Difference between revisions of "1960 AHSME Problems"

(Created page with "== Problem 1== If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: <math>\textbf{(A)}10\qquad \textbf{(B )}9 \qquad \textbf{(C )}2\qq...")
 
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== Problem 9==
 
== Problem 9==
  
The fraction <math>\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}</math> is (with suitable restrictions of the values of <math>a, b</math>, and <math>c</math>):
+
The fraction <math>\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}</math> is (with suitable restrictions of the values of a, b, and c):
  
<math>\text{(A) irreducible}\qquad
+
<math>\text{(A) irreducible}\qquad</math>
\text{(B) reducible to negative 1}\qquad
+
 
\text{(C) reducible to a polynomial of three terms}\qquad
+
<math>\text{(B) reducible to negative 1}\qquad</math>
\text{(D) reducible to}\frac{a-b+c}{a+b-c}\qquad
+
 
\text{(E) reducible to}\frac{a+b-c}{a-b+c} </math>  
+
<math>\text{(C) reducible to a polynomial of three terms}\qquad</math>
 
+
 
 +
<math>\text{(D) reducible to} \frac{a-b+c}{a+b-c}\qquad</math>
 +
 
 +
<math>\text{(E) reducible to} \frac{a+b-c}{a-b+c}</math>
  
 
[[1960 AHSME Problems/Problem 9|Solution]]
 
[[1960 AHSME Problems/Problem 9|Solution]]
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Given the following six statements:
 
Given the following six statements:
<math>\text{(1) All women are good drivers}\
+
<cmath>\text{(1) All women are good drivers} \
\text{(2) Some women are good drivers}\
+
\text{(2) Some women are good drivers} \
\text{(3) No men are good drivers}\
+
\text{(3) No men are good drivers} \
\text{(4) All men are bad drivers}\
+
\text{(4) All men are bad drivers} \
\text{(5) At least one man is a bad driver}\
+
\text{(5) At least one man is a bad driver} \
\text{(6) All men are good drivers.}</math><math>
+
\text{(6) All men are good drivers.}</cmath>
 +
 
 +
 
 +
The statement that negates statement <math>(6)</math> is:
  
The statement that negates statement </math>\text{(6)}<math> is:
 
  
</math>\textbf{(A)}(1)\qquad
+
<math>\textbf{(A )}(1)\qquad
 
\textbf{(B )}(2)\qquad
 
\textbf{(B )}(2)\qquad
 
\textbf{(C )}(3)\qquad
 
\textbf{(C )}(3)\qquad
 
\textbf{(D )}(4)\qquad
 
\textbf{(D )}(4)\qquad
\textbf{(E )}(5)     <math>
+
\textbf{(E )}(5)</math>
 
    
 
    
  
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== Problem 11==
 
== Problem 11==
  
For a given value of </math>k<math> the product of the roots of </math>x^2-3kx+2k^2-1=0<math>
+
For a given value of <math>k</math> the product of the roots of <math>x^2-3kx+2k^2-1=0</math>
is </math>7<math>. The roots may be characterized as:
+
is <math>7</math>. The roots may be characterized as:
  
</math>\textbf{(A)}\text{integral and positive} \qquad
+
<math>\textbf{(A)}\text{integral and positive} \qquad
 
\textbf{(B )}\text{integral and negative} \qquad
 
\textbf{(B )}\text{integral and negative} \qquad
 
\textbf{(C )}\text{rational, but not integral} \qquad
 
\textbf{(C )}\text{rational, but not integral} \qquad
 
\textbf{(D )}\text{irrational} \qquad
 
\textbf{(D )}\text{irrational} \qquad
\textbf{(E )} \text{imaginary}    <math>
+
\textbf{(E )} \text{imaginary}    </math>
 
    
 
    
  
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== Problem 12==
 
== Problem 12==
  
The locus of the centers of all circles of given radius </math>a<math>, in the same plane, passing through a fixed point, is:
+
The locus of the centers of all circles of given radius <math>a</math>, in the same plane, passing through a fixed point, is:
  
</math>\textbf{(A)}\text{a point}\qquad
+
<math>\textbf{(A)}\text{a point}\qquad
 
\textbf{(B )}\text{ a straight line}\qquad
 
\textbf{(B )}\text{ a straight line}\qquad
 
\textbf{(C )}\text{two straight lines}\qquad
 
\textbf{(C )}\text{two straight lines}\qquad
 
\textbf{(D )}\text{a circle}\qquad  
 
\textbf{(D )}\text{a circle}\qquad  
\textbf{(E )}\text{two circles}    <math>
+
\textbf{(E )}\text{two circles}    </math>
 
    
 
    
  
