Question bank AMC12 1963

Which one of the following points is *not* on the graph of $y=\dfrac{x}{x+1}?$

Let $n=x-y^{x-y}.$ Find $n$ when $x=2$ and $y=-2.$

If the reciprocal of $x+1$ is $x-1,$ then $x$ equals:

For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions?

If $x$ and $\log_{10}x$ are real numbers and $\log_{10}x\lt0,$ then:

Triangle $BAD$ is right-angled at $B.$ On $AD$ there is a point $C$ for which $AC=CD$ and $AB=BC.$ The magnitude of angle $DAB,$ in degrees, is:

Given the four equations:

$$
\begin{aligned} (1)\quad&3y-2x=12,\\ (2)\quad&-2x-3y=10,\\ (3)\quad&3y+2x=12,\\ (4)\quad&2y+3x=10. \end{aligned}
$$

The pair representing perpendicular lines is:

The smallest positive integer $x$ for which $1260x=N^3,$ where $N$ is an integer, is:

In the expansion of $\left(a-\dfrac1{\sqrt a}\right)^7$ the coefficient of $a^{-\frac{1}{2}}$ is:

Point $P$ is taken interior to a square with side-length $a$ and such that it is equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals:

The arithmetic mean of a set of $50$ numbers is $38.$ If two numbers of the set, namely $45$ and $55,$ are discarded, the arithmetic mean of the remaining set of numbers is:

Three vertices of parallelogram $PQRS$ are $P(-3,-2),$ $Q(1,-5),$ $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is:

If $2^a+2^b=3^c+3^d,$ the number of integers $a,$ $b,$ $c,$ $d$ which can possibly be negative is, at most:

Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0.$ If, when the roots of the equations are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals:

A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is:

Three numbers $a,$ $b,$ $c,$ none zero, form an arithmetic progression. Increasing $a$ by $1$ or increasing $c$ by $2$ results in a geometric progression. Then $b$ equals:

The expression

$$
\frac{\dfrac{a}{a+y}+\dfrac{y}{a-y}} {\dfrac{y}{a+y}-\dfrac{a}{a-y}},
$$

$a$ real, $a\ne0,$ has the value $-1$ for:

Chord $EF$ is the perpendicular bisector of chord $BC,$ intersecting it in $M.$ Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A.$ Then, for any selection of $U,$ as described, triangle $EUM$ is similar to triangle:

In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red. Thereafter, $7$ out of every $8$ counted were red. If, in all, $90\%$ or more of the balls counted were red, the maximum value of $n$ is:

Two men at points $R$ and $S,$ $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at $4\dfrac12$ miles per hour; the man at $S$ walks at $3\dfrac14$ miles per hour for the first hour, at $3\dfrac34$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is: