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上海市交大附中自招试卷数学试题(含答案)

上海市交大附中自招试卷数学试题(含答案)
上海市交大附中自招试卷数学试题(含答案)

交大附中自主招生试卷

2019.03

第一部分

1. 已知13x x +=-,求3311000x x

++. 2. 11(1)

x x x t x x x x +++=++有增根,求所有可能的t 之和. 3. AB ∥CD ,15AB =,10CD =,3AD =,4CB =,求ABCD S .

4. 346y x x =-+,若a x b ≤≤时,其中x 的最小值为a ,最大值为b ,求a b +.

5. 22(2)y x m =-+,若抛物线与x 轴交点与顶点组成正三角形,求m 的值.

6. DE 为?BC

的切线,正方形ABCD 边长为200,?BC 以BC 为直径的半圆,求DE 的长. 7. 在直角坐标系中,正ABC ?,(2,0)B ,9(,0)2C 过点O 作直线DMN ,OM MN =, 求M 的横坐标.

8. 四圆相切⊙B 与⊙C 半径相同,⊙A 过⊙D 圆心,⊙A 的半径为9,求⊙B 的半径.

9. 横纵坐标均为整数的点为整点,(

12m a <<),y mx a =+(1100x ≤≤),不经过整 点,求a 可取到的最大值.

10. G 为重心,DE 过重心,1ABC S ?=,求ADE S ?的最值,并证明结论.

第二部分(科学素养)

1. 已知直角三角形三边长为整数,有一条边长为85,求另两边长(写出10组).

2. 阅读材料,根据凸函数的定义和性质解三道小题,其中第(3)小题为不等式证明

(1)14

b =;(2)13b =.(注:选(1)做对得10分,选(2)做对得20分) 3. 请用最优美的语言赞美仰晖班(80字左右)(17分)

4. 附加题(25分)

(2 points ) solve the following system of equations for 2122.2221

w x y z w x y z w w x y z w x y z +++=??+++=??+++=??+++=? (4 points )

Compute 98∞

(6 points )Solve the equation 1=.Express your answer as a reduced fraction with the numerator written in their prime factorization.

The gauss function []x denotes the greatest less than or equal to x

A )(3 points )Compute 2018!2015!2017!2016!+??

??+??

B )(4points )Let real numbers 12,,,n x x x ??? be the solutions of the equation 23[]40x x --=,

find the value of 22212n x x x ++???+ C )(6 points )Find all ordered triples (,,)a b c of positive real that satisfy :[]3a bc =,[]4a b c =,and []5ab c =

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