Difference between revisions of "1974 AHSME Problems"

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{{AHSME Problems
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==Problem 1==
 
==Problem 1==
 
If <math> x\not=0 </math> or <math> 4 </math> and <math> y\not=0 </math> or <math> 6 </math>, then <math> \frac{2}{x}+\frac{3}{y}=\frac{1}{2} </math> is equivalent to
 
If <math> x\not=0 </math> or <math> 4 </math> and <math> y\not=0 </math> or <math> 6 </math>, then <math> \frac{2}{x}+\frac{3}{y}=\frac{1}{2} </math> is equivalent to

Latest revision as of 13:04, 19 February 2020

1974 AHSME (Answer Key)
Printable version: Wiki | AoPS ResourcesPDF

Instructions

  1. This is a 30-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 5 points for each correct answer, 2 points for each problem left unanswered, and 0 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 90 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

Problem 1

If $x\not=0$ or $4$ and $y\not=0$ or $6$, then $\frac{2}{x}+\frac{3}{y}=\frac{1}{2}$ is equivalent to

$\mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \  } \frac{x}{2}+\frac{y}{3}=2 \qquad$

$\mathrm{(D) \  } \frac{4y}{y-6}=x \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 2

Let $x_1$ and $x_2$ be such that $x_1\not=x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals

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

Solution

Problem 3

The coefficient of $x^7$ in the polynomial expansion of

\[(1+2x-x^2)^4\]

is

$\mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \  } 6 \qquad \mathrm{(D) \  } -12 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 4

What is the remainder when $x^{51}+51$ is divided by $x+1$?

$\mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 49 \qquad \mathrm{(D) \  } 50 \qquad \mathrm{(E) \  }51$

Solution

Problem 5

Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^\circ$ and $\measuredangle ADC=68^\circ$, find $\measuredangle EBC$.

$\mathrm{(A)\ } 66^\circ \qquad \mathrm{(B) \ }68^\circ \qquad \mathrm{(C) \  } 70^\circ \qquad \mathrm{(D) \  } 88^\circ \qquad \mathrm{(E) \  }92^\circ$

Solution

Problem 6

For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$' then

$\mathrm{(A)\ } \text{``*" is commutative but not associative} \qquad$

$\mathrm{(B) \ }\text{``*" is associative but not commutative} \qquad$

$\mathrm{(C) \  } \text{``*" is neither commutative nor associative} \qquad$

$\mathrm{(D) \  } \text{``*" is commutative and associative} \qquad$

$\mathrm{(E) \  }\text{none of these} \qquad$

Solution

Problem 7

A town's population increased by $1,200$ people, and then this new population decreased by $11\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?

$\mathrm{(A)\ } 1,200 \qquad \mathrm{(B) \ }11,200 \qquad \mathrm{(C) \  } 9,968 \qquad \mathrm{(D) \  } 10,000 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 8

What is the smallest prime number dividing the sum $3^{11}+5^{13}$?

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 5 \qquad \mathrm{(D) \  } 3^{11}+5^{13} \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 9

The integers greater than one are arranged in five columns as follows:

\[\begin{tabular}{c c c c c}\ & 2 & 3 & 4 & 5\\ 9 & 8 & 7 & 6 &\ \\ \ & 10 & 11 & 12 & 13\\ 17 & 16 & 15 & 14 &\ \\ \ & . & . & . & .\\ \end{tabular}\]

(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)

In which column will the number $1,000$ fall?

$\mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \  } \text{third} \qquad \mathrm{(D) \  } \text{fourth} \qquad \mathrm{(E) \  }\text{fifth}$

Solution

Problem 10

What is the smallest integral value of $k$ such that

\[2x(kx-4)-x^2+6=0\]

has no real roots?

