Harshