高中数学-高考试题(上海交通大学 1931年代数式)

计算题（1931年上海交通大学）

Factor the following:

4(a-b)²-12(a-b)(e+9e²)

答案解析

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讨论

高中数学

Factor the following:4a-16b+4a+1.

Find the equation of the ruled surface whose generators are the system of the lines x-2y =4kx,k(x-2y)=4 and discuss the surface.

Find the equation of the sphere which passes through (1,5,-3),(-3,0,0) which center on the 3x+y+z=0,x+2y +1 = 0.

Find the equation of the plane passing through the line (x-x1)/a1 =(y-y1)/b1 =(z-z1)/c1 which is parallel to the (x-x2)/a2 =(y-y2)/b2 =(z-z2)/c2 .

Find the equation of the projection of the linex=z+2,y=2z-4 upon the plane x+y- z = 0.

Find the locus of the point of intersection of lines drawn through the foci of an ellipse parallel to conjugate diameters.

Chords of the circle p=29cosθ which pass through the pole are extended a distance 2b. Find the locus of their extremities.

The point of contact of a tangent to an hyperbola is midway between the points in which the tangent meets the asymptotes.

Given the parabola y²=4x and the line x=2+ecosα,y=-4+ecosβ，find the condition which cosα and cosβ must satisfy if the line meets the parabola in but one point.

A boy standing cft. behind and opposite the middle of a foothall goal sees that the angle of elevation of the nearer is A and the angle of elevation of the farther one is B. Show that the length of the field is c(tanAcotB-1).

线性代数

析a2b+ab2-a2c+ac2-2abc-b2c+bc2之因式.

化(5x²-4x+16)/((x²-x+1)²(x-3))为部分分式.

设 A,B 为 x 的两个有理整式，请用辗转相除法说明并证明何种情况为互质，何种情况下有公因式.有公因式时，说明求最高公因式之方法并证明之.

A polynomial P with integer coefficients is square-free if it is not expressible in the form P=Q² R, where Q and R are polynomials with integer coefficients and Q is not constant. For a positive integer n, let Pn be the set of polynomials of the form1+a1 x+a2 x²+⋯+an xnwith a1,a2,⋯,an∈{0,1}. Prove that there exists an integer N so that, for all integers n>N, more than 99% of the polynomials in Pn are square-free.【译】我们称整系数多项式P是无平方因子的，如果其不能表示为P=Q² R的形式，这里Q,R为整系数多项式且Q不为常数.对于正整数n，记Pn为如下形式的多项式组成的集合：1+a1 x+a2 x²+⋯+an xn这里a1,a2,⋯,an∈{0,1}.证明：存在整数N，使得对任意的整数n≥N,Pn中超过99%的多项式都是无平方因子的.

设20阶实矩阵A满足eA=I20，且A在复数域上的所有特征值模长均不超过20，则这样的互不相似的A有______个.

有理数加群(Q,+)，记所有分母不超过10的有理数构成的子集为G，其对应的陪集GZ记为G ̅，则Q/Z包含G ̅的最小子群的阶为______.

分解因式：x2-4xy+4y2-4z2.

甲乙两容器内都盛有酒精，甲有v1公斤，乙有v2公斤.甲中纯酒精与水（重量）之比为m1:n1，乙中纯酒精与水之比为m2:n2，问将二者混合后所得液体中纯酒精与水之比是多少？

将多项式x5y-9xy5分别在下列范围内分解因式：1. 有理数范围； 2. 实数范围；3. 复数范围．

如果n是正整数，那么1/8[1-(-1)n](n2-1)的值【】

代数式

已知正整数n,恰有36个不同的质数整除n，对k=1,2,3,4,5，记[(k-1)n/5,kn/5]中互质的整数个数为Cn，已知C1,C2,C3,C4,C5不完全相同.求证：(Ci - Cj)2 ≥236.

安徽省代数式

安徽省代数式

分解因式：a2-2ab+b2-6a+6b+5.

分解因式x2y - 2y3.

试分 ab(x²- y²)+ xy (a²-b²)为因数.

北京师范大学代数式

试将下列繁分数简单之：

有人定酒二坛，两坛所盛斤数不等，原定一盛甲酒每斤8角，一盛乙酒每斤5角.今误将甲坛盛乙酒，乙坛盛甲酒，酒商要求加洋一元五角.问两坛斤数各几何?

若a:b=c:d，试证(a2+b2):a3/(a+b)=(c2+d2):c3/(c+d).