Difference between revisions of "1962 AHSME Problems"

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{{AHSC 40 Problems
|year = 1962
== Problem 1 ==
== Problem 1 ==
The expression <math>\frac{1^{4y-1}}{5^{-1}+3^{-1}}</math> is equal to:  
The expression <math>\frac{1^{4y-1}}{5^{-1}+3^{-1}}</math> is equal to:  

Revision as of 14:35, 19 February 2020

1962 AHSC (Answer Key)
Printable version: Wiki | AoPS ResourcesPDF


  1. This is a 40-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. You will have ? minutes working time to complete the test.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Problem 1

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

$\textbf{(A)}\ \frac{4y-1}{8} \qquad  \textbf{(B)}\ 8 \qquad  \textbf{(C)}\ \frac{15}{2} \qquad  \textbf{(D)}\ \frac{15}{8}\qquad \textbf{(E)}\ \frac{1}{8}$


Problem 2

The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to:

$\textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad  \textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad  \textbf{(C)}\ \frac{\sqrt{-3}}{6}\qquad \textbf{(D)}\ \frac{5\sqrt{3}}{6}\qquad \textbf{(E)}\ 1$


Problem 3

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

$\textbf{(A)}\ - 2 \qquad  \textbf{(B)}\ 0 \qquad  \textbf{(C)}\ 2 \qquad  \textbf{(D)}\ 4 \qquad  \textbf{(E)}\ \text{undetermined}$


Problem 4

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

$\textbf{(A)}\ 4 \qquad  \textbf{(B)}\ \frac{5}{3} \qquad  \textbf{(C)}\ \frac{3}{2} \qquad  \textbf{(D)}\ \frac{3}{5} \qquad  \textbf{(E)}\ \frac{1}{4}$


Problem 5

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

$\textbf{(A)}\ \pi + 2 \qquad  \textbf{(B)}\ \frac{2 \pi + 1}{2} \qquad  \textbf{(C)}\ \pi \qquad  \textbf{(D)}\ \frac{2\pi-1}{2}\qquad \textbf{(E)}\ \pi-2$


Problem 6

A square and an equilateral triangle have equal perimeters. The area of the triangle is $9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is:

$\textbf{(A)}\ \frac{9}{2} \qquad  \textbf{(B)}\ 2 \sqrt{5} \qquad  \textbf{(C)}\ 4 \sqrt{2} \qquad  \textbf{(D)}\ \frac{9\sqrt{2}}{2}\qquad \textbf{(E)}\ \text{none of these}$


Problem 7

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:

$\textbf{(A)}\ \frac {1}{2} (90 - A) \qquad  \textbf{(B)}\ 90 - A \qquad  \textbf{(C)}\ \frac {1}{2} (180 - A) \qquad \\ \textbf{(D)}\ 180-A\qquad \textbf{(E)}\ 180-2A$


Problem 8

Given the set of $n$ numbers; $n > 1$, of which one is $1 - \frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is:

$\textbf{(A)}\ 1 \qquad  \textbf{(B)}\ n - \frac {1}{n} \qquad  \textbf{(C)}\ n - \frac {1}{n^2} \qquad  \textbf{(D)}\ 1-\frac{1}{n^2}\qquad \textbf{(E)}\ 1-\frac{1}{n}-\frac{1}{n^2}$


Problem 9

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

$\textbf{(A)}\ \text{more than 5} \qquad  \textbf{(B)}\ 5 \qquad  \textbf{(C)}\ 4 \qquad  \textbf{(D)}\ 3 \qquad  \textbf{(E)}\ 2$


Problem 10

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:

$\textbf{(A)}\ 5 \qquad  \textbf{(B)}\ 4 \frac{1}{2} \qquad  \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2 \qquad  \textbf{(E)}\ 1$


Problem 11

The difference between the larger root and the smaller root of $x^2 - px + (p^2 - 1)/4 = 0$ is:

$\textbf{(A)}\ 0 \qquad  \textbf{(B)}\ 1 \qquad  \textbf{(C)}\ 2 \qquad  \textbf{(D)}\ p \qquad  \textbf{(E)}\ p+1$


