Question bank AMC12 1960

If $2$ is a solution (root) of $x^3+hx+10=0,$ then $h$ equals:

It takes $5$ seconds for a clock to strike $6$ o’clock beginning at $6{:}00$ o’clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o’clock?

Applied to a bill for $\$10{,}000$ the difference between a discount of $40\%$ and two successive discounts of $36\%$ and $4\%,$ expressed in dollars, is:

Each of two angles of a triangle is $60^\circ$ and the included side is $4$ inches. The area of the triangle, in square inches, is:

The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is:

The circumference of a circle is $100$ inches. The side of a square inscribed in this circle, expressed in inches, is:

Circle I passes through the center of, and is tangent to, circle II. The area of circle I is $4$ square inches. Then the area of circle II, in square inches, is:

The number $2.5252525\ldots$ can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is:

The fraction $$
\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}
$$ is (with suitable restrictions on the values of $a,$ $b,$ and $c$):

Given the following six statements:

$(1)$ All women are good drivers.

$(2)$ Some women are good drivers.

$(3)$ No men are good drivers.

$(4)$ All men are bad drivers.

$(5)$ At least one man is a bad driver.

$(6)$ All men are good drivers.

The statement that negates statement $(6)$ is:

For a given value of $k,$ the product of the roots of $$
x^2-3kx+2k^2-1=0
$$ is $7.$ The roots may be characterized as:

The locus of the centers of all circles of given radius $a,$ in the same plane, passing through a fixed point, is:

The polygon(s) formed by $y=3x+2,$ $y=-3x+2,$ and $y=-2,$ is (are):

If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [the symbol $a\ne0$ means that $a$ is different from zero]:

Triangle I is equilateral with side $A,$ perimeter $P,$ area $K,$ and circumradius $R$ (radius of the circumscribed circle). Triangle II is equilateral with side $a,$ perimeter $p,$ area $k,$ and circumradius $r.$ If $A$ is different from $a,$ then:

In the numeration system with base $5,$ counting is as follows: $1,$ $2,$ $3,$ $4,$ $10,$ $11,$ $12,$ $13,$ $14,$ $20,$ $\ldots$ The number whose description in the decimal system is $69,$ when described in the base $5$ system, is a number with:

The formula $N=8\cdot10^8\cdot x^{-\frac{3}{2}}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars. The lowest income, in dollars, of the wealthiest $800$ individuals is at least:

The pair of equations $3^{x+y}=81$ and $81^{x-y}=3$ has:

Consider equation I: $x+y+z=46,$ where $x,$ $y,$ and $z$ are positive integers, and equation II: $x+y+z+w=46,$ where $x,$ $y,$ $z,$ and $w$ are positive integers. Then:

The coefficient of $x^7$ in the expansion of $$
\left(\frac{x^2}{2}-\frac2x\right)^8
$$ is: