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Question bank AMC12 1952

Exam papers

50 questions

Question 1

If the radius of a circle is a rational number, its area is given by a number which is:
  1. Rational
  2. Irrational
  3. Integral
  4. A perfect square
  5. None of these

Question 2

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:
  1. $75\%$
  2. $74\%$
  3. $72\%$
  4. $77\%$
  5. None of these

Question 3

The expression $a^3-a^{-3}$ equals:
  1. $\left(a-\dfrac1a\right)\left(a^2+1+\dfrac1{a^2}\right)$
  2. $\left(\dfrac1a-a\right)\left(a^2-1+\dfrac1{a^2}\right)$
  3. $\left(a-\dfrac1a\right)\left(a^2-2+\dfrac1{a^2}\right)$
  4. $\left(\dfrac1a-a\right)\left(\dfrac1{a^2}+1+a^2\right)$
  5. 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:
  1. $C=10+3P$
  2. $C=10P+3$
  3. $C=10+3(P-1)$
  4. $C=9+3P$
  5. $C=10P-7$

Question 5

The points $(6,12)$ and $(0,-6)$ are connected by a straight line. Another point on this line is:
  1. $(3,3)$
  2. $(2,1)$
  3. $(7,16)$
  4. $(-1,-4)$
  5. $(-3,-8)$

Question 6

The difference of the roots of $x^2-7x-9=0$ is:
  1. $+7$
  2. $+\dfrac72$
  3. $+9$
  4. $2\sqrt{85}$
  5. $\sqrt{85}$

Question 7

When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to:
  1. $x+y$
  2. $\dfrac{xy}{x+y}$
  3. $xy$
  4. $\dfrac1{xy}$
  5. $\dfrac{x+y}{xy}$

Question 8

Two equal circles in the same plane cannot have the following number of common tangents:
  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. None of these

Question 9

If $m=\dfrac{cab}{a-b},$ then $b$ equals:
  1. $\dfrac{m(a-b)}{ca}$
  2. $\dfrac{cab-ma}{-m}$
  3. $\dfrac1{1+c}$
  4. $\dfrac{ma}{m+ca}$
  5. $\dfrac{m+ca}{ma}$

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:
  1. $12\dfrac12$ mph
  2. $13\dfrac13$ mph
  3. $14\dfrac12$ mph
  4. $15$ mph
  5. None of these

Question 11

If $y=f(x)=\dfrac{x+2}{x-1},$ then it is incorrect to say:
  1. $x=\dfrac{y+2}{y-1}$
  2. $f(0)=-2$
  3. $f(1)=0$
  4. $f(-2)=0$
  5. $f(y)=x$

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:
  1. $3$ or $1\dfrac12$
  2. $1$
  3. $2\dfrac12$
  4. $6$
  5. $9$ or $3$

Question 13

The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when:
  1. $x=-p$
  2. $x=\dfrac p2$
  3. $x=-2p$
  4. $x=\dfrac{p^2}{4q}$
  5. $x=-\dfrac p2$

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:
  1. No loss or gain
  2. Loss of $\$1000$
  3. Gain of $\$1000$
  4. Gain of $\$2000$
  5. None of these

Question 15

The sides of a triangle are in the ratio $6:8:9.$ Then:
  1. The triangle is obtuse
  2. The angles are in the ratio $6:8:9$
  3. The triangle is acute
  4. The angle opposite the largest side is double the angle
    opposite the smallest side
  5. 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:
  1. $9\%$
  2. $10\%$
  3. $11\%$
  4. $11\dfrac19\%$
  5. $9\dfrac1{11}\%$

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:
  1. $20$
  2. $100$
  3. $125$
  4. $80$
  5. $120$

Question 18

$\log p+\log q=\log(p+q)$ only if:
  1. $p=q=0$
  2. $p=\dfrac{q^2}{1-q}$
  3. $p=q=1$
  4. $p=\dfrac q{q-1}$
  5. $p=\dfrac q{q+1}$

Question 19

Angle $B$ of triangle $ABC$ is trisected by $BD$ and $BE,$ which meet $AC$ at $D$ and $E,$ respectively. Then:
  1. $\dfrac{AD}{EC}=\dfrac{AE}{DC}$
  2. $\dfrac{AD}{EC}=\dfrac{AB}{BC}$
  3. $\dfrac{AD}{EC}=\dfrac{BD}{BE}$
  4. $\dfrac{AD}{EC}=\dfrac{AB\cdot BD}{BE\cdot
    BC}$
  5. $\dfrac{AD}{EC}=\dfrac{AE\cdot BD}{DC\cdot BE}$

Question 20

If $\dfrac xy=\dfrac34,$ then the incorrect expression in the following is:
  1. $\dfrac{x+y}{y}=\dfrac74$
  2. $\dfrac{y}{y-x}=\dfrac41$
  3. $\dfrac{x+2y}{x}=\dfrac{11}3$
  4. $\dfrac{x}{2y}=\dfrac38$
  5. $\dfrac{x-y}{y}=\dfrac14$