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== Problem 13==
 
== Problem 13==
  
The polygon(s) formed by </math>y=3x+2, y=-3x+2<math>, and </math>y=-2<math>, is (are):
+
The polygon(s) formed by <math>y=3x+2, y=-3x+2</math>, and <math>y=-2</math>, is (are):
  
</math>\textbf{(A)}\text{An equilateral triangle}\qquad
+
<math>\textbf{(A)}\text{An equilateral triangle}\qquad
 
\textbf{(B )}\text{an isosceles triangle}\qquad
 
\textbf{(B )}\text{an isosceles triangle}\qquad
 
\textbf{(C )}\text{a right triangle}\qquad  
 
\textbf{(C )}\text{a right triangle}\qquad  
 
\textbf{(D )}\text{a triangle and a trapezoid}\qquad
 
\textbf{(D )}\text{a triangle and a trapezoid}\qquad
\textbf{(E )}\text{a quadrilateral}    <math>
+
\textbf{(E )}\text{a quadrilateral}    </math>
 
    
 
    
  
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== Problem 14==
 
== Problem 14==
  
If </math>a<math> and </math>b<math> are real numbers, the equation </math>3x-5+a=bx+1<math> has a unique solution </math>x<math> [The symbol </math>a \neq 0<math> means that </math>a<math> is different from zero]:
+
If <math>a</math> and <math>b</math> are real numbers, the equation <math>3x-5+a=bx+1</math> has a unique solution <math>x</math> [The symbol <math>a \neq 0</math> means that <math>a</math> is different from zero]:
  
</math>\textbf{(A)}\text{for all a and b} \qquad
+
<math>\textbf{(A)}\text{for all a and b} \qquad
 
\textbf{(B )}\text{if a }\neq\text{2b}\qquad
 
\textbf{(B )}\text{if a }\neq\text{2b}\qquad
 
\textbf{(C )}\text{if a }\neq 6\qquad
 
\textbf{(C )}\text{if a }\neq 6\qquad
 
\textbf{(D )}\text{if b }\neq 0\qquad
 
\textbf{(D )}\text{if b }\neq 0\qquad
\textbf{(E )}\text{if b }\neq 3    <math>
+
\textbf{(E )}\text{if b }\neq 3    </math>
 
    
 
    
  
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== Problem 15==
 
== Problem 15==
  
Triangle </math>I<math> is equilateral with side </math>A<math>, perimeter </math>P<math>, area </math>K<math>, and circumradius </math>R<math> (radius of the circumscribed circle).  
+
Triangle <math>I</math> is equilateral with side <math>A</math>, perimeter <math>P</math>, area <math>K</math>, and circumradius <math>R</math> (radius of the circumscribed circle).  
Triangle </math>II<math> is equilateral with side </math>a<math>, perimeter </math>p<math>, area </math>k<math>, and circumradius </math>r<math>. If </math>A<math> is different from </math>a<math>, then:  
+
Triangle <math>II</math> is equilateral with side <math>a</math>, perimeter <math>p</math>, area <math>k</math>, and circumradius <math>r</math>. If <math>A</math> is different from <math>a</math>, then:  
  
</math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad
+
<math>\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad
 
\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad
 
\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad
 
\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad
 
\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad
 
\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad
 
\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad
\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}    <math>
+
\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}    </math>
 
    
 
    
  
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== Problem 16==
 
== Problem 16==
  
In the numeration system with base </math>5<math>, counting is as follows: </math>1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\ldots<math>.
+
In the numeration system with base <math>5</math>, counting is as follows: <math>1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\ldots</math>.
The number whose description in the decimal system is </math>69<math>, when described in the base </math>5<math> system, is a number with:
+
The number whose description in the decimal system is <math>69</math>, when described in the base <math>5</math> system, is a number with:
  
</math>\textbf{(A)}\ \text{two consecutive digits} \qquad
+
<math>\textbf{(A)}\ \text{two consecutive digits} \qquad
 
\textbf{(B)}\ \text{two non-consecutive digits} \qquad
 
\textbf{(B)}\ \text{two non-consecutive digits} \qquad
 
\textbf{(C)}\ \text{three consecutive digits} \qquad
 
\textbf{(C)}\ \text{three consecutive digits} \qquad
 
\textbf{(D)}\ \text{three non-consecutive digits} \qquad
 
\textbf{(D)}\ \text{three non-consecutive digits} \qquad
\textbf{(E)}\ \text{four digits}    <math>
+
\textbf{(E)}\ \text{four digits}    </math>
 
    
 
    
  
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== Problem 17==
 
== Problem 17==
  
The formula </math>N=8 \times 10^{8} \times x^{-3/2}<math> gives, for a certain group, the number of individuals whose income exceeds </math>x<math> dollars.  
+
The formula <math>N=8 \times 10^{8} \times x^{-3/2}</math> gives, for a certain group, the number of individuals whose income exceeds <math>x</math> dollars.  
The lowest income, in dollars, of the wealthiest </math>800<math> individuals is at least:
+
The lowest income, in dollars, of the wealthiest <math>800</math> individuals is at least:
  