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

Solution

Problem 11

If $(a, b)$ and $(c, d)$ are two points on the line whose equation is $y=mx+k$, then the distance between $(a, b)$ and $(c, d)$, in terms of $a, c,$ and $m$ is

$\mathrm{(A)\ } |a-c|\sqrt{1+m^2} \qquad \mathrm{(B) \ }|a+c|\sqrt{1+m^2} \qquad \mathrm{(C) \  } \frac{|a-c|}{\sqrt{1+m^2}} \qquad$

$\mathrm{(D) \  } |a-c|(1+m^2) \qquad \mathrm{(E) \  }|a-c|\,|m|$

Solution

Problem 12

If $g(x)=1-x^2$ and $f(g(x))=\frac{1-x^2}{x^2}$ when $x\not=0$, then $f(1/2)$ equals

$\mathrm{(A)\ } 3/4 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \  } 3 \qquad \mathrm{(D) \  } \sqrt{2}/2 \qquad \mathrm{(E) \  }\sqrt{2}$

Solution

Problem 13

Which of the following is equivalent to "If P is true, then Q is false."?

$\mathrm{(A)\ } \text{``P is true or Q is false."} \qquad$

$\mathrm{(B) \ }\text{``If Q is false then P is true."} \qquad$

$\mathrm{(C) \  } \text{``If P is false then Q is true."} \qquad$

$\mathrm{(D) \  } \text{``If Q is true then P is false."} \qquad$

$\mathrm{(E) \  }\text{``If Q is true then P is true."} \qquad$

Solution

Problem 14

Which statement is correct?

$\mathrm{(A)\ } \text{If } x<0, \text{then } x^2>x. \qquad \mathrm{(B) \ } \text{If } x^2>0, \text{then } x>0.$

$\qquad \mathrm{(C) \  } \text{If } x^2>x, \text{then } x>0. \qquad \mathrm{(D) \  } \text{If } x^2>x, \text{then } x<0.$

$\qquad \mathrm{(E) \  }\text{If } x<1, \text{then } x^2<x.$

Solution

Problem 15

If $x<-2$, then $|1-|1+x||$ equals

$\mathrm{(A)\ } 2+x \qquad \mathrm{(B) \ }-2-x \qquad \mathrm{(C) \  } x \qquad \mathrm{(D) \  } -x \qquad \mathrm{(E) \  }-2$

Solution

Problem 16

A circle of radius $r$ is inscribed in a right isosceles triangle, and a circle of radius $R$ is circumscribed about the triangle. Then $R/r$ equals

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

Solution

Problem 17

If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals

$\mathrm{(A)\ } -1024 \qquad \mathrm{(B) \ }-1024i \qquad \mathrm{(C) \  } 0 \qquad \mathrm{(D) \  } 1024 \qquad \mathrm{(E) \  }1024i$

Solution

Problem 18

If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals

$\mathrm{(A)\ } pq \qquad \mathrm{(B) \ }\frac{3p+q}{5} \qquad \mathrm{(C) \  } \frac{1+3pq}{p+q} \qquad \mathrm{(D) \  } \frac{3pq}{1+3pq} \qquad \mathrm{(E) \  }p^2+q^2$

Solution

Problem 19

In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is

[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((.82,0)--(1,1)--(0,.76)--cycle); label("A", (0,0), S); label("B", (1,0), S); label("C", (1,1), N); label("D", (0,1), N); label("M", (0,.76), W); label("N", (.82,0), S);[/asy]

$\mathrm{(A)\ } 2\sqrt{3}-3 \qquad \mathrm{(B) \ }1-\frac{\sqrt{3}}{3} \qquad \mathrm{(C) \  } \frac{\sqrt{3}}{4} \qquad \mathrm{(D) \  } \frac{\sqrt{2}}{3} \qquad \mathrm{(E) \  }4-2\sqrt{3}$

Solution

Problem 20

Let

\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\]

Then

$\mathrm{(A)\ } T<1 \qquad \mathrm{(B) \ }T=1 \qquad \mathrm{(C) \  } 1<T<2 \qquad \mathrm{(D) \  } T>2 \qquad$

$\mathrm{(E) \  }T=\frac{1}{(3-\sqrt{8})(\sqrt{8}-\sqrt{7})(\sqrt{7}-\sqrt{6})(\sqrt{6}-\sqrt{5})(\sqrt{5}-2)}$

Solution

Problem 21

In a geometric series of positive terms the difference between the fifth and fourth terms is $576$, and the difference between the second and first terms is $9$. What is the sum of the first five terms of this series?

$\mathrm{(A)\ } 1061 \qquad \mathrm{(B) \ }1023 \qquad \mathrm{(C) \  } 1024 \qquad \mathrm{(D) \  } 768 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 22

The minimum of $\sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2}$ is attained when $A$ is

$\mathrm{(A)\ } -180^\circ \qquad \mathrm{(B) \ }60^\circ \qquad \mathrm{(C) \  } 120^\circ \qquad \mathrm{(D) \  } 0^\circ \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 23

In the adjoining figure $TP$ and $T'Q$ are parallel tangents to a circle of radius $r$, with $T$ and $T'$ the points of tangency. $PT''Q$ is a third tangent with $T'''$ as a point of tangency. If $TP=4$ and $T'Q=9$ then $r$ is

[asy] unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label("O",O,W); label("T",T,N); label("T'",T1,S); label("T''",T2,NE); label("P",P,NE); label("Q",Q,S); draw(O--T2); label("$r$",midpoint(O--T2),NW); draw(T--P); label("4",midpoint(T--P),N); draw(T1--Q); label("9",midpoint(T1--Q),S); draw(P--Q);[/asy]

$\mathrm{(A)\ } 25/6 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \  } 25/4 \qquad$

$\mathrm{(D) \  } \text{a number other than }25/6, 6, 25/4 \qquad$

$\mathrm{(E) \  }\text{not determinable from the given information}$

Solution

Problem 24

A fair die is rolled six times. The probability of rolling at least a five at least five times is

$\mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{12}{729} \qquad \mathrm{(C) \  } \frac{2}{729} \qquad \mathrm{(D) \  } \frac{3}{729} \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 25

In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to

[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75));[/asy]

$\mathrm{(A)\ } k \qquad \mathrm{(B) \ }\frac{6k}{5} \qquad \mathrm{(C) \  } \frac{9k}{8} \qquad \mathrm{(D) \  } \frac{5k}{4} \qquad \mathrm{(E) \  }2k$

Solution

Problem 26

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is

$\mathrm{(A)\ } 100 \qquad \mathrm{(B) \ }125 \qquad \mathrm{(C) \  } 123 \qquad \mathrm{(D) \  } 30 \qquad \mathrm{(E) \  }\text{none of these}$

Solution

Problem 27

If $f(x)=3x+2$ for all real $x$, then the statement: "$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$" is true when

$\mathrm{(A)}\ b\le a/3\qquad\mathrm{(B)}\ b > a/3\qquad\mathrm{(C)}\ a\le b/3\qquad\mathrm{(D)}\ a > b/3\\ \qquad\mathrm{(E)}\ \text{The statement is never true.}$

Solution

Problem 28

Which of the following is satisfied by all numbers $x$ of the form

\[x=\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{25}}{3^{25}}\]

where $a_1$ is $0$ or $2$, $a_2$ is $0$ or $2$,...,$a_{25}$ is $0$ or $2$?

$\mathrm{(A)\ } 0\le x<1/3 \qquad \mathrm{(B) \ } 1/3\le x<2/3 \qquad \mathrm{(C) \  } 2/3\le x<1 \qquad$

$\mathrm{(D) \  } 0\le x<1/3 \text{ or }2/3\le x<1 \qquad \mathrm{(E) \  }1/2\le x\le 3/4$

Solution

Problem 29

For $p=1, 2, \cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\cdots+S_{10}$ is

$\mathrm{(A)\ } 80000 \qquad \mathrm{(B) \ }80200 \qquad \mathrm{(C) \  } 80400 \qquad \mathrm{(D) \  } 80600 \qquad \mathrm{(E) \  }80800$

Solution

Problem 30

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $R$ is the ratio of the lesser part to the greater part, then the value of

\[R^{[R^{(R^2+R^{-1})}+R^{-1}]}+R^{-1}\]

is

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }2R \qquad \mathrm{(C) \  } R^{-1} \qquad \mathrm{(D) \  } 2+R^{-1} \qquad \mathrm{(E) \  }2+R$

Solution

See also

1974 AHSME (ProblemsAnswer KeyResources)
Preceded by
1973 AHSME
Followed by
1975 AHSME
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
All AHSME Problems and Solutions


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