1951 AMC 12

Question 1

The percent that $M$ is greater than $N$ is:

$\dfrac{100(M-N)}{M}$
$\dfrac{100(M-N)}{N}$
$\dfrac{M-N}{N}$
$\dfrac{M-N}{M}$
$\dfrac{100(M+N)}{N}$

Question 2

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:

$\dfrac{x^2}{2}$
$2x^2$
$\dfrac{2x^2}{9}$
$\dfrac{x^2}{18}$
$\dfrac{x^2}{72}$

Question 3

If the length of a diagonal of a square is $a+b,$ then the area of the square is:

$(a+b)^2$
$\dfrac12(a+b)^2$
$a^2+b^2$
$\dfrac12(a^2+b^2)$
None of these

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:

$360$
$460$
$490$
$590$
$720$

Question 5

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:

$A$ comes out even
$A$ makes $\$1100$ on the deal
$A$ makes $\$1000$ on the deal
$A$ loses $\$900$ on the deal
$A$ loses $\$1000$ on the deal

Question 6

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

The volume of the box
The square root of the volume
Twice the volume
The square of the volume
The cube of the volume

Question 7

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:

Greater by $0.18$
The same
Less
$10$ times as great
Correctly described by both $(A)$ and $(D)$

Question 8

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

$10\%$
$9\%$
$11\dfrac19\%$
$11\%$
None of these answers

Question 9

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:

Infinite
$\left(5\dfrac14\right)a$
$2a$
$6a$
$\left(4\dfrac12\right)a$

Question 10

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

Doubling the base of a given rectangle doubles the area.
Doubling the altitude of a triangle doubles the area.
Doubling the radius of a given circle doubles
the area.
Doubling the divisor of a fraction and dividing its numerator
by $2$ changes the quotient.
Doubling a given quantity may make it less than it originally
was.

Question 11

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:

$\dfrac{a^2}{(1-r)^2}$
$\dfrac{a^2}{1+r^2}$
$\dfrac{a^2}{1-r^2}$
$\dfrac{4a^2}{1+r^2}$
None of these

Question 12

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

$30^\circ$
$5^\circ$
$22\dfrac12^\circ$
$7\dfrac12^\circ$
$28^\circ$

Question 13

$A$ 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:

$13\dfrac12$
$4\dfrac12$
$6$
$3$
None of these answers

Question 14

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

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

Question 15

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

Question 16

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:

Have a maximum
Have a minimum
Be tangent to the $x$-axis
Be tangent to the $y$-axis
Lie in one quadrant only

Question 17

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

$x+y=0$
$3xy=10$
$x=5y$
$3x+y=10$
$\dfrac{x}{y}=\sqrt3$

Question 18

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:

Any odd number
Some odd number
Any even number
Some even number
Zero

Question 19

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:

$7$ only
$11$ only
$13$ only
$101$
$1001$

Question 20

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

$x^{-2}+2x^{-1}y^{-1}+y^{-2}$
$x^{-2}+2^{-1}x^{-1}y^{-1}+y^{-2}$
$x^{-1}y^{-1}$
$x^{-2}+y^{-2}$
$\dfrac{1}{x^{-1}y^{-1}}$