</math>\textbf{(A)}\ 10^4\qquad
+
<math>\textbf{(A)}\ 10^4\qquad
 
\textbf{(B)}\ 10^6\qquad
 
\textbf{(B)}\ 10^6\qquad
 
\textbf{(C)}\ 10^8\qquad
 
\textbf{(C)}\ 10^8\qquad
 
\textbf{(D)}\ 10^{12} \qquad
 
\textbf{(D)}\ 10^{12} \qquad
\textbf{(E)}\ 10^{16}    <math>
+
\textbf{(E)}\ 10^{16}    </math>
 
    
 
    
  
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== Problem 18==
 
== Problem 18==
  
The pair of equations </math>3^{x+y}=81<math> and </math>81^{x-y}=3<math> has:
+
The pair of equations <math>3^{x+y}=81</math> and <math>81^{x-y}=3</math> has:
  
</math>\textbf{(A)}\ \text{no common solution} \qquad
+
<math>\textbf{(A)}\ \text{no common solution} \qquad
 
\textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad
 
\textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad
 
\textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad
 
\textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad
 
\textbf{(D)}\text{ a common solution in positive and negative integers} \qquad
 
\textbf{(D)}\text{ a common solution in positive and negative integers} \qquad
\textbf{(E)}\ \text{none of these}    <math>
+
\textbf{(E)}\ \text{none of these}    </math>
 
    
 
    
  
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== Problem 19==
 
== Problem 19==
  
Consider equation </math>I: x+y+z=46<math> where </math>x, y<math>, and </math>z<math> are positive integers, and equation </math>II: x+y+z+w=46<math>,  
+
Consider equation <math>I: x+y+z=46</math> where <math>x, y</math>, and <math>z</math> are positive integers, and equation <math>II: x+y+z+w=46</math>,  
where </math>x, y, z<math>, and </math>w<math> are positive integers. Then
+
where <math>x, y, z</math>, and <math>w</math> are positive integers. Then
  
</math>\text{(A)  I can be solved in consecutive integers} \qquad
+
<math>\text{(A)  I can be solved in consecutive integers} \qquad
 
\text{(B) I can be solved in consecutive even integers} \qquad
 
\text{(B) I can be solved in consecutive even integers} \qquad
 
\text{(C) II can be solved in consecutive integers} \qquad
 
\text{(C) II can be solved in consecutive integers} \qquad
 
\text{(D) II can be solved in consecutive even integers} \qquad
 
\text{(D) II can be solved in consecutive even integers} \qquad
\text{(E) II can be solved in consecutive odd integers}    <math>
+
\text{(E) II can be solved in consecutive odd integers}    </math>
 
    
 
    
  
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== Problem 20==
 
== Problem 20==
  
The coefficient of </math>x^7<math> in the expansion of </math>(\frac{x^2}{2}-\frac{2}{x})^8<math> is:
+
The coefficient of <math>x^7</math> in the expansion of <math>(\frac{x^2}{2}-\frac{2}{x})^8</math> is:
  
</math>\textbf{(A)}\ 56\qquad
+
<math>\textbf{(A)}\ 56\qquad
 
\textbf{(B)}\ -56\qquad
 
\textbf{(B)}\ -56\qquad
 
\textbf{(C)}\ 14\qquad
 
\textbf{(C)}\ 14\qquad
 
\textbf{(D)}\ -14\qquad
 
\textbf{(D)}\ -14\qquad
\textbf{(E)}\ 0    <math>
+
\textbf{(E)}\ 0    </math>
 
    
 
    
  
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== Problem 21==
 
== Problem 21==
  
The diagonal of square </math>I<math> is </math>a+b<math>. The perimeter of square </math>II<math> with twice the area of </math>I<math> is:
+
The diagonal of square <math>I</math> is <math>a+b</math>. The perimeter of square <math>II</math> with twice the area of <math>I</math> is:
  
</math>\textbf{(A)}\ (a+b)^2\qquad
+
<math>\textbf{(A)}\ (a+b)^2\qquad
 
\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad
 
\textbf{(B)}\ \sqrt{2}(a+b)^2\qquad
 
\textbf{(C)}\ 2(a+b)\qquad
 
\textbf{(C)}\ 2(a+b)\qquad
 
\textbf{(D)}\ \sqrt{8}(a+b) \qquad
 
\textbf{(D)}\ \sqrt{8}(a+b) \qquad
\textbf{(E)}\ 4(a+b)    <math>
+
\textbf{(E)}\ 4(a+b)    </math>
 
    
 
    
  
