1952 AMC 12

Question 1

If the radius of a circle is a rational number, its area is given by a number which is:

Rational
Irrational
Integral
A perfect square
None of these

Two high school classes took the same test. One class of $20$ students made an average grade of $80\%$; the other class of $30$ students made an average grade of $70\%.$ The average grade for all students in both classes is:

$75\%$
$74\%$
$72\%$
$77\%$
None of these

Question 3

The expression $a^3-a^{-3}$ equals:

$\left(a-\dfrac1a\right)\left(a^2+1+\dfrac1{a^2}\right)$
$\left(\dfrac1a-a\right)\left(a^2-1+\dfrac1{a^2}\right)$
$\left(a-\dfrac1a\right)\left(a^2-2+\dfrac1{a^2}\right)$
$\left(\dfrac1a-a\right)\left(\dfrac1{a^2}+1+a^2\right)$
None of these

Question 4

The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is:

$C=10+3P$
$C=10P+3$
$C=10+3(P-1)$
$C=9+3P$
$C=10P-7$

Question 5

The points $(6,12)$ and $(0,-6)$ are connected by a straight line. Another point on this line is:

$(3,3)$
$(2,1)$
$(7,16)$
$(-1,-4)$
$(-3,-8)$

Question 6

The difference of the roots of $x^2-7x-9=0$ is:

$+7$
$+\dfrac72$
$+9$
$2\sqrt{85}$
$\sqrt{85}$

Question 7

When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to:

$x+y$
$\dfrac{xy}{x+y}$
$xy$
$\dfrac1{xy}$
$\dfrac{x+y}{xy}$

Question 8

Two equal circles in the same plane cannot have the following number of common tangents:

$1$
$2$
$3$
$4$
None of these

Question 9

If $m=\dfrac{cab}{a-b},$ then $b$ equals:

$\dfrac{m(a-b)}{ca}$
$\dfrac{cab-ma}{-m}$
$\dfrac1{1+c}$
$\dfrac{ma}{m+ca}$
$\dfrac{m+ca}{ma}$

Question 10

An automobile went up a hill at a speed of $10$ miles an hour and down the same distance at a speed of $20$ miles an hour. The average speed for the round trip was:

$12\dfrac12$ mph
$13\dfrac13$ mph
$14\dfrac12$ mph
$15$ mph
None of these

Question 11

If $y=f(x)=\dfrac{x+2}{x-1},$ then it is incorrect to say:

$x=\dfrac{y+2}{y-1}$
$f(0)=-2$
$f(1)=0$
$f(-2)=0$
$f(y)=x$

Question 12

The sum to infinity of the terms of an infinite geometric progression is $6.$ The sum of the first two terms is $4\dfrac12.$ The first term of the progression is:

$3$ or $1\dfrac12$
$1$
$2\dfrac12$
$6$
$9$ or $3$

Question 13

The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:

$x=-p$
$x=\dfrac p2$
$x=-2p$
$x=\dfrac{p^2}{4q}$
$x=-\dfrac p2$

Question 14

A house and store were sold for $\$12000$ each. The house was sold at a loss of $20\%$ of the cost, and the store at a gain of $20\%$ of the cost. The entire transaction resulted in:

No loss or gain
Loss of $\$1000$
Gain of $\$1000$
Gain of $\$2000$
None of these

Question 15

The sides of a triangle are in the ratio $6:8:9.$ Then:

The triangle is obtuse
The angles are in the ratio $6:8:9$
The triangle is acute
The angle opposite the largest side is double the angle
opposite the smallest side
None of these

Question 16

If the base of a rectangle is increased by $10\%$ and the area is unchanged, then the altitude is decreased by:

$9\%$
$10\%$
$11\%$
$11\dfrac19\%$
$9\dfrac1{11}\%$

Question 17

A merchant bought some goods at a discount of $20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\%$ of the marked price and still make a profit of $20\%$ of the selling price. The percent of the list price at which he should mark them is:

$20$
$100$
$125$
$80$
$120$

Question 18

$\log p+\log q=\log(p+q)$ only if:

$p=q=0$
$p=\dfrac{q^2}{1-q}$
$p=q=1$
$p=\dfrac q{q-1}$
$p=\dfrac q{q+1}$

Question 19

Angle $B$ of triangle $ABC$ is trisected by $BD$ and $BE,$ which meet $AC$ at $D$ and $E,$ respectively. Then:

$\dfrac{AD}{EC}=\dfrac{AE}{DC}$
$\dfrac{AD}{EC}=\dfrac{AB}{BC}$
$\dfrac{AD}{EC}=\dfrac{BD}{BE}$
$\dfrac{AD}{EC}=\dfrac{AB\cdot BD}{BE\cdot
BC}$
$\dfrac{AD}{EC}=\dfrac{AE\cdot BD}{DC\cdot BE}$

Question 20

If $\dfrac xy=\dfrac34,$ then the incorrect expression in the following is:

$\dfrac{x+y}{y}=\dfrac74$
$\dfrac{y}{y-x}=\dfrac41$
$\dfrac{x+2y}{x}=\dfrac{11}3$
$\dfrac{x}{2y}=\dfrac38$
$\dfrac{x-y}{y}=\dfrac14$