Problem 12

When $\left ( 1 - \frac{1}{a} \right ) ^6$ is expanded the sum of the last three coefficients is:

$\textbf{(A)}\ 22 \qquad  \textbf{(B)}\ 11 \qquad  \textbf{(C)}\ 10 \qquad  \textbf{(D)}\ -10 \qquad  \textbf{(E)}\ -11$


Problem 13

$R$ varies directly as $S$ and inverse as $T$. When $R = \frac43$ and $T = \frac {9}{14}, S = \frac37$. Find $S$ when $R = \sqrt {48}$ and $T = \sqrt {75}$.

$\textbf{(A)}\ 28 \qquad  \textbf{(B)}\ 30 \qquad  \textbf{(C)}\ 40 \qquad  \textbf{(D)}\ 42 \qquad  \textbf{(E)}\ 60$


Problem 14

Let $s$ be the limiting sum of the geometric series $4- \frac{8}{3} + \frac{16}{9} - \dots$, as the number of terms increases without bound. Then $s$ equals:

$\textbf{(A)}\ \text{a number between 0 and 1} \qquad  \textbf{(B)}\ 2.4 \qquad  \textbf{(C)}\ 2.5 \qquad  \textbf{(D)}\ 3.6\qquad \textbf{(E)}\ 12$


Problem 15

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:

$\textbf{(A)}\ \text{a circle} \qquad  \textbf{(B)}\ \text{a parabola} \qquad  \textbf{(C)}\ \text{an ellipse} \qquad  \textbf{(D)}\ \text{a straight line}\qquad \textbf{(E)}\ \text{a curve here not listed}$


Problem 16

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:

$\textbf{(A)}\ \frac{9}{2} \qquad  \textbf{(B)}\ 36 \qquad  \textbf{(C)}\ \frac{135}{2} \qquad  \textbf{(D)}\ 9\sqrt{10}\qquad \textbf{(E)}\ \frac{27\sqrt{10}}{4}$


Problem 17

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

$\textbf{(A)}\ \frac{b}{2} \qquad  \textbf{(B)}\ \frac{2b}{3}\qquad  \textbf{(C)}\ b \qquad  \textbf{(D)}\ \frac{3b}{2}\qquad \textbf{(E)}\ 2b$


Problem 18

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

$\textbf{(A)}\ 3r^2 \qquad  \textbf{(B)}\ 2r^2 \qquad  \textbf{(C)}\ \frac{3r^2 \sqrt{3}}{4} \qquad  \textbf{(D)}\ r^2\sqrt{3}\qquad \textbf{(E)}\ 3r^2\sqrt{3}$


Problem 19

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:

$\textbf{(A)}\ - 4 \qquad  \textbf{(B)}\ - 2 \qquad  \textbf{(C)}\ 0 \qquad  \textbf{(D)}\ 1 \qquad  \textbf{(E)}\ 2$


Problem 20

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

$\textbf{(A)}\ 108 \qquad  \textbf{(B)}\ 90 \qquad  \textbf{(C)}\ 72 \qquad  \textbf{(D)}\ 54 \qquad  \textbf{(E)}\ 36$


Problem 21

It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \sqrt{-1})$. The value of $s$ is:

$\textbf{(A)}\ \text{undetermined} \qquad  \textbf{(B)}\ 5 \qquad  \textbf{(C)}\ 6 \qquad  \textbf{(D)}\ -13\qquad \textbf{(E)}\ 26$


Problem 22

The number $121_b$, written in the integral base $b$, is the square of an integer, for

$\textbf{(A)}\ b = 10,\text{ only} \qquad  \textbf{(B)}\ b = 10 \text{ and } b = 5, \text{ only} \qquad \\ \textbf{(C)}\ 2\leq b\leq 10\qquad \textbf{(D)}\ b > 2\qquad \textbf{(E)}\ \text{no value of }b$


Problem 23

In $\triangle ABC$, $CD$ is the altitude to $AB$ and $AE$ is the altitude to $BC$. If the lengths of $AB, CD$, and $AE$ are known, the length of $DB$ is:

$\textbf{(A)}\ \text{not determined by the information given} \qquad \\ \textbf{(B)}\ \text{determined only if A is an acute angle} \qquad \\ \textbf{(C)}\ \text{determined only if B is an acute angle} \qquad \\ \textbf{(D)}\ \text{determined only in ABC is an acute triangle} \qquad \\ \textbf{(E)}\ \text{none of these is correct}$


Problem 24

Three machines $\text{P, Q, and R,}$ working together, can do a job in $x$ hours. When working alone, $\text{P}$ needs an additional $6$ hours to do the job; $\text{Q}$, one additional hour; and $R$, $x$ additional hours. The value of $x$ is:

$\textbf{(A)}\ \frac23 \qquad  \textbf{(B)}\ \frac{11}{12} \qquad  \textbf{(C)}\ \frac32 \qquad  \textbf{(D)}\ 2\qquad \textbf{(E)}\ 3$


Problem 25

Given square $ABCD$ with side $8$ feet. A circle is drawn through vertices $A$ and $D$ and tangent to side $BC$. The radius of the circle, in feet, is:

$\textbf{(A)}\ 4 \qquad  \textbf{(B)}\ 4 \sqrt{2} \qquad  \textbf{(C)}\ 5 \qquad  \textbf{(D)}\ 5 \sqrt{2} \qquad  \textbf{(E)}\ 6$


Problem 26

For any real value of $x$ the maximum value of $8x - 3x^2$ is:

$\textbf{(A)}\ 0 \qquad  \textbf{(B)}\ \frac83 \qquad  \textbf{(C)}\ 4 \qquad  \textbf{(D)}\ 5 \qquad  \textbf{(E)}\ \frac{16}{3}$


Problem 27

Let $a @ b$ represent the operation on two numbers, $a$ and $b$, which selects the larger of the two numbers, with $a@a = a$. Let $a ! b$ represent the operator which selects the smaller of the two numbers, with $a ! a = a$. Which of the following three rules is (are) correct?

$\textbf{(1)}\ a@b = b@a \qquad \\ \textbf{(2)}\ a@(b@c) = (a@b)@c \qquad \\ \textbf{(3)}\ a ! (b@c) = (a ! b) @ (a ! c)$

$\textbf{(A)}\ (1)\text{ only} \qquad  \textbf{(B)}\ (2) \text{ only} \qquad  \textbf{(C)}\ \text{(1) and (2) only}\qquad \textbf{(D)}\ \text{(1) and (3) only}\qquad \textbf{(E)}\ \text{all three}$


Problem 28

The set of $x$-values satisfying the equation $x^{\log_{10} x} = \frac{x^3}{100}$ consists of:

$\textbf{(A)}\ \frac{1}{10} \qquad  \textbf{(B)}\ \text{10, only} \qquad  \textbf{(C)}\ \text{100, only} \qquad  \textbf{(D)}\ \text{10 or 100, only}\qquad \textbf{(E)}\ \text{more than two real numbers.}$


Problem 29

Which of the following sets of $x$-values satisfy the inequality $2x^2 + x < 6$?

$\textbf{(A)}\ - 2 < x < \frac{3}{2} \qquad  \textbf{(B)}\ x > \frac32 \text{ or }x < - 2 \qquad  \textbf{(C)}\ x <\frac{3}2\qquad\\ \textbf{(D)}\ \frac{3}2 < x < 2\qquad \textbf{(E)}\ x <-2$


Problem 30

Consider the statements:

$\textbf{(1)}\ \text{p and q are both true}\qquad\\ \textbf{(2)}\ \text{p is true and q is false}\qquad\\ \textbf{(3)}\ \text{p is false and q is true}\qquad\\ \textbf{(4)}\ \text{p is false and q is false.}$

How many of these imply the negative of the statement "$p$ and $q$ are both true?"

$\textbf{(A)}\ 0 \qquad  \textbf{(B)}\ 1 \qquad  \textbf{(C)}\ 2 \qquad  \textbf{(D)}\ 3 \qquad  \textbf{(E)}\ 4$


Problem 31

The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there?

$\textbf{(A)}\ 1 \qquad  \textbf{(B)}\ 2 \qquad  \textbf{(C)}\ 3 \qquad  \textbf{(D)}\ 4 \qquad  \textbf{(E)}\ \infty$


Problem 32

If $x_{k+1} = x_k + \frac12$ for $k=1, 2, \dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \dots + x_n$.

$\textbf{(A)}\ \frac{n+1}{2} \qquad  \textbf{(B)}\ \frac{n+3}{2} \qquad  \textbf{(C)}\ \frac{n^2-1}{2} \qquad  \textbf{(D)}\ \frac{n^2+n}{4}\qquad \textbf{(E)}\ \frac{n^2+3n}{4}$


Problem 33

The set of $x$-values satisfying the inequality $2 \leq |x-1| \leq 5$ is:

$\textbf{(A)}\ -4\leq x\leq-1\text{ or }3\leq x\leq 6\qquad \textbf{(B)}\ 3\leq x\leq 6\text{ or }-6\leq x\leq-3\qquad\\ \textbf{(C)}\ x\leq-1\text{ or }x\geq 3\qquad \textbf{(D)}\ -1\leq x\leq 3\qquad \textbf{(E)}\ -4\leq x\leq 6$


Problem 34

For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots?

$\textbf{(A)}\ \text{none}\qquad \textbf{(B)}\ -2<K<1\qquad \textbf{(C)}\ -2\sqrt{2}< K < 2\sqrt{2}\qquad\\ \textbf{(D)}\ K>1\text{ or }K<-2\qquad \textbf{(E)}\ \text{all}$


Problem 35

A man on his way to dinner short after 6: 00 p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before 7: 00 p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is:

$\textbf{(A)}\ 36 \frac23 \qquad  \textbf{(B)}\ 40 \qquad  \textbf{(C)}\ 42 \qquad  \textbf{(D)}\ 42.4 \qquad  \textbf{(E)}\ 45$


Problem 36

If both $x$ and $y$ are both integers, how many pairs of solutions are there of the equation $(x-8)(x-10) = 2^y$?

$\textbf{(A)}\ 0 \qquad  \textbf{(B)}\ 1 \qquad  \textbf{(C)}\ 2 \qquad  \textbf{(D)}\ 3 \qquad  \textbf{(E)}\ \text{more than 3}$


Problem 37

$ABCD$ is a square with side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. In square units this maximum area is:

$\textbf{(A)}\ \frac12 \qquad  \textbf{(B)}\ \frac {9}{16} \qquad  \textbf{(C)}\ \frac{19}{32} \qquad  \textbf{(D)}\ \frac{5}{8}\qquad \textbf{(E)}\ \frac{2}3$


Problem 38

The population of Nosuch Junction at one time was a perfect square. Later, with an increase of $100$, the population was one more than a perfect square. Now, with an additional increase of $100$, the population is again a perfect square.

The original population is a multiple of:

$\textbf{(A)}\ 3 \qquad  \textbf{(B)}\ 7 \qquad  \textbf{(C)}\ 9 \qquad  \textbf{(D)}\ 11 \qquad  \textbf{(E)}\ 17$


Problem 39

Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \sqrt{15}$ square inches. The length of the third median in inches, is:

$\textbf{(A)}\ 4 \qquad  \textbf{(B)}\ 3 \sqrt{3} \qquad  \textbf{(C)}\ 3 \sqrt{6} \qquad  \textbf{(D)}\ 6\sqrt{3}\qquad \textbf{(E)}\ 6\sqrt{6}$


Problem 40

The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is:

$\textbf{(A)}\ \frac{1}9\qquad \textbf{(B)}\ \frac{10}{81}\qquad \textbf{(C)}\ \frac{1}8\qquad \textbf{(D)}\ \frac{17}{72}\qquad \textbf{(E)}\ \infty$


See also

1962 AHSME (ProblemsAnswer KeyResources)
Preceded by
1961 AHSME
Followed by
1963 AHSME
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