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== Problem 22==
 
== Problem 22==
  
The equality </math>(x+m)^2-(x+n)^2=(m-n)^2<math>, where </math>m<math> and </math>n<math> are unequal non-zero constants, is satisfied by </math>x=am+bn<math>, where:
+
The equality <math>(x+m)^2-(x+n)^2=(m-n)^2</math>, where <math>m</math> and <math>n</math> are unequal non-zero constants, is satisfied by <math>x=am+bn</math>, where:
  
</math>\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad
+
<math>\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad
 
\textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad
 
\textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad
 
\textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad
 
\textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad
 
\textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad
 
\textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad
\textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}    <math>
+
\textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}    </math>
 
    
 
    
  
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== Problem 23==
 
== Problem 23==
  
The radius </math>R<math> of a cylindrical box is </math>8<math> inches, the height </math>H<math> is </math>3<math> inches.  
+
The radius <math>R</math> of a cylindrical box is <math>8</math> inches, the height <math>H</math> is <math>3</math> inches.  
The volume </math>V = \pi R^2H<math> is to be increased by the same fixed positive amount when </math>R<math>  
+
The volume <math>V = \pi R^2H</math> is to be increased by the same fixed positive amount when <math>R</math>  
is increased by </math>x<math> inches as when </math>H<math> is increased by </math>x<math> inches. This condition is satisfied by:
+
is increased by <math>x</math> inches as when <math>H</math> is increased by <math>x</math> inches. This condition is satisfied by:
  
</math>\textbf{(A)}\ \text{no real value of} \text{ } x\qquad
+
<math>\textbf{(A)}\ \text{no real value of} \text{ } x\qquad
 
\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad
 
\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad
 
\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad
 
\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad
 
\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad
 
\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad
\textbf{(E)}\ \text{two real values of} \text{ } x    <math>
+
\textbf{(E)}\ \text{two real values of} \text{ } x    </math>
 
    
 
    
  
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== Problem 24==
 
== Problem 24==
  
If </math>\log_{2x}216 = x<math>, where </math>x<math> is real, then </math>x<math> is:
+
If <math>\log_{2x}216 = x</math>, where <math>x</math> is real, then <math>x</math> is:
  
</math>\textbf{(A)}\ \text{A non-square, non-cube integer} \qquad
+
<math>\textbf{(A)}\ \text{A non-square, non-cube integer} \qquad
 
\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad
 
\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad
 
\textbf{(C)}\ \text{An irrational number} \qquad
 
\textbf{(C)}\ \text{An irrational number} \qquad
 
\textbf{(D)}\ \text{A perfect square}\qquad
 
\textbf{(D)}\ \text{A perfect square}\qquad
\textbf{(E)}\ \text{A perfect cube}    <math>
+
\textbf{(E)}\ \text{A perfect cube}    </math>
 
    
 
    
  
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== Problem 25==
 
== Problem 25==
  
Let </math>m<math> and </math>n<math> be any two odd numbers, with </math>n<math> less than </math>m<math>.  
+
Let <math>m</math> and <math>n</math> be any two odd numbers, with <math>n</math> less than <math>m</math>.  
The largest integer which divides all possible numbers of the form </math>m^2-n^2<math> is:
+
The largest integer which divides all possible numbers of the form <math>m^2-n^2</math> is:
  
</math>\textbf{(A)}\ 2\qquad
+
<math>\textbf{(A)}\ 2\qquad
 
\textbf{(B)}\ 4\qquad
 
\textbf{(B)}\ 4\qquad
 
\textbf{(C)}\ 6\qquad
 
\textbf{(C)}\ 6\qquad
 
\textbf{(D)}\ 8\qquad
 
\textbf{(D)}\ 8\qquad
\textbf{(E)}\ 16    <math>
+
\textbf{(E)}\ 16    </math>
 
    
 
    
  
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== Problem 26==
 
== Problem 26==
  
Find the set of </math>x<math>-values satisfying the inequality </math>|\frac{5-x}{3}|<2<math>. [The symbol </math>|a|<math> means </math>+a<math> if </math>a<math> is positive,  
+
Find the set of <math>x</math>-values satisfying the inequality <math>|\frac{5-x}{3}|<2</math>. [The symbol <math>|a|</math> means <math>+a</math> if <math>a</math> is positive,  
</math>-a<math> if </math>a<math> is negative,</math>0<math> if </math>a<math> is zero. The notation </math>1<a<2<math> means that a can have any value between </math>1<math> and </math>2<math>, excluding </math>1<math> and </math>2<math>. ]
+
<math>-a</math> if <math>a</math> is negative,<math>0</math> if <math>a</math> is zero. The notation <math>1<a<2</math> means that a can have any value between <math>1</math> and <math>2</math>, excluding <math>1</math> and <math>2</math>. ]
  
