1951 AMC 12 真题（P版）

1

\(M\) 比 \(N\) 大的百分率为：

A\(\dfrac{100(M-N)}{M}\) B\(\dfrac{100(M-N)}{N}\) C\(\dfrac{M-N}{N}\) D\(\dfrac{M-N}{M}\) E\(\dfrac{100(M+N)}{N}\)

English

The percent that \(M\) is greater than \(N\) is:

(A) \(\dfrac{100(M-N)}{M}\)(B) \(\dfrac{100(M-N)}{N}\)(C) \(\dfrac{M-N}{N}\)(D) \(\dfrac{M-N}{M}\)(E) \(\dfrac{100(M+N)}{N}\)

2

一块长方形田地的宽是长的一半，四周共用 \(x\) 码栅栏围住。用 \(x\) 表示其面积为：

A\(\dfrac{x^2}{2}\) B\(2x^2\) C\(\dfrac{2x^2}{9}\) D\(\dfrac{x^2}{18}\) E\(\dfrac{x^2}{72}\)

English

A rectangular field is half as wide as it is long and is completely enclosed by \(x\) yards of fencing. The area in terms of \(x\) is:

(A) \(\dfrac{x^2}{2}\)(B) \(2x^2\)(C) \(\dfrac{2x^2}{9}\)(D) \(\dfrac{x^2}{18}\)(E) \(\dfrac{x^2}{72}\)

3

若正方形的对角线长为 \(a+b\)，则正方形的面积为：

A\((a+b)^2\) B\(\dfrac12(a+b)^2\) C\(a^2+b^2\) D\(\dfrac12(a^2+b^2)\) E以上答案均不正确

English

If the length of a diagonal of a square is \(a+b,\) then the area of the square is:

(A) \((a+b)^2\)(B) \(\dfrac12(a+b)^2\)(C) \(a^2+b^2\)(D) \(\dfrac12(a^2+b^2)\)(E) None of these

4

一座平顶长方体谷仓宽 \(10\) 码、长 \(13\) 码、高 \(5\) 码。谷仓内外墙面和天花板都要粉刷，但屋顶和地面不刷。需要粉刷的总面积为多少平方码？

A\(360\) B\(460\) C\(490\) D\(590\) E\(720\)

English

A barn with a flat roof is rectangular in shape, \(10\) yd. wide, \(13\) yd. long, and \(5\) yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of square yards to be painted is:

(A) \(360\)(B) \(460\)(C) \(490\)(D) \(590\)(E) \(720\)

5

\(A\) 先生拥有一栋价值 \(\$10{,}000\) 的房子。他以高于房屋价值 \(10\%\) 的价格卖给 \(B\) 先生。随后，\(B\) 先生以亏损 \(10\%\) 的价格将房子卖回给 \(A\) 先生。则：

A\(A\) 收支相抵 B\(A\) 在交易中赚了 \(\$1100\) C\(A\) 在交易中赚了 \(\$1000\) D\(A\) 在交易中亏了 \(\$900\) E\(A\) 在交易中亏了 \(\$1000\)

English

Mr. \(A\) owns a home worth \(\$10{,}000.\) He sells it to Mr. \(B\) at a \(10\%\) profit based on the worth of the house. Mr. \(B\) sells the house back to Mr. \(A\) at a \(10\%\) loss. Then:

(A) \(A\) comes out even(B) \(A\) makes \(\$1100\) on the deal(C) \(A\) makes \(\$1000\) on the deal(D) \(A\) loses \(\$900\) on the deal(E) \(A\) loses \(\$1000\) on the deal

6

已知一个长方体盒子的底面、侧面和正面面积。这三个面积的乘积等于：

A盒子的体积 B体积的平方根 C体积的两倍 D体积的平方 E体积的立方

English

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to:

(A) The volume of the box(B) The square root of the volume(C) Twice the volume(D) The square of the volume(E) The cube of the volume

7

测量一条长 \(10''\) 的线段时误差为 \(0.02''\)，而测量一条长 \(100''\) 的线段时误差仅为 \(0.2''\)。与第一次测量的相对误差相比，第二次测量的相对误差：

A大 \(0.18\) B相同 C较小 D是其 \(10\) 倍 E\((A)\) 和 \((D)\) 的描述都正确

English

An error of \(0.02''\) is made in the measurement of a line \(10''\) long, while an error of only \(0.2''\) is made in a measurement of a line \(100''\) long. In comparison with the relative error of the first measurement, the relative error of the second measurement is:

(A) Greater by \(0.18\)(B) The same(C) Less(D) \(10\) times as great(E) Correctly described by both \((A)\) and \((D)\)

8

一件商品的价格降低了 \(10\%\)。要恢复原价，降价后的价格必须增加：

A\(10\%\) B\(9\%\) C\(11\dfrac19\%\) D\(11\%\) E以上答案均不正确

English

The price of an article is cut \(10\%.\) To restore it to its former value, the new price must be increased by:

(A) \(10\%\)(B) \(9\%\)(C) \(11\dfrac19\%\)(D) \(11\%\)(E) None of these answers

9

先画一个边长为 \(a\) 的等边三角形，再连接其三边中点，得到一个新的等边三角形；随后连接第二个三角形的三边中点，得到第三个等边三角形；如此无限继续。所有这些三角形周长之和的极限为：

A无穷大 B\(\left(5\dfrac14\right)a\) C\(2a\) D\(6a\) E\(\left(4\dfrac12\right)a\)

English

An equilateral triangle is drawn with a side of length \(a.\) A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the triangles thus drawn is:

(A) Infinite(B) \(\left(5\dfrac14\right)a\)(C) \(2a\)(D) \(6a\)(E) \(\left(4\dfrac12\right)a\)

10

下列说法中，不正确的是：

A将一个给定长方形的底边加倍，面积也加倍。 B将三角形的高加倍，面积也加倍。 C将一个给定圆的半径加倍，面积也加倍。 D将一个分式的除数加倍，并将其分子除以 \(2\)，会改变商。 E将一个给定的量加倍，可能使结果小于原来的量。

English

Of the following statements, the one that is incorrect is:

(A) Doubling the base of a given rectangle doubles the area.(B) Doubling the altitude of a triangle doubles the area.(C) Doubling the radius of a given circle doubles the area.(D) Doubling the divisor of a fraction and dividing its numerator by \(2\) changes the quotient.(E) Doubling a given quantity may make it less than it originally was.

11

一个无穷等比数列的和为 \(\dfrac{a}{1-r}\)，其中 \(a\) 表示首项，\(-1\lt r\lt1\) 表示公比。该数列各项平方之和为：

A\(\dfrac{a^2}{(1-r)^2}\) B\(\dfrac{a^2}{1+r^2}\) C\(\dfrac{a^2}{1-r^2}\) D\(\dfrac{4a^2}{1+r^2}\) E以上答案均不正确

English

The limit of the sum of an infinite number of terms in a geometric progression is \(\dfrac{a}{1-r},\) where \(a\) denotes the first term and \(-1\lt r\lt1\) denotes the common ratio. The limit of the sum of their squares is:

(A) \(\dfrac{a^2}{(1-r)^2}\)(B) \(\dfrac{a^2}{1+r^2}\)(C) \(\dfrac{a^2}{1-r^2}\)(D) \(\dfrac{4a^2}{1+r^2}\)(E) None of these

12

在 \(2{:}15\) 时，钟表的时针与分针所成的角为：

A\(30^\circ\) B\(5^\circ\) C\(22\dfrac12^\circ\) D\(7\dfrac12^\circ\) E\(28^\circ\)

English

At \(2{:}15\) o’clock, the hour and minute hands of a clock form an angle of:

(A) \(30^\circ\)(B) \(5^\circ\)(C) \(22\dfrac12^\circ\)(D) \(7\dfrac12^\circ\)(E) \(28^\circ\)

13

\(A\) 可在 \(9\) 天内完成一项工作。\(B\) 的效率比 \(A\) 高 \(50\%\)。\(B\) 完成同一项工作所需的天数为：

A\(13\dfrac12\) B\(4\dfrac12\) C\(6\) D\(3\) E以上答案均不正确

English

\(A\) can do a piece of work in \(9\) days. \(B\) is \(50\%\) more efficient than \(A.\) The number of days it takes \(B\) to do the same piece of work is:

(A) \(13\dfrac12\)(B) \(4\dfrac12\)(C) \(6\)(D) \(3\)(E) None of these answers

14

关于几何证明，请指出下列哪一项说法不正确：

A有些陈述无需证明即可接受。 B在某些情况下，证明某些命题时存在不止一种正确的步骤顺序。 C证明中使用的每一个术语都必须事先定义。 D如果已知条件中含有一个不真实的命题，就不可能通过正确推理得出真实的结论。 E只要存在两个或更多互相矛盾的命题，就可以使用间接证明。

English

In connection with proof in geometry, indicate which one of the following statements is incorrect:

(A) Some statements are accepted without being proved.(B) In some instances there is more than one correct order in proving certain propositions.(C) Every term used in a proof must have been defined previously.(D) It is not possible to arrive by correct reasoning at a true conclusion if, in the given, there is an untrue proposition.(E) Indirect proof can be used whenever there are two or more contrary propositions.

15

对于 \(n\) 的一切整数值，表达式 \(n^3-n\) 恒能被下列哪个最大的数整除？

A\(2\) B\(3\) C\(4\) D\(5\) E\(6\)

English

The largest number by which the expression \(n^3-n\) is divisible for all possible integral values of \(n\) is:

(A) \(2\)(B) \(3\)(C) \(4\)(D) \(5\)(E) \(6\)

16

对二次方程 \[ f(x)=ax^2+bx+c=0 \] 使用求根公式时，若恰有 \(c=\dfrac{b^2}{4a}\)，则 \(y=f(x)\) 的图像一定：

A有最大值 B有最小值 C与 \(x\) 轴相切 D与 \(y\) 轴相切 E只位于一个象限内

English

If in applying the quadratic formula to a quadratic equation \[ f(x)=ax^2+bx+c=0, \] it happens that \(c=\dfrac{b^2}{4a},\) then the graph of \(y=f(x)\) will certainly:

(A) Have a maximum(B) Have a minimum(C) Be tangent to the \(x\)-axis(D) Be tangent to the \(y\)-axis(E) Lie in one quadrant only

17

请指出下列哪个方程中，\(y\) 与 \(x\) 既不成正比，也不成反比：

A\(x+y=0\) B\(3xy=10\) C\(x=5y\) D\(3x+y=10\) E\(\dfrac{x}{y}=\sqrt3\)

English

Indicate in which one of the following equations \(y\) is neither directly nor inversely proportional to \(x\):

(A) \(x+y=0\)(B) \(3xy=10\)(C) \(x=5y\)(D) \(3x+y=10\)(E) \(\dfrac{x}{y}=\sqrt3\)

18

要将表达式 \(21x^2+ax+21\) 分解为两个系数为整数的线性不可约二项式之积。当 \(a\) 是下列哪一项时，可以完成这种分解？

A任意奇数 B某些奇数 C任意偶数 D某些偶数 E零

English

The expression \(21x^2+ax+21\) is to be factored into two linear prime binomial factors with integer coefficients. This can be done if \(a\) is:

(A) Any odd number(B) Some odd number(C) Any even number(D) Some even number(E) Zero

19

将一个三位数重复书写一次，组成一个六位数，例如 \(256{,}256\)、\(678{,}678\) 等。任何这种形式的数都一定能被下列哪个数整除？

A仅 \(7\) B仅 \(11\) C仅 \(13\) D\(101\) E\(1001\)

English

A six-place number is formed by repeating a three-place number; for example, \(256{,}256,\) or \(678{,}678,\) etc. Any number of this form is always exactly divisible by:

(A) \(7\) only(B) \(11\) only(C) \(13\) only(D) \(101\)(E) \(1001\)

20

将表达式 \((x+y)^{-1}(x^{-1}+y^{-1})\) 化简，并用负指数表示，结果等于：

A\(x^{-2}+2x^{-1}y^{-1}+y^{-2}\) B\(x^{-2}+2^{-1}x^{-1}y^{-1}+y^{-2}\) C\(x^{-1}y^{-1}\) D\(x^{-2}+y^{-2}\) E\(\dfrac{1}{x^{-1}y^{-1}}\)

English

When simplified and expressed with negative exponents, the expression \((x+y)^{-1}(x^{-1}+y^{-1})\) is equal to:

(A) \(x^{-2}+2x^{-1}y^{-1}+y^{-2}\)(B) \(x^{-2}+2^{-1}x^{-1}y^{-1}+y^{-2}\)(C) \(x^{-1}y^{-1}\)(D) \(x^{-2}+y^{-2}\)(E) \(\dfrac{1}{x^{-1}y^{-1}}\)

21

已知 \(x\gt0\)、\(y\gt0\)、\(x\gt y\)，且 \(z\ne0\)。下列不等式中，不一定正确的是：

A\(x+z\gt y+z\) B\(x-z\gt y-z\) C\(xz\gt yz\) D\(\dfrac{x}{z^2}\gt\dfrac{y}{z^2}\) E\(xz^2\gt yz^2\)

English

Given \(x\gt0,\) \(y\gt0,\) \(x\gt y,\) and \(z\ne0.\) The inequality which is not always correct is:

(A) \(x+z\gt y+z\)(B) \(x-z\gt y-z\)(C) \(xz\gt yz\)(D) \(\dfrac{x}{z^2}\gt\dfrac{y}{z^2}\)(E) \(xz^2\gt yz^2\)

22

方程 \(\log_{10}(a^2-15a)=2\) 中 \(a\) 的值为：

A\(\dfrac{15\pm\sqrt{233}}{2}\) B\(20\)、\(-5\) C\(\dfrac{15\pm\sqrt{305}}{2}\) D\(\pm20\) E以上答案均不正确

English

The values of \(a\) in the equation \(\log_{10}(a^2-15a)=2\) are:

(A) \(\dfrac{15\pm\sqrt{233}}{2}\)(B) \(20,\) \(-5\)(C) \(\dfrac{15\pm\sqrt{305}}{2}\)(D) \(\pm20\)(E) None of these

23

一个圆柱形盒子的半径为 \(8\) 英寸，高为 \(3\) 英寸。分别给半径或高增加相同的长度，若要使两种情况下体积的非零增量相等，则应增加多少英寸？

A\(1\) B\(5\dfrac13\) C任意数 D不存在 E以上答案均不正确

English

The radius of a cylindrical box is \(8\) inches and the height is \(3\) inches. The number of inches that may be added to either the radius or the height to give the same nonzero increase in volume is:

(A) \(1\)(B) \(5\dfrac13\)(C) Any number(D) Non-existent(E) None of these

24

化简 \(\dfrac{2^{n+4}-2(2^n)}{2(2^{n+3})}\)，结果为：

A\(2^{n+1}-\dfrac18\) B\(-2^{n+1}\) C\(1-2^n\) D\(\dfrac78\) E\(\dfrac74\)

English

When simplified, \(\dfrac{2^{n+4}-2(2^n)}{2(2^{n+3})}\) is:

(A) \(2^{n+1}-\dfrac18\)(B) \(-2^{n+1}\)(C) \(1-2^n\)(D) \(\dfrac78\)(E) \(\dfrac74\)

25

一个正方形的面积数值等于其周长数值；一个等边三角形的面积数值也等于其周长数值。比较它们的边心距，前者的边心距：

A等于后者 B是后者的 \(\dfrac43\) 倍 C是后者的 \(\dfrac{2}{\sqrt3}\) 倍 D是后者的 \(\dfrac{\sqrt2}{\sqrt3}\) 倍 E与后者的关系无法确定

English

The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:

(A) Equal to the second(B) \(\dfrac43\) times the second(C) \(\dfrac{2}{\sqrt3}\) times the second(D) \(\dfrac{\sqrt2}{\sqrt3}\) times the second(E) Indeterminately related to the second

26

在方程 \[ \frac{x(x-1)-(m+1)}{(x-1)(m-1)}=\frac{x}{m} \] 中，当下列哪一条件成立时，两根相等？

A\(m=1\) B\(m=\dfrac12\) C\(m=0\) D\(m=-1\) E\(m=-\dfrac12\)

English

In the equation \[ \frac{x(x-1)-(m+1)}{(x-1)(m-1)}=\frac{x}{m}, \] the roots are equal when:

(A) \(m=1\)(B) \(m=\dfrac12\)(C) \(m=0\)(D) \(m=-1\)(E) \(m=-\dfrac12\)

27

过三角形内一点，从三个顶点分别向对边画直线，将原三角形分成六个三角形区域。则：

A每一对相对的三角形都相似 B每一对相对的三角形都全等 C每一对相对的三角形面积都相等 D形成三个相似的四边形 E以上关系均不成立

English

Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides, forming six triangular sections. Then:

(A) The triangles are similar in opposite pairs(B) The triangles are congruent in opposite pairs(C) The triangles are equal in area in opposite pairs(D) Three similar quadrilaterals are formed(E) None of the above relations is true

28

风对帆的压力 \(P\) 与帆的面积 \(A\) 及风速 \(V\) 的平方成联合正比。当风速为每小时 \(16\) 英里时，每平方英尺帆面所受压力为 \(1\) 磅。当一平方码帆面所受压力为 \(36\) 磅时，风速为：

A\(10\dfrac23\) 英里/小时 B\(96\) 英里/小时 C\(32\) 英里/小时 D\(1\dfrac13\) 英里/小时 E\(16\) 英里/小时

English

The pressure \(P\) of wind on a sail varies jointly as the area \(A\) of the sail and the square of the velocity \(V\) of the wind. The pressure on a square foot is \(1\) pound when the velocity is \(16\) miles per hour. The velocity of the wind when the pressure on a square yard is \(36\) pounds is:

(A) \(10\dfrac23\) mph(B) \(96\) mph(C) \(32\) mph(D) \(1\dfrac13\) mph(E) \(16\) mph

29

下列各组数据中，唯一不能确定三角形形状的是：

A两边之比及其夹角 B三条高之比 C三条中线之比 D一条高与其对应底边之比 E两个角

English

Of the following sets of data, the only one that does not determine the shape of a triangle is:

(A) The ratio of two sides and the included angle(B) The ratios of the three altitudes(C) The ratios of the three medians(D) The ratio of the altitude to the corresponding base(E) Two angles

30

两根高分别为 \(20''\) 和 \(80''\) 的杆相距 \(100''\)。分别连接每根杆的顶端与另一根杆的底端，则两条连线交点的高度为：

A\(50''\) B\(40''\) C\(16''\) D\(60''\) E以上答案均不正确

English

If two poles \(20''\) and \(80''\) high are \(100''\) apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:

(A) \(50''\)(B) \(40''\)(C) \(16''\)(D) \(60''\)(E) None of these

31

一次聚会结束时，人们共握手 \(28\) 次。假设每位参加者对其他所有人都同样礼貌，则在场人数为：

A\(14\) B\(28\) C\(56\) D\(8\) E\(7\)

English

A total of \(28\) handshakes was exchanged at the conclusion of a party. Assuming that each participant was equally polite toward all the others, the number of people present was:

(A) \(14\)(B) \(28\)(C) \(56\)(D) \(8\)(E) \(7\)

32

若 \(\triangle ABC\) 内接于以 \(\overline{AB}\) 为直径的半圆，则 \(\overline{AC}+\overline{BC}\) 必须：

A等于 \(\overline{AB}\) B等于 \(\overline{AB}\sqrt2\) C\(\ge \overline{AB}\sqrt2\) D\(\le \overline{AB}\sqrt2\) E\(\overline{AB}^{\,2}\)

English

If \(\triangle ABC\) is inscribed in a semicircle whose diameter is \(\overline{AB},\) then \(\overline{AC}+\overline{BC}\) must be:

(A) Equal to \(\overline{AB}\)(B) Equal to \(\overline{AB}\sqrt2\)(C) \(\ge \overline{AB}\sqrt2\)(D) \(\le \overline{AB}\sqrt2\)(E) \(\overline{AB}^{\,2}\)

33

下列每一对方程的图像交点横坐标都可以给出方程 \(x^2-2x=0\) 的根，唯独哪一对不能？

A\(y=x^2\)，\(y=2x\) B\(y=x^2-2x\)，\(y=0\) C\(y=x\)，\(y=x-2\) D\(y=x^2-2x+1\)，\(y=1\) E\(y=x^2-1\)，\(y=2x-1\)

English

The roots of the equation \(x^2-2x=0\) can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:

(A) \(y=x^2,\) \(y=2x\)(B) \(y=x^2-2x,\) \(y=0\)(C) \(y=x,\) \(y=x-2\)(D) \(y=x^2-2x+1,\) \(y=1\)(E) \(y=x^2-1,\) \(y=2x-1\)

34

\(10^{\log_{10}7}\) 的值为：

A\(7\) B\(1\) C\(10\) D\(\log_{10}7\) E\(\log_7 10\)

English

The value of \(10^{\log_{10}7}\) is:

(A) \(7\)(B) \(1\)(C) \(10\)(D) \(\log_{10}7\)(E) \(\log_7 10\)

35

若 \(a^x=c^q=b\)，且 \(c^y=a^z=d\)，则：

A\(xy=qz\) B\(\dfrac{x}{y}=\dfrac{q}{z}\) C\(x+y=q+z\) D\(x-y=q-z\) E\(x^y=q^z\)

English

If \(a^x=c^q=b\) and \(c^y=a^z=d,\) then:

(A) \(xy=qz\)(B) \(\dfrac{x}{y}=\dfrac{q}{z}\)(C) \(x+y=q+z\)(D) \(x-y=q-z\)(E) \(x^y=q^z\)

36

下列用于证明一个几何图形是轨迹的方法中，哪一种不正确？

A轨迹上的每一点都满足条件，轨迹外的每一点都不满足条件。 B每个不满足条件的点都不在轨迹上，且轨迹上的每一点都满足条件。 C每个满足条件的点都在轨迹上，且轨迹上的每一点都满足条件。 D轨迹外的每一点都不满足条件，且每个不满足条件的点都不在轨迹上。 E每个满足条件的点都在轨迹上，且每个不满足条件的点都不在轨迹上。

English

Which of the following methods of proving a geometric figure a locus is not correct?

(A) Every point on the locus satisfies the conditions and every point not on the locus does not satisfy the conditions.(B) Every point not satisfying the conditions is not on the locus and every point on the locus does satisfy the conditions.(C) Every point satisfying the conditions is on the locus and every point on the locus satisfies the conditions.(D) Every point not on the locus does not satisfy the conditions and every point not satisfying the conditions is not on the locus.(E) Every point satisfying the conditions is on the locus and every point not satisfying the conditions is not on the locus.

37

有一个数，除以 \(10\) 余 \(9\)，除以 \(9\) 余 \(8\)，除以 \(8\) 余 \(7\)，依此类推，直到除以 \(2\) 余 \(1\)。这个数是：

A\(59\) B\(419\) C\(1259\) D\(2519\) E以上答案均不正确

English

A number which when divided by \(10\) leaves a remainder of \(9,\) when divided by \(9\) leaves a remainder of \(8,\) by \(8\) leaves a remainder of \(7,\) etc., down to where, when divided by \(2,\) it leaves a remainder of \(1,\) is:

(A) \(59\)(B) \(419\)(C) \(1259\)(D) \(2519\)(E) None of these answers

38

一条铁路要越过一座山，需要上升 \(600\) 英尺。延长轨道并使其绕过山峰，可以降低坡度。要将坡度从 \(3\%\) 降至 \(2\%\)，大约需要增加多长的轨道？

A\(10{,}000\) 英尺 B\(20{,}000\) 英尺 C\(30{,}000\) 英尺 D\(12{,}000\) 英尺 E以上答案均不正确

English

A rise of \(600\) feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from \(3\%\) to \(2\%\) is approximately:

(A) \(10{,}000\) ft.(B) \(20{,}000\) ft.(C) \(30{,}000\) ft.(D) \(12{,}000\) ft.(E) None of these

39

将一块石头投入井中，在投下后 \(7.7\) 秒听到石头撞击井底的声音。假设石头在 \(t\) 秒内下落 \(16t^2\) 英尺，声速为每秒 \(1120\) 英尺。井深为：

A\(784\) 英尺 B\(342\) 英尺 C\(1568\) 英尺 D\(156.8\) 英尺 E以上答案均不正确

English

A stone is dropped into a well and the report of the stone striking the bottom is heard \(7.7\) seconds after it is dropped. Assume that the stone falls \(16t^2\) feet in \(t\) seconds and that the velocity of sound is \(1120\) feet per second. The depth of the well is:

(A) \(784\) ft.(B) \(342\) ft.(C) \(1568\) ft.(D) \(156.8\) ft.(E) None of these

40

表达式 \[ \begin{aligned} &\left(\frac{(x+1)^2(x^2-x+1)^2}{(x^3+1)^2}\right)^2\\ &\quad{}\cdot \left(\frac{(x-1)^2(x^2+x+1)^2}{(x^3-1)^2}\right)^2 \end{aligned} \] 等于：

A\((x+1)^4\) B\((x^3+1)^4\) C\(1\) D\(\big[(x^3+1)(x^3-1)\big]^2\) E\(\big[(x^3-1)^2\big]^2\)

English

The expression \[ \begin{aligned} &\left(\frac{(x+1)^2(x^2-x+1)^2}{(x^3+1)^2}\right)^2\\ &\quad{}\cdot \left(\frac{(x-1)^2(x^2+x+1)^2}{(x^3-1)^2}\right)^2 \end{aligned} \] equals:

(A) \((x+1)^4\)(B) \((x^3+1)^4\)(C) \(1\)(D) \(\big[(x^3+1)(x^3-1)\big]^2\)(E) \(\big[(x^3-1)^2\big]^2\)

41

表中 \(x\) 与 \(y\) 之间的关系可用下列哪个公式表示？

A\(y=2x-4\) B\(y=x^2-3x+2\) C\(y=x^3-3x^2+2x\) D\(y=x^2-4x\) E\(y=x^2-4\)

English

The formula expressing the relationship between \(x\) and \(y\) in the table is:

(A) \(y=2x-4\)(B) \(y=x^2-3x+2\)(C) \(y=x^3-3x^2+2x\)(D) \(y=x^2-4x\)(E) \(y=x^2-4\)

42

若 \(x\) 等于 \[ \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}\text{，}\] 则：

A\(x=1\) B\(0\lt x\lt1\) C\(1\lt x\lt2\) D\(x\) 是无穷大 E\(x\gt2\)，但为有限值

English

If \(x\) equals \[ \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}, \] then:

(A) \(x=1\)(B) \(0\lt x\lt1\)(C) \(1\lt x\lt2\)(D) \(x\) is infinite(E) \(x\gt2\) but finite

43

下列说法中，唯一不正确的是：

A不等式两边同时增加、减少、乘以或除以（除数不为零）同一个正数后，不等式仍然成立。 B两个不相等正数的算术平均数大于其几何平均数。 C两个正数的和一定时，二者相等时乘积最大。 D若 \(a\) 和 \(b\) 是不相等的正数，则 \(\dfrac12(a^2+b^2)\) 大于 \(\left[\dfrac12(a+b)\right]^2\)。 E两个正数的乘积一定时，二者相等时和最大。

English

Of the following statements, the only one that is incorrect is:

(A) An inequality will remain true after each side is increased, decreased, multiplied, or divided (zero excluded) by the same positive quantity.(B) The arithmetic mean of two unequal positive quantities is greater than their geometric mean.(C) If the sum of two positive quantities is given, their product is largest when they are equal.(D) If \(a\) and \(b\) are positive and unequal, \(\dfrac12(a^2+b^2)\) is greater than \(\left[\dfrac12(a+b)\right]^2.\)(E) If the product of two positive quantities is given, their sum is greatest when they are equal.

44

若 \(\dfrac{xy}{x+y}=a\)、\(\dfrac{xz}{x+z}=b\)、\(\dfrac{yz}{y+z}=c\)，其中 \(a\)、\(b\)、\(c\) 均不为零，则 \(x\) 等于：

A\(\dfrac{abc}{ab+ac+bc}\) B\(\dfrac{2abc}{ab+bc+ac}\) C\(\dfrac{2abc}{ab+ac-bc}\) D\(\dfrac{2abc}{ab+bc-ac}\) E\(\dfrac{2abc}{ac+bc-ab}\)

English

If \(\dfrac{xy}{x+y}=a,\) \(\dfrac{xz}{x+z}=b,\) and \(\dfrac{yz}{y+z}=c,\) where \(a,\) \(b,\) \(c\) are other than zero, then \(x\) equals:

(A) \(\dfrac{abc}{ab+ac+bc}\)(B) \(\dfrac{2abc}{ab+bc+ac}\)(C) \(\dfrac{2abc}{ab+ac-bc}\)(D) \(\dfrac{2abc}{ab+bc-ac}\)(E) \(\dfrac{2abc}{ac+bc-ab}\)

