1954 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(50)

1

Angle Formed by Clock's Hands

The times between

7

8

o'clock, correct to the nearest minute, when the hands of a clock will form an angle of

84

<span class='latex-bold'>(A)</span>\ \text{7: 23 and 7: 53} \qquad <span class='latex-bold'>(B)</span>\ \text{7: 20 and 7: 50} \qquad <span class='latex-bold'>(C)</span>\ \text{7: 22 and 7: 53} \\ <span class='latex-bold'>(D)</span>\ \text{7: 23 and 7: 52} \qquad <span class='latex-bold'>(E)</span>\ \text{7: 21 and 7: 49}

1

Difference of Squares

The difference of the squares of two odd numbers is always divisible by

8

a>b

, and 2a\plus{}1 and 2b\plus{}1 are the odd numbers, to prove the given statement we put the difference of the squares in the form: (A)\ (2a\plus{}1)^2\minus{}(2b\plus{}1)^2 \\ (B)\ 4a^2\minus{}4b^2\plus{}4a\minus{}4b \\ (C)\ 4[a(a\plus{}1)\minus{}b(b\plus{}1)] \\ (D)\ 4(a\minus{}b)(a\plus{}b\plus{}1) \\ (E)\ 4(a^2\plus{}a\minus{}b^2\minus{}b)

1

Train hits Accident

A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at

\frac{3}{4}

of its former rate and arrives

3 \frac{1}{2}

hours late. Had the accident happened

90

miles farther along the line, it would have arrived only

3

hours late. The length of the trip in miles was:

<span class='latex-bold'>(A)</span>\ 400 \qquad <span class='latex-bold'>(B)</span>\ 465 \qquad <span class='latex-bold'>(C)</span>\ 600 \qquad <span class='latex-bold'>(D)</span>\ 640 \qquad <span class='latex-bold'>(E)</span>\ 550

1

Perpendicular from Midpoint

At the midpoint of line segment

AB

p

units long, a perpendicular

MR

is erected with length

q

units. An arc is described from

R

with a radius equal to

\frac{1}{2}AB

AB

T

AT

TB

are the roots of: (A)\ x^2\plus{}px\plus{}q^2\equal{}0 \\ (B)\ x^2\minus{}px\plus{}q^2\equal{}0 \\ (C)\ x^2\plus{}px\minus{}q^2\equal{}0 \\ (D)\ x^2\minus{}px\minus{}q^2\equal{}0 \\ (E)\ x^2\minus{}px\plus{}q\equal{}0

1

Points of Tangency

In the diagram, if points

A

B

C

are points of tangency, then

x

equals: [asy]unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3;
pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8);
draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0));
label("

A

",A,SW); label("

B

",B,SE); label("

C

",C,N); label("

\frac{3}{8}

",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label("

\frac{1}{2}

",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label("

x

",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label("

60^{\circ}

",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]

<span class='latex-bold'>(A)</span>\ \frac {3}{16}" \qquad <span class='latex-bold'>(B)</span>\ \frac {1}{8}" \qquad <span class='latex-bold'>(C)</span>\ \frac {1}{32}" \qquad <span class='latex-bold'>(D)</span>\ \frac {3}{32}" \qquad <span class='latex-bold'>(E)</span>\ \frac {1}{16}"

1

A Rhombus

ABCD

, line segments are drawn within the rhombus, parallel to diagonal

BD

, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex

A

. The graph is:

<span class='latex-bold'>(A)</span>\ \text{A straight line passing through the origin.} \\ <span class='latex-bold'>(B)</span>\ \text{A straight line cutting across the upper right quadrant.} \\ <span class='latex-bold'>(C)</span>\ \text{Two line segments forming an upright V.} \\ <span class='latex-bold'>(D)</span>\ \text{Two line segments forming an inverted V.} \\ <span class='latex-bold'>(E)</span>\ \text{None of these.}

1

Year a Man Was Born in

A man born in the first half of the nineteenth century was

x

years old in the year

x^2

. He was born in:

<span class='latex-bold'>(A)</span>\ 1849 \qquad <span class='latex-bold'>(B)</span>\ 1825 \qquad <span class='latex-bold'>(C)</span>\ 1812 \qquad <span class='latex-bold'>(D)</span>\ 1836 \qquad <span class='latex-bold'>(E)</span>\ 1806

1

Perimeter of Right Triangle

The hypotenuse of a right triangle is

10

inches and the radius of the inscribed circle is

1

inch. The perimeter of the triangle in inches is:

