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Difference between revisions of "1966 AHSME Problems/Problem 22"

(Created page with "== Problem == Consider the statements: (I)<math>\sqrt{a^2+b^2}=0</math>, (II) <math>\sqrt{a^2+b^2}=ab</math>, (III) <math>\sqrt{a^2+b^2}=a+b</math>, (IV) <math>\sqrt{a^2+b^...")
 
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== Problem ==
 
== Problem ==
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Consider the statements: (I)<math>\sqrt{a^2+b^2}=0</math>,  (II) <math>\sqrt{a^2+b^2}=ab</math>,  (III) <math>\sqrt{a^2+b^2}=a+b</math>,  (IV) <math>\sqrt{a^2+b^2}=a\cdot b</math>, where we allow <math>a</math> and <math>b</math> to be real or complex numbers. Those statements for which there exist solutions other than <math>a=0</math> and <math>b=0</math>, are:
+
Consider the statements: (I)<math>\sqrt{a^2+b^2}=0</math>,  (II) <math>\sqrt{a^2+b^2}=ab</math>,  (III) <math>\sqrt{a^2+b^2}=a+b</math>,  (IV) <math>\sqrt{a^2+b^2}=a - b</math>, where we allow <math>a</math> and <math>b</math> to be real or complex numbers. Those statements for which there exist solutions other than <math>a=0</math> and <math>b=0</math>, are:
  
 
<math>\text{(A)  (I),(II),(III),(IV)}\quad \text{(B)  (II),(III),(IV) only} \quad \text{(C)  (I),(III),(IV) only}\quad \text{(D)  (III),(IV) only} \quad \text{(E)  (I) only}</math>
 
<math>\text{(A)  (I),(II),(III),(IV)}\quad \text{(B)  (II),(III),(IV) only} \quad \text{(C)  (I),(III),(IV) only}\quad \text{(D)  (III),(IV) only} \quad \text{(E)  (I) only}</math>
  
 
== Solution ==
 
== Solution ==
−
 
+
<math>\fbox{A}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 21:10, 11 March 2024

Problem

Consider the statements: (I)$\sqrt{a^2+b^2}=0$, (II) $\sqrt{a^2+b^2}=ab$, (III) $\sqrt{a^2+b^2}=a+b$, (IV) $\sqrt{a^2+b^2}=a - b$, where we allow $a$ and $b$ to be real or complex numbers. Those statements for which there exist solutions other than $a=0$ and $b=0$, are:

$\text{(A) (I),(II),(III),(IV)}\quad \text{(B) (II),(III),(IV) only} \quad \text{(C) (I),(III),(IV) only}\quad \text{(D) (III),(IV) only} \quad \text{(E) (I) only}$

Solution

$\fbox{A}$

See also

1966 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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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