</math>\textbf{(A)}\ 1 < x < 11\qquad
+
<math>\textbf{(A)}\ 1 < x < 11\qquad
 
\textbf{(B)}\ -1 < x < 11\qquad
 
\textbf{(B)}\ -1 < x < 11\qquad
 
\textbf{(C)}\ x< 11\qquad
 
\textbf{(C)}\ x< 11\qquad
 
\textbf{(D)}\ x>11\qquad
 
\textbf{(D)}\ x>11\qquad
\textbf{(E)}\ |x| < 6    <math>
+
\textbf{(E)}\ |x| < 6    </math>
 
    
 
    
  
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== Problem 27==
 
== Problem 27==
  
Let </math>S<math> be the sum of the interior angles of a polygon </math>P<math> for which each interior angle is </math>7\frac{1}{2}<math> times the  
+
Let <math>S</math> be the sum of the interior angles of a polygon <math>P</math> for which each interior angle is <math>7\frac{1}{2}</math> times the  
 
exterior angle at the same vertex. Then
 
exterior angle at the same vertex. Then
  
</math>\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad
+
<math>\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad
 
\textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad
 
\textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad
 
\textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad
 
\textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad
 
\textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad
 
\textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad
\textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular}    <math>
+
\textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular}    </math>
 
    
 
    
  
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== Problem 28==
 
== Problem 28==
  
The equation </math>x-\frac{7}{x-3}=3-\frac{7}{x-3}<math> has:
+
The equation <math>x-\frac{7}{x-3}=3-\frac{7}{x-3}</math> has:
  
</math>\textbf{(A)}\ \text{infinitely many integral roots}\qquad
+
<math>\textbf{(A)}\ \text{infinitely many integral roots}\qquad
 
\textbf{(B)}\ \text{no root}\qquad
 
\textbf{(B)}\ \text{no root}\qquad
 
\textbf{(C)}\ \text{one integral root}\qquad  
 
\textbf{(C)}\ \text{one integral root}\qquad  
 
\textbf{(D)}\ \text{two equal integral roots} \qquad
 
\textbf{(D)}\ \text{two equal integral roots} \qquad
\textbf{(E)}\ \text{two equal non-integral roots}    <math>
+
\textbf{(E)}\ \text{two equal non-integral roots}    </math>
 
    
 
    
  
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== Problem 29==
 
== Problem 29==
  
Five times </math>A<math>'s money added to </math>B<math>'s money is more than </math>\texdollar{51.00}<math>. Three times </math>A<math>'s money minus </math>B<math>'s money is </math>\textdollar{21.00}<math>.
+
Five times <math>A</math>'s money added to <math>B</math>'s money is more than <math>\texdollar{51.00}</math>. Three times <math>A</math>'s money minus <math>B</math>'s money is <math>\textdollar{21.00}</math>.
If </math>a<math> represents </math>A<math>'s money in dollars and </math>b<math> represents </math>B<math>'s money in dollars, then:
+
If <math>a</math> represents <math>A</math>'s money in dollars and <math>b</math> represents <math>B</math>'s money in dollars, then:
  
</math>\textbf{(A)}\ a>9, b>6 \qquad
+
<math>\textbf{(A)}\ a>9, b>6 \qquad
 
\textbf{(B)}\ a>9, b<6 \qquad
 
\textbf{(B)}\ a>9, b<6 \qquad
 
\textbf{(C)}\ a>9, b=6\qquad
 
\textbf{(C)}\ a>9, b=6\qquad
 
\textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad
 
\textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad
\textbf{(E)}\ 2a=3b    <math>
+
\textbf{(E)}\ 2a=3b    </math>
 
    
 
    
  
Line 399: Line 404:
 
== Problem 30==
 
== Problem 30==
  
Given the line </math>3x+5y=15<math> and a point on this line equidistant from the coordinate axes. Such a point exists in:
+
Given the line <math>3x+5y=15</math> and a point on this line equidistant from the coordinate axes. Such a point exists in:
  
</math>\textbf{(A)}\ \text{none of the quadrants}\qquad
+
<math>\textbf{(A)}\ \text{none of the quadrants}\qquad
 
\textbf{(B)}\ \text{quadrant I only}\qquad
 
\textbf{(B)}\ \text{quadrant I only}\qquad
 
\textbf{(C)}\ \text{quadrants I, II only}\qquad  
 
\textbf{(C)}\ \text{quadrants I, II only}\qquad  
 
\textbf{(D)}\ \text{quadrants I, II, III only} \qquad
 
\textbf{(D)}\ \text{quadrants I, II, III only} \qquad
\textbf{(E)}\ \text{each of the quadrants}    <math>
+
\textbf{(E)}\ \text{each of the quadrants}    </math>
 
    
 
    
  
Line 412: Line 417:
 
== Problem 31==
 
== Problem 31==
  
For </math>x^2+2x+5<math> to be a factor of </math>x^4+px^2+q<math>, the values of </math>p<math> and </math>q<math> must be, respectively:
+
For <math>x^2+2x+5</math> to be a factor of <math>x^4+px^2+q</math>, the values of <math>p</math> and <math>q</math> must be, respectively:
  
</math>\textbf{(A)}\ -2, 5\qquad
+
<math>\textbf{(A)}\ -2, 5\qquad
 
\textbf{(B)}\ 5, 25\qquad
 
\textbf{(B)}\ 5, 25\qquad
 
\textbf{(C)}\ 10, 20\qquad
 
\textbf{(C)}\ 10, 20\qquad
 
\textbf{(D)}\ 6, 25\qquad
 
\textbf{(D)}\ 6, 25\qquad
\textbf{(E)}\ 14, 25    <math>
+
\textbf{(E)}\ 14, 25    </math>
 
    
 
    
  
Line 425: Line 430:
 
== Problem 32==
 
== Problem 32==
  
In this figure the center of the circle is </math>O<math>. </math>AB \perp BC<math>, </math>ADOE<math> is a straight line, </math>AP = AD<math>, and </math>AB$ has a length twice the radius. Then:
+
In this figure the center of the circle is <math>O</math>. <math>AB \perp BC</math>, <math>ADOE</math> is a straight line, <math>AP = AD</math>, and <math>AB</math> has a length twice the radius. Then:
  
 
<asy>
 
<asy>

Revision as of 00:21, 11 October 2014

Problem 1

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

$\textbf{(A)}10\qquad \textbf{(B )}9 \qquad \textbf{(C )}2\qquad \textbf{(D )}-2\qquad \textbf{(E )}-9$

Solution

Problem 2

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?

$\textbf{(A)}9\frac{1}{5}\qquad \textbf{(B )}10\qquad \textbf{(C )}11\qquad \textbf{(D )}14\frac{2}{5}\qquad \textbf{(E )}\text{none of these}$


Solution

Problem 3

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

$\textbf{(A)}0\qquad \textbf{(B )}144\qquad \textbf{(C )}256\qquad \textbf{(D )}400\qquad \textbf{(E )}416$


Solution

Problem 4

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:

$\textbf{(A)}8\sqrt{3}\qquad \textbf{(B )}8\qquad \textbf{(C )}4\sqrt{3}\qquad \textbf{(D )}4\qquad \textbf{(E )}2\sqrt{3}$


Solution

Problem 5

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

$\textbf{(A)}\text{infinitely many}\qquad \textbf{(B )}\text{four}\qquad \textbf{(C )}\text{two}\qquad \textbf{(D )}\text{one}\qquad \textbf{(E )}\text{none}$


Solution

Problem 6

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

$\textbf{(A)}\frac{25\sqrt{2}}{\pi}\qquad \textbf{(B )}\frac{50\sqrt{2}}{\pi}\qquad \textbf{(C )}\frac{100}{\pi}\qquad \textbf{(D )}\frac{100\sqrt{2}}{\pi}\qquad \textbf{(E )}50\sqrt{2}$


Solution

Problem 7

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:

$\textbf{(A)}8\qquad \textbf{(B )}8\sqrt{2}\qquad \textbf{(C )}8\sqrt{\pi}\qquad \textbf{(D )}16\qquad \textbf{(E )}16\sqrt{2}$


Solution

Problem 8

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:

$\textbf{(A)}7\qquad \textbf{(B)} 29\qquad \textbf{(C )}141\qquad \textbf{(D )}349\qquad \textbf{(E )}\text{none of these}$


Solution

Problem 9

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

$\text{(A) irreducible}\qquad$

$\text{(B) reducible to negative 1}\qquad$

$\text{(C) reducible to a polynomial of three terms}\qquad$

$\text{(D) reducible to} \frac{a-b+c}{a+b-c}\qquad$

$\text{(E) reducible to} \frac{a+b-c}{a-b+c}$

Solution

Problem 10

Given the following six statements: \[\text{(1) All women are good drivers} \\ \text{(2) Some women are good drivers} \\ \text{(3) No men are good drivers} \\ \text{(4) All men are bad drivers} \\ \text{(5) At least one man is a bad driver} \\ \text{(6) All men are good drivers.}\]


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


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


Solution

Problem 11

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:

$\textbf{(A)}\text{integral and positive} \qquad \textbf{(B )}\text{integral and negative} \qquad \textbf{(C )}\text{rational, but not integral} \qquad \textbf{(D )}\text{irrational} \qquad \textbf{(E )} \text{imaginary}$