<span class='latex-bold'>(A)</span>\ 15 \qquad <span class='latex-bold'>(B)</span>\ 22 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 26 \qquad <span class='latex-bold'>(E)</span>\ 30

1

Considering Graphs

Consider the graphs of (1): y\equal{}x^2\minus{}\frac{1}{2}x\plus{}2 and (2) y\equal{}x^2\plus{}\frac{1}{2}x\plus{}2 on the same set of axis. These parabolas are exactly the same shape. Then:

<span class='latex-bold'>(A)</span>\ \text{the graphs coincide.} \\ <span class='latex-bold'>(B)</span>\ \text{the graph of (1) is lower than the graph of (2).} \\ <span class='latex-bold'>(C)</span>\ \text{the graph of (1) is to the left of the graph of (2).} \\ <span class='latex-bold'>(D)</span>\ \text{the graph of (1) is to the right of the graph of (2).} \\ <span class='latex-bold'>(E)</span>\ \text{the graph of (1) is higher than the graph of (2).}

1

Sum of Roots

The sum of all the roots of 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0 is: (A)\ 8 \qquad (B)\ 2 \qquad (C)\ \minus{}8 \qquad (D)\ \minus{}2 \qquad (E)\ 0

1

Solving an Equation

If \left (a\plus{}\frac{1}{a} \right )^2\equal{}3, then a^3\plus{}\frac{1}{a^3} equals:

<span class='latex-bold'>(A)</span>\ \frac{10\sqrt{3}}{3} \qquad <span class='latex-bold'>(B)</span>\ 3\sqrt{3} \qquad <span class='latex-bold'>(C)</span>\ 0 \qquad <span class='latex-bold'>(D)</span>\ 7\sqrt{7} \qquad <span class='latex-bold'>(E)</span>\ 6\sqrt{3}

1

Midpoint of Line Segment

The locus of the midpoint of a line segment that is drawn from a given external point

P

to a given circle with center

O

r

<span class='latex-bold'>(A)</span>\ \text{a straight line perpendicular to }\overline{PO} \\ <span class='latex-bold'>(B)</span>\ \text{a straight line parallel to } \overline{PO} \\ <span class='latex-bold'>(C)</span>\ \text{a circle with center }P\text{ and radius }r \\ <span class='latex-bold'>(D)</span>\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r \\ <span class='latex-bold'>(E)</span>\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r

1

Logarithms

If \log 2\equal{}.3010 and \log 3\equal{}.4771, the value of

x

when 3^{x\plus{}3}\equal{}135 is approximately:

<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 1.47 \qquad <span class='latex-bold'>(C)</span>\ 1.67 \qquad <span class='latex-bold'>(D)</span>\ 1.78 \qquad <span class='latex-bold'>(E)</span>\ 1.63

1

Angles Bisected

PQR

\overline{RS}

\angle R

PQ

D

\angle n

a right angle, then: [asy]path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false) { pair M,N; path mark; M=t*0.03*unit(A-B)+B; N=t*0.03*unit(C-B)+B; if(flip) mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B)); else mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B)); return mark; }
unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt));
pair P=(0,0), R=(3,2), Q=(4,0); pair S0=bisectorpoint(P,R,Q); pair Sp=extension(P,Q,S0,R); pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp); pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R);
draw(P--R--Q); draw(R--Sp); draw(P--D--M);
draw(anglemark2(Sp,P,R,17)); label("

p

",P+(0.35,0.1)); draw(anglemark2(R,Q,P,11)); label("

q

",Q+(-0.17,0.1)); draw(anglemark2(R,Np,D,8,true)); label("

n

",Np+(+0.12,0.07)); draw(anglemark2(R,M,D,13,true)); label("

m

",M+(+0.25,0.03)); draw(anglemark2(M,D,P,29)); label("

d

",D+(-0.75,0.095));
pen f=fontsize(10pt); label("

R

",R,N,f); label("

P

",P,S,f); label("

S

",Sp,S,f); label("

Q

",Q,S,f); label("

D

",D,S,f);[/asy] (A)\ \angle m \equal{} \frac {1}{2}(\angle p \minus{} \angle q) \qquad (B)\ \angle m \equal{} \frac {1}{2}(\angle p \plus{} \angle q) (C)\ \angle d \equal{} \frac {1}{2} (\angle q \plus{} \angle p) \qquad (D)\ \angle d \equal{} \frac {1}{2}\angle m \qquad (E)\ \text{none of these is correct}

1

Boat in Water

A boat has a speed of

15

mph in still water. In a stream that has a current of

5

mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is:

<span class='latex-bold'>(A)</span>\ \frac{5}{4} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{1} \qquad <span class='latex-bold'>(C)</span>\ \frac{8}{9} \qquad <span class='latex-bold'>(D)</span>\ \frac{7}{8} \qquad <span class='latex-bold'>(E)</span>\ \frac{9}{8}

1

Right Triangle

In the right triangle shown the sum of the distances

BM

MA

is equal to the sum of the distances

BC

CA

. If MB \equal{} x, CB \equal{} h, and CA \equal{} d, then

x

equals: [asy]size(200); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; draw((0,0)--(8,0)--(0,5)--cycle); label("C",(0,0),SW); label("A",(8,0),SE); label("M",(0,5),N); dot((0,3.5)); label("B",(0,3.5),W); label("

x

",(0,4.25),W); label("

h

",(0,1),W); label("

d

",(4,0),S);[/asy] (A)\ \frac {hd}{2h \plus{} d} \qquad (B)\ d \minus{} h \qquad (C)\ \frac {1}{2}d \qquad (D)\ h \plus{} d \minus{} \sqrt {2d} \qquad (E)\ \sqrt {h^2 \plus{} d^2} \minus{} h

1

Simple Fraction

\frac{1}{3}

<span class='latex-bold'>(A)</span>\ \text{equals 0.33333333} \qquad <span class='latex-bold'>(B)</span>\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\ <span class='latex-bold'>(C)</span>\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^9} \\ <span class='latex-bold'>(D)</span>\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\ <span class='latex-bold'>(E)</span>\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^9}

1

Charging Loans

\$6

\$120

. The borrower receives

\$114

and repays the loan in

12

installments of

\$10

a month. The interest rate is approximately:

<span class='latex-bold'>(A)</span>\ 5 \% \qquad <span class='latex-bold'>(B)</span>\ 6 \% \qquad <span class='latex-bold'>(C)</span>\ 7 \% \qquad <span class='latex-bold'>(D)</span>\ 9\% \qquad <span class='latex-bold'>(E)</span>\ 15 \%

1

Factoring

The factors of x^4\plus{}64 are: (A)\ (x^2\plus{}8)^2 \qquad (B)\ (x^2\plus{}8)(x^2\minus{}8) \qquad (C)\ (x^2\plus{}2x\plus{}4)(x^2\minus{}8x\plus{}16) \\ (D)\ (x^2\minus{}4x\plus{}8)(x^2\minus{}4x\minus{}8) \qquad (E)\ (x^2\minus{}4x\plus{}8)(x^2\plus{}4x\plus{}8)

1

Triangles and Angles

ABC

, AB\equal{}AC, \angle A\equal{}40^\circ. Point

O

is within the triangle with

\angle OBC \cong \angle OCA

. The number of degrees in angle

BOC

<span class='latex-bold'>(A)</span>\ 110 \qquad <span class='latex-bold'>(B)</span>\ 35 \qquad <span class='latex-bold'>(C)</span>\ 140 \qquad <span class='latex-bold'>(D)</span>\ 55 \qquad <span class='latex-bold'>(E)</span>\ 70

1

Doing a Job Together and Alone

A

B

together can do a job in

2

B

C

can do it in four days; and

A

C

2\frac{2}{5}

days. The number of days required for

A

to do the job alone is:

<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 2.8

1

Legs and Hypotenuse

If the ratio of the legs of a right triangle is

1: 2

, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:

<span class='latex-bold'>(A)</span>\ 1: 4 \qquad <span class='latex-bold'>(B)</span>\ 1: \sqrt{2} \qquad <span class='latex-bold'>(C)</span>\ 1: 2 \qquad <span class='latex-bold'>(D)</span>\ 1: \sqrt{5} \qquad <span class='latex-bold'>(E)</span>\ 1: 5

1

Ratios

If \frac{m}{n}\equal{}\frac{4}{3} and \frac{r}{t}\equal{}\frac{9}{14}, the value of \frac{3mr\minus{}nt}{4nt\minus{}7mr} is: (A)\ \minus{}5 \frac{1}{2} \qquad (B)\ \minus{}\frac{11}{14} \qquad (C)\ \minus{}1\frac{1}{4} \qquad (D)\ \frac{11}{14} \qquad (E)\ \minus{}\frac{2}{3}

1

Right Circular Cone

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:

<span class='latex-bold'>(A)</span>\ \frac{1}{1} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(C)</span>\ \frac{2}{3} \qquad <span class='latex-bold'>(D)</span>\ \frac{2}{1} \qquad <span class='latex-bold'>(E)</span>\ \sqrt{\frac{5}{4}}

1

Money Made on Article

If the margin made on an article costing

C

dollars and selling for

S

dollars is M\equal{}\frac{1}{n}C, then the margin is given by: (A)\ M\equal{}\frac{1}{n\minus{}1}S \qquad (B)\ M\equal{}\frac{1}{n}S \qquad (C)\ M\equal{}\frac{n}{n\plus{}1}S \\ (D)\ M\equal{}\frac{1}{n\plus{}1}S \qquad (E)\ M\equal{}\frac{n}{n\minus{}1}S

1

Lines and Circles

The straight line

\overline{AB}

C

so that AC\equal{}3CB. Circles are described on

\overline{AC}

\overline{CB}

as diameters and a common tangent meets

AB

D

BD

<span class='latex-bold'>(A)</span>\ \text{diameter of the smaller circle} \\ <span class='latex-bold'>(B)</span>\ \text{radius of the smaller circle} \\ <span class='latex-bold'>(C)</span>\ \text{radius of the larger circle} \\ <span class='latex-bold'>(D)</span>\ CB\sqrt{3}\\ <span class='latex-bold'>(E)</span>\ \text{the difference of the two radii}

1

Roots of Equation

The two roots of the equation a(b\minus{}c)x^2\plus{}b(c\minus{}a)x\plus{}c(a\minus{}b)\equal{}0 are

1

and: (A)\ \frac{b(c\minus{}a)}{a(b\minus{}c)} \qquad (B)\ \frac{a(b\minus{}c)}{c(a\minus{}b)} \qquad (C)\ \frac{a(b\minus{}c)}{b(c\minus{}a)} \qquad (D)\ \frac{c(a\minus{}b)}{a(b\minus{}c)} \qquad (E)\ \frac{c(a\minus{}b)}{b(c\minus{}a)}

1

Real and Equal Roots

k

for which the equation 2x^2\minus{}kx\plus{}x\plus{}8\equal{}0 will have real and equal roots are: (A)\ 9 \text{ and }\minus{}7 \qquad (B)\ \text{only }\minus{}7 \qquad (C)\ \text{9 and 7} \\ (D)\ \minus{}9 \text{ and }\minus{}7 \qquad (E)\ \text{only 9}

1

Evaluating Expressions

\frac{2x^2-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}

cannot be evaluated for

x=-1

x=2

, since division by zero is not allowed. For other values of

x

:

<span class='latex-bold'>(A)</span>\ \text{The expression takes on many different values.} \\ <span class='latex-bold'>(B)</span>\ \text{The expression has only the value 2.} \\ <span class='latex-bold'>(C)</span>\ \text{The expression has only the value 1.} \\ <span class='latex-bold'>(D)</span>\ \text{The expression always has a value between } -1 \text{ and } +2. \\ <span class='latex-bold'>(E)</span>\ \text{The expression has a value greater than 2 or less than } -1.

1

Roots of Equation

The roots of the equation 2\sqrt {x} \plus{} 2x^{ \minus{} \frac {1}{2}} \equal{} 5 can be found by solving: (A)\ 16x^2 \minus{} 92x \plus{} 1 \equal{} 0 \qquad (B)\ 4x^2 \minus{} 25x \plus{} 4 \equal{} 0 \qquad (C)\ 4x^2 \minus{} 17x \plus{} 4 \equal{} 0 \\ (D)\ 2x^2 \minus{} 21x \plus{} 2 \equal{} 0 \qquad (E)\ 4x^2 \minus{} 25x \minus{} 4 \equal{} 0

1

Another Cubic

The equation x^3\plus{}6x^2\plus{}11x\plus{}6\equal{}0 has:

<span class='latex-bold'>(A)</span>\ \text{no negative real roots} \qquad <span class='latex-bold'>(B)</span>\ \text{no positive real roots} \qquad <span class='latex-bold'>(C)</span>\ \text{no real roots} \\ <span class='latex-bold'>(D)</span>\ \text{1 positive and 2 negative roots} \qquad <span class='latex-bold'>(E)</span>\ \text{2 positive and 1 negative root}

1

A Function

If f(x) \equal{} 5x^2 \minus{} 2x \minus{} 1, then f(x \plus{} h) \minus{} f(x) equals: (A)\ 5h^2 \minus{} 2h \qquad (B)\ 10xh \minus{} 4x \plus{} 2 \qquad (C)\ 10xh \minus{} 2x \minus{} 2 \\ (D)\ h(10x \plus{} 5h \minus{} 2) \qquad (E)\ 3h

1

Inscribed Circle

If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle:

<span class='latex-bold'>(A)</span>\ \text{are always equal to }60^\circ \\ <span class='latex-bold'>(B)</span>\ \text{are always one obtuse angle and two unequal acute angles} \\ <span class='latex-bold'>(C)</span>\ \text{are always one obtuse angle and two equal acute angles} \\ <span class='latex-bold'>(D)</span>\ \text{are always acute angles} \\ <span class='latex-bold'>(E)</span>\ \text{are always unequal to each other}

1

An Inequality

Of the following sets, the one that includes all values of

x

which will satisfy 2x \minus{} 3 > 7 \minus{} x is: (A)\ x > 4 \qquad (B)\ x < \frac {10}{3} \qquad (C)\ x \equal{} \frac {10}{3} \qquad (D)\ x > \frac {10}{3} \qquad (E)\ x < 0

1

Graph of a Cubic

The graph of the function f(x) \equal{} 2x^3 \minus{} 7 goes:

<span class='latex-bold'>(A)</span>\ \text{up to the right and down to the left} \\ <span class='latex-bold'>(B)</span>\ \text{down to the right and up to the left} \\ <span class='latex-bold'>(C)</span>\ \text{up to the right and up to the left} \\ <span class='latex-bold'>(D)</span>\ \text{down to the right and down to the left} \\ <span class='latex-bold'>(E)</span>\ \text{none of these ways.}

1

Simple Logarithms

\log 125

<span class='latex-bold'>(A)</span>\ 100 \log 1.25 \qquad <span class='latex-bold'>(B)</span>\ 5 \log 3 \qquad <span class='latex-bold'>(C)</span>\ 3 \log 25

(D)\ 3 \minus{} 3\log 2 \qquad (E)\ (\log 25)(\log 5)

1

Simplifying an Expression

When simplified \sqrt{1\plus{} \left (\frac{x^4\minus{}1}{2x^2} \right )^2} equals: (A)\ \frac{x^4\plus{}2x^2\minus{}1}{2x^2} \qquad (B)\ \frac{x^4\minus{}1}{2x^2} \qquad (C)\ \frac{\sqrt{x^2\plus{}1}}{2} \\ (D)\ \frac{x^2}{\sqrt{2}} \qquad (E)\ \frac{x^2}{2}\plus{}\frac{1}{2x^2}

1

Inscribed Angles

A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be:

<span class='latex-bold'>(A)</span>\ 180^\circ \qquad <span class='latex-bold'>(B)</span>\ 540^\circ \qquad <span class='latex-bold'>(C)</span>\ 360^\circ \qquad <span class='latex-bold'>(D)</span>\ 450^\circ \qquad <span class='latex-bold'>(E)</span>\ 1080^\circ

1

Solving a System of Equations

The solution of the equations \begin{align*} 2x-3y&=7 \\ 4x-6y &=20 \\ \end{align*} is:

<span class='latex-bold'>(A)</span>\ x=18, y=12 \qquad <span class='latex-bold'>(B)</span>\ x=0, y=0 \qquad <span class='latex-bold'>(C)</span>\ \text{There is no solution} \\ <span class='latex-bold'>(D)</span>\ \text{There are an unlimited number of solutions} \qquad <span class='latex-bold'>(E)</span>\ x=8, y=5

1

Selling Dresses

A merchant placed on display some dresses, each with a marked price. He then posted a sign “

\frac{1}{3}

off on these dresses.” The cost of the dresses was

\frac{3}{4}

of the price at which he actually sold them. Then the ratio of the cost to the marked price was:

<span class='latex-bold'>(A)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{4} \qquad <span class='latex-bold'>(D)</span>\ \frac{2}{3} \qquad <span class='latex-bold'>(E)</span>\ \frac{3}{4}

1

Comparing Side Lengths

The base of a triangle is twice as long as a side of a square and their areas are the same. Then the ratio of the altitude of the triangle to the side of the square is:

<span class='latex-bold'>(A)</span>\ \frac{1}{4} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(C)</span>\ 1 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 4

1

Buying a Dress

A housewife saved

\$2.50

in buying a dress on sale. If she spent

\$25

for the dress, she saved about:

<span class='latex-bold'>(A)</span>\ 8 \% \qquad <span class='latex-bold'>(B)</span>\ 9 \% \qquad <span class='latex-bold'>(C)</span>\ 10 \% \qquad <span class='latex-bold'>(D)</span>\ 11 \% \qquad <span class='latex-bold'>(E)</span>\ 12 \%

1

Sum of Coefficients

The sum of the numerical coefficients in the expansion of the binomial (a\plus{}b)^8 is:

<span class='latex-bold'>(A)</span>\ 32 \qquad <span class='latex-bold'>(B)</span>\ 16 \qquad <span class='latex-bold'>(C)</span>\ 64 \qquad <span class='latex-bold'>(D)</span>\ 48 \qquad <span class='latex-bold'>(E)</span>\ 7

1

Point and a Secant

P

is outside a circle and is

13

inches from the center. A secant from

P

cuts the circle at

Q

R

so that the external segment of the secant

PQ

9

QR

7

inches. The radius of the circle is:

<span class='latex-bold'>(A)</span>\ 3" \qquad <span class='latex-bold'>(B)</span>\ 4" \qquad <span class='latex-bold'>(C)</span>\ 5" \qquad <span class='latex-bold'>(D)</span>\ 6" \qquad <span class='latex-bold'>(E)</span>\ 7"

1

Simplifying an Expression

The value of \frac{1}{16}a^0\plus{}\left (\frac{1}{16a} \right )^0\minus{} \left (64^{\minus{}\frac{1}{2}} \right )\minus{} (\minus{}32)^{\minus{}\frac{4}{5}} is:

<span class='latex-bold'>(A)</span>\ 1 \frac{13}{16} \qquad <span class='latex-bold'>(B)</span>\ 1 \frac{3}{16} \qquad <span class='latex-bold'>(C)</span>\ 1 \qquad <span class='latex-bold'>(D)</span>\ \frac{7}{8} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{16}

1

Inscribing a Hexagon in a Circle

A regular hexagon is inscribed in a circle of radius

10

inches. Its area is:

<span class='latex-bold'>(A)</span>\ 150\sqrt{3} \text{ sq. in.} \qquad <span class='latex-bold'>(B)</span>\ \text{150 sq. in.} \qquad <span class='latex-bold'>(C)</span>\ 25\sqrt{3} \text{ sq. in.} \qquad <span class='latex-bold'>(D)</span>\ \text{600 sq. in.} \qquad <span class='latex-bold'>(E)</span>\ 300\sqrt{3} \text{ sq. in.}

1

"Highest Common Divisor"

If the Highest Common Divisor of

6432

132

is diminished by

8

, it will equal: (A)\ \minus{}6 \qquad (B)\ 6 \qquad (C)\ \minus{}2 \qquad (D)\ 3 \qquad (E)\ 4

1

Proportions

x

varies as the cube of

y

y

varies as the fifth root of

z

x

n

z

n

<span class='latex-bold'>(A)</span>\ \frac{1}{15} \qquad <span class='latex-bold'>(B)</span>\ \frac{5}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{3}{5} \qquad <span class='latex-bold'>(D)</span>\ 15 \qquad <span class='latex-bold'>(E)</span>\ 8

1

Eliminating Fractions

The equation \frac{2x^2}{x\minus{}1}\minus{}\frac{2x\plus{}7}{3}\plus{}\frac{4\minus{}6x}{x\minus{}1}\plus{}1\equal{}0 can be transformed by eliminating fractions to the equation x^2\minus{}5x\plus{}4\equal{}0. The roots of the latter equation are

4

1

. Then the roots of the first equation are:

<span class='latex-bold'>(A)</span>\ 4 \text{ and }1 \qquad <span class='latex-bold'>(B)</span>\ \text{only }1 \qquad <span class='latex-bold'>(C)</span>\ \text{only }4 \qquad <span class='latex-bold'>(D)</span>\ \text{neither 4 nor 1} \qquad <span class='latex-bold'>(E)</span>\ \text{4 and some other root}

1

Squaring an Expression

The square of 5\minus{}\sqrt{y^2\minus{}25} is: (A)\ y^2\minus{}5\sqrt{y^2\minus{}25} \qquad (B)\ \minus{}y^2 \qquad (C)\ y^2 \\ (D)\ (5\minus{}y)^2 \qquad (E)\ y^2\minus{}10\sqrt{y^2\minus{}25}