其中SKIPIF1<0.(1)求SKIPIF1<0的坐標(biāo)和數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)設(shè)SKIPIF1<0的面積為SKIPIF1<0,SKIPIF1<0為數(shù)列SKIPIF1<0的前n項(xiàng)和,是否存在實(shí)數(shù)M,使得SKIPIF1<0對于一切SKIPIF1<0恒成立,若存在求出M的最小值,不存在說明理由.22.已知直線SKIPIF1<0是雙曲線的漸近線，且雙曲線過點(diǎn)SKIPIF1