45

已知 \(\log 8=0.9031\) 和 \(\log 9=0.9542\)，则下列唯一不能在不查表的情况下求出的对数是：

A\(\log 17\) B\(\log(\frac{5}{4})\) C\(\log 15\) D\(\log 600\) E\(\log 0.4\)

English

If you are given \(\log 8=0.9031\) and \(\log 9=0.9542,\) then the only logarithm that cannot be found without the use of tables is:

(A) \(\log 17\)(B) \(\log(\frac{5}{4})\)(C) \(\log 15\)(D) \(\log 600\)(E) \(\log 0.4\)

46

\(\overline{AB}\) 是圆心为 \(O\) 的圆的一条固定直径。从圆上任意一点 \(C\) 作弦 \(\overline{CD}\)，使其垂直于 \(\overline{AB}\)。当 \(C\) 在一个半圆上移动时，角 \(OCD\) 的平分线与圆相交于一点，该点始终：

A平分弧 \(AB\) B三等分弧 \(AB\) C位置随之变化 D到 \(\overline{AB}\) 的距离与到 \(D\) 的距离相等 E到 \(B\) 与 \(C\) 的距离相等

English

\(\overline{AB}\) is a fixed diameter of a circle whose center is \(O.\) From \(C,\) any point on the circle, a chord \(\overline{CD}\) is drawn perpendicular to \(\overline{AB}.\) Then, as \(C\) moves over a semicircle, the bisector of angle \(OCD\) cuts the circle in a point that always:

(A) Bisects the arc \(AB\)(B) Trisects the arc \(AB\)(C) Varies(D) Is as far from \(\overline{AB}\) as from \(D\)(E) Is equidistant from \(B\) and \(C\)

47

若 \(r\) 和 \(s\) 是方程 \(ax^2+bx+c=0\) 的根，则 \(\dfrac1{r^2}+\dfrac1{s^2}\) 的值为：

A\(b^2-4ac\) B\(\dfrac{b^2-4ac}{2a}\) C\(\dfrac{b^2-4ac}{c^2}\) D\(\dfrac{b^2-2ac}{c^2}\) E以上答案均不正确

English

If \(r\) and \(s\) are the roots of the equation \(ax^2+bx+c=0,\) the value of \(\dfrac1{r^2}+\dfrac1{s^2}\) is:

(A) \(b^2-4ac\)(B) \(\dfrac{b^2-4ac}{2a}\)(C) \(\dfrac{b^2-4ac}{c^2}\)(D) \(\dfrac{b^2-2ac}{c^2}\)(E) None of these

48

内接于半圆的正方形面积与内接于整圆的正方形面积之比为：

A\(1:2\) B\(2:3\) C\(2:5\) D\(3:4\) E\(3:5\)

English

The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:

(A) \(1:2\)(B) \(2:3\)(C) \(2:5\)(D) \(3:4\)(E) \(3:5\)

49

从一个直角三角形的两个锐角顶点所作的中线长分别为 \(5\) 和 \(\sqrt{40}\)。斜边长为：

A\(10\) B\(2\sqrt{40}\) C\(\sqrt{13}\) D\(2\sqrt{13}\) E以上答案均不正确

English

The medians of a right triangle which are drawn from the vertices of the acute angles are \(5\) and \(\sqrt{40}.\) The value of the hypotenuse is:

(A) \(10\)(B) \(2\sqrt{40}\)(C) \(\sqrt{13}\)(D) \(2\sqrt{13}\)(E) None of these

50

汤姆、迪克和哈里一起开始一段 \(100\) 英里的旅程。汤姆与哈里乘汽车，以每小时 \(25\) 英里的速度前进；迪克则以每小时 \(5\) 英里的速度步行。行驶一段距离后，哈里下车，改以每小时 \(5\) 英里的速度继续步行；汤姆掉头去接迪克，并使迪克与哈里同时到达终点。整个旅程所需的小时数为：

A\(5\) B\(6\) C\(7\) D\(8\) E以上答案均不正确

English

Tom, Dick, and Harry started out on a \(100\)-mile journey. Tom and Harry went by automobile at the rate of \(25\) mph, while Dick walked at the rate of \(5\) mph. After a certain distance, Harry got off and walked on at \(5\) mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:

(A) \(5\)(B) \(6\)(C) \(7\)(D) \(8\)(E) None of these answers