高中数学-高考试题(天津大学 1949年代数式)

计算题（1949年天津大学）

析(3x²+x-2)/(x-2)²(1-2x)为部分分式.

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讨论

高中数学

求圆锥曲线 x² +y² = 49 及 x² +y² - 20y +90 =0之公切线的长.

求圆锥曲线 2x²-8xy - 4y² - 4y +1=0 之焦点及准线.

某人在高处望见正东海面上一船首，其俯角为 30°，当船向正南行 a 里后，求得船首俯角为 15°，问此人之视点高出海面若干?

设一四面体有一三面角与另一四面体的一三面角对称，求证：其体积之比等于此两三面角三棱分别的乘积之比.

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Find the distance from the point (-1, -4) to the straight line which is drawn through (-2,6) and the perpendicular to the line joining (3,6) and (-5,-1).

Two towers, A and B, on the shore of a lake can be observed from only one point C on the opposite shore. The lines joining the bases of two towers subtend anangle of 63°42' at C. The heights of the towers are 132 feet and 89 feet, and the angle of elevation of the tops as seen from C are 8°13' and 7°21' respectively.Find the distance AB.

The sides of a triangle are 149,163 and 222. To find(1) The area of the triangle.(2) The radius of the inscribed circle.(3) The angles of the triangle.

线性代数

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设x+y+z =0,证明: x³ +y³ +z³ = 3xyz

试求适合5x+3y>121,7/4 x+y=42 二式之x,y之值之界限.

劈生数(因式分解)：3a²+a-6a²b-2b

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劈生数(因式分解)：a4+a²b²+b4

解明：(6x+5)/(8x-15)-(1+8x)/15=(1-x)/3+(3-x)/5

代数式

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若 x³ + 3px² + 3qx +r 及 x² + 2px +q 有一个一次公因子，试问 p,g,r 之间应有何种关系？又若有两个一次公因子，则其关系又若何?

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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【】