Solution

Problem 12

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

$\textbf{(A)}\text{a point}\qquad \textbf{(B )}\text{ a straight line}\qquad \textbf{(C )}\text{two straight lines}\qquad \textbf{(D )}\text{a circle}\qquad  \textbf{(E )}\text{two circles}$


Solution

Problem 13

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

$\textbf{(A)}\text{An equilateral triangle}\qquad \textbf{(B )}\text{an isosceles triangle}\qquad \textbf{(C )}\text{a right triangle}\qquad  \textbf{(D )}\text{a triangle and a trapezoid}\qquad \textbf{(E )}\text{a quadrilateral}$


Solution

Problem 14

If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [The symbol $a \neq 0$ means that $a$ is different from zero]:

$\textbf{(A)}\text{for all a and b} \qquad \textbf{(B )}\text{if a }\neq\text{2b}\qquad \textbf{(C )}\text{if a }\neq 6\qquad \textbf{(D )}\text{if b }\neq 0\qquad \textbf{(E )}\text{if b }\neq 3$


Solution

Problem 15

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:

$\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad \textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad \textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad \textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}$


Solution

Problem 16

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:

$\textbf{(A)}\ \text{two consecutive digits} \qquad \textbf{(B)}\ \text{two non-consecutive digits} \qquad \textbf{(C)}\ \text{three consecutive digits} \qquad \textbf{(D)}\ \text{three non-consecutive digits} \qquad \textbf{(E)}\ \text{four digits}$


Solution

Problem 17

The formula $N=8 \times 10^{8} \times x^{-3/2}$ 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:

$\textbf{(A)}\ 10^4\qquad \textbf{(B)}\ 10^6\qquad \textbf{(C)}\ 10^8\qquad \textbf{(D)}\ 10^{12} \qquad \textbf{(E)}\ 10^{16}$


Solution

Problem 18

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

$\textbf{(A)}\ \text{no common solution} \qquad \textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad \textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad \textbf{(D)}\text{ a common solution in positive and negative integers} \qquad \textbf{(E)}\ \text{none of these}$


Solution

Problem 19

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

$\text{(A)  I can be solved in consecutive integers} \qquad \text{(B) I can be solved in consecutive even integers} \qquad \text{(C) II can be solved in consecutive integers} \qquad \text{(D) II can be solved in consecutive even integers} \qquad \text{(E) II can be solved in consecutive odd integers}$


Solution

Problem 20

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

$\textbf{(A)}\ 56\qquad \textbf{(B)}\ -56\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ -14\qquad \textbf{(E)}\ 0$


Solution

Problem 21

The diagonal of square $I$ is $a+b$. The perimeter of square $II$ with twice the area of $I$ is:

$\textbf{(A)}\ (a+b)^2\qquad \textbf{(B)}\ \sqrt{2}(a+b)^2\qquad \textbf{(C)}\ 2(a+b)\qquad \textbf{(D)}\ \sqrt{8}(a+b) \qquad \textbf{(E)}\ 4(a+b)$


Solution

Problem 22

The equality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are unequal non-zero constants, is satisfied by $x=am+bn$, where:

$\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}$


Solution

Problem 23

The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. The volume $V = \pi R^2H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by:

$\textbf{(A)}\ \text{no real value of} \text{ } x\qquad \textbf{(B)}\ \text{one integral value of} \text{ } x\qquad \textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad \textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad \textbf{(E)}\ \text{two real values of} \text{ } x$


Solution

Problem 24

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is:

$\textbf{(A)}\ \text{A non-square, non-cube integer} \qquad \textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad \textbf{(C)}\ \text{An irrational number} \qquad \textbf{(D)}\ \text{A perfect square}\qquad \textbf{(E)}\ \text{A perfect cube}$


Solution

Problem 25

Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$


Solution

Problem 26

Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, $-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ]

$\textbf{(A)}\ 1 < x < 11\qquad \textbf{(B)}\ -1 < x < 11\qquad \textbf{(C)}\ x< 11\qquad \textbf{(D)}\ x>11\qquad \textbf{(E)}\ |x| < 6$


Solution

Problem 27

Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then

$\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad \textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular}$


Solution

Problem 28

The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has:

$\textbf{(A)}\ \text{infinitely many integral roots}\qquad \textbf{(B)}\ \text{no root}\qquad \textbf{(C)}\ \text{one integral root}\qquad  \textbf{(D)}\ \text{two equal integral roots} \qquad \textbf{(E)}\ \text{two equal non-integral roots}$


Solution

Problem 29

Five times $A$'s money added to $B$'s money is more than $\texdollar{51.00}$ (Error compiling LaTeX. Unknown error_msg). Three times $A$'s money minus $B$'s money is $\textdollar{21.00}$. If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:

$\textbf{(A)}\ a>9, b>6 \qquad \textbf{(B)}\ a>9, b<6 \qquad \textbf{(C)}\ a>9, b=6\qquad \textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad \textbf{(E)}\ 2a=3b$


Solution

Problem 30

Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in:

$\textbf{(A)}\ \text{none of the quadrants}\qquad \textbf{(B)}\ \text{quadrant I only}\qquad \textbf{(C)}\ \text{quadrants I, II only}\qquad  \textbf{(D)}\ \text{quadrants I, II, III only} \qquad \textbf{(E)}\ \text{each of the quadrants}$


Solution

Problem 31

For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively:

$\textbf{(A)}\ -2, 5\qquad \textbf{(B)}\ 5, 25\qquad \textbf{(C)}\ 10, 20\qquad \textbf{(D)}\ 6, 25\qquad \textbf{(E)}\ 14, 25$


Solution

Problem 32

In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:

[asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); real e=350,c=55; pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot); path tangent=A--B; pair P=waypoint(tangent,abs(A-D)/abs(A-B)); draw(unitcircle^^C--B--A--E); dot(A^^B^^C^^D^^E^^P,linewidth(2)); label("$O$",O,dir(290)); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,dir(120)); label("$E$",E,SE); label("$P$",P,SW);[/asy]

$\textbf{(A)} AP^2 = PB \times AB\qquad \textbf{(B)}\ AP \times DO = PB \times AD\qquad \textbf{(C)}\ AB^2 = AD \times DE\qquad \textbf{(D)}\ AB \times AD = OB \times AO\qquad \textbf{(E)}\ \text{none of these}$


Solution

Problem 33

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times\ldots \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4,\ldots, 59$. Let $N$ be the number of primes appearing in this sequence. Then $N$ is:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 57\qquad \textbf{(E)}\ 58$


Solution

Problem 34

Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.

$\textbf{(A)}\ 24\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 18$


Solution

Problem 35

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values:

$\textbf{(A)}\ \text{zero}\qquad \textbf{(B)}\ \text{one}\qquad \textbf{(C)}\ \text{two}\qquad \textbf{(D)}\ \text{three}\qquad \textbf{(E)}\ \text{infinitely many}$


Solution

Problem 36

Let $s_1, s_2, s_3$ be the respective sums of $n, 2n, 3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on:

$\textbf{(A)}\ a\text{ }\text{and}\text{ }d\qquad \textbf{(B)}\ d\text{ }\text{and}\text{ }n\qquad \textbf{(C)}\ a\text{ }\text{and}\text{ }n\qquad \textbf{(D)}\ a, d,\text{ }\text{and}\text{ }n\qquad  \textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n$


Solution

Problem 37

The base of a triangle is of length $b$, and the latitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:

$\textbf{(A)}\ \frac{bx}{h}(h-x)\qquad \textbf{(B)}\ \frac{hx}{b}(b-x)\qquad \textbf{(C)}\ \frac{bx}{h}(h-2x)\qquad \textbf{(D)}\ x(b-x)\qquad \textbf{(E)}\ x(h-x)$


Solution

Problem 38

In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\triangle ABC$, in which is inscribed equilateral $\triangle DEF$. Designate $\angle BFD$ by $a$, $\angle ADE$ by $b$, and $\angle FEC$ by $c$. Then:

[asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); pair A=(5,12),B=origin,C=(10,0),D=(5/3,4),E=(10-5*.45,12*.45),F=(6,0); draw(A--B--C--cycle^^D--E--F--cycle); draw(anglemark(E,D,A,1,45)^^anglemark(F,E,C,1,45)^^anglemark(D,F,B,1,45)); label("$b$",(D.x+.2,D.y+.25),dir(30)); label("$c$",(E.x,E.y-.4),S); label("$a$",(F.x-.4,F.y+.1),dir(150)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,dir(150)); label("$E$",E,dir(60)); label("$F$",F,S);[/asy]

$\textbf{(A)}\ b=\frac{a+c}{2}\qquad \textbf{(B)}\ b=\frac{a-c}{2}\qquad \textbf{(C)}\ a=\frac{b-c}{2} \qquad \textbf{(D)}\ a=\frac{b+c}{2}\qquad \textbf{(E)}\ \text{none of these}$


Solution

Problem 39

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be:

$\textbf{(A)}\ \text{both rational}\qquad \textbf{(B)}\ \text{both real but not rational}\qquad \textbf{(C)}\ \text{both not real}\qquad  \textbf{(D)}\ \text{one real, one not real}\qquad \textbf{(E)}\ \text{one real, one not real or both not real}$


Solution

Problem 40

Given right $\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse:

$\textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad \textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad \textbf{(C)}\ 6\sqrt{3}-8\qquad \textbf{(D)}\ \frac{5\sqrt{10}}{6}\qquad  \textbf{(E)}\ \frac{25}{12}$

Solution