Question bank AMC12 1962

The expression $\dfrac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to:

The expression $\sqrt{\dfrac43}-\sqrt{\dfrac34}$ is equal to:

The first three terms of an arithmetic progression are $x-1,$ $x+1,$ $2x+3,$ in the order shown. The value of $x$ is:

If $8^x=32,$ then $x$ equals:

If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is:

A square and an equilateral triangle have equal perimeters. The area of the triangle is $9\sqrt3$ square inches. Expressed in inches the diagonal of the square is:

Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at $D.$ Then, if all measurements are in degrees, angle $BDC$ equals:

Given the set of $n$ numbers, $n\gt1,$ of which one is $1-\dfrac1n$ and all the others are $1.$ The arithmetic mean of the $n$ numbers is:

When $x^9-x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:

A man drives $150$ miles to the seashore in $3$ hours and $20$ minutes. He returns from the shore to the starting point in $4$ hours and $10$ minutes. Let $r$ be the average rate for the entire trip. Then the average rate for the trip going exceeds $r,$ in miles per hour, by:

The difference between the larger root and the smaller root of $$
x^2-px+\frac{p^2-1}{4}=0
$$ is:

When $\left(1-\dfrac1a\right)^6$ is expanded, the sum of the last three coefficients is:

$R$ varies directly as $S$ and inversely as $T.$ When $R=\dfrac43$ and $T=\dfrac9{14},$ $S=\dfrac37.$ Find $S$ when $R=\sqrt{48}$ and $T=\sqrt{75}.$

Let $s$ be the limiting sum of the geometric series $4-\dfrac83+\dfrac{16}{9}-\cdots,$ as the number of terms increases without bound. Then $s$ equals:

Given triangle $ABC$ with base $AB$ fixed in length and position. As the vertex $C$ moves on a straight line, the intersection point of the three medians moves on:

Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1.$ Expressed in square inches the area of $R_2$ is:

If $a=\log_8 225$ and $b=\log_2 15,$ then $a,$ in terms of $b,$ is:

A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is:

If the parabola $y=ax^2+bx+c$ passes through the points $(-1,12),$ $(0,5),$ and $(2,-3),$ the value of $a+b+c$ is:

The angles of a pentagon are in arithmetic progression. One of the angles, in degrees, must be: