Difference between revisions of "2022 AMC 10A Problems/Problem 6"

Revision as of 16:21, 3 July 2025

Problem

Which expression is equal to $\left|a-2-\sqrt{(a-1)^2}\right|$ for $a<0?$

$\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

Solution 1

We have $\begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|a-2-|a-1|\right| \\ &=\left|a-2-(1-a)\right| \\ &=\left|2a-3\right| \\ &=\boxed{\textbf{(A) } 3-2a}. \end{align*}$ ~MRENTHUSIASM

Solution 2

Assume that $a=-1.$ Then, the given expression simplifies to $5$ : $\begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|-1-2-\sqrt{(-1-1)^2}\right| \\ &= \left|-1-2-\sqrt{4}\right| \\ &= \left|-1-2-2\right| \\ &= 5. \end{align*}$ Then, we test each of the answer choices to see which one is equal to $5$ :

$\textbf{(A) } 3-2a = 3-2\cdot(-1) = 3+2 = 5.$

$\textbf{(B) } 1-a = 1-(-1) = 2 \neq 5.$

$\textbf{(C) } 1 \neq 5.$

$\textbf{(D) } a+1 = -1+1 = 0 \neq 5.$

$\textbf{(E) } 3 \neq 5.$

The only answer choice equal to $5$ for $a=-1$ is $\boxed{\textbf{(A) } 3-2a}.$

-MathWizard09

Solution 3

The given function is continuous, so assume that $a=0.$ Then, the given expression simplifies to $3.$

We test each of the answer choices and get $\textbf{(A) } 3-2a$ or $\textbf{(E) } 3.$

We test $x = - 1000$ and get $\left|-1000-2- \text{positive} \right| \ne 3 \implies \boxed{\textbf{(A) } 3-2a}.$

vladimir.shelomovskii@gmail.com, vvsss

Solution 4

We know that ,

$\begin{aligned} & |x|=x, \text { if } x>0 \\ & =-x, \text { if } x<0 \\ & \text { Now, }\left|a-2-\sqrt{(a-1)^2}\right|=\left|a-2-\sqrt{(1-a)^2}\right|, \text { as } a<0 \\ & =|a-2-(1-a)| \\ & =|2 a-3| \\ & =-(2 a-3) \text { as } a<0 \\ & =3-2 a\end{aligned}$ \

So, the correct choice is option (A).

~KENJAKURA

Video Solution 1 (Quick and Easy)

https://youtu.be/ZWHzdrW4rvw

~Education, the Study of Everything

Video Solution 2

https://youtu.be/XWTTtL9kW98

@@ Line 52: / Line 52: @@
 We know that ,
-\(\begin{aligned} & |x|=x, \text { if } x>0 \\ & =-x, \text { if } x<0 \\ & \text { Now, }\left|a-2-\sqrt{(a-1)^2}\right|=\left|a-2-\sqrt{(1-a)^2}\right|, \text { as }  a<0 \\ & =|a-2-(1-a)| \\ & =|2 a-3| \\ & =-(2 a-3) \text { as } a<0 \\ & =3-2 a\end{aligned}\) \\
+\(\begin{aligned} & |x|=x, \text { if } x>0 \\ & =-x, \text { if } x<0 \\ & \text { Now, }\left|a-2-\sqrt{(a-1)^2}\right|=\left|a-2-\sqrt{(1-a)^2}\right|, \text { as }  a<0 \\ & =|a-2-(1-a)| \\ & =|2 a-3| \\ & =-(2 a-3) \text { as } a<0 \\ & =3-2 a\end{aligned}\) \
 So, the correct choice is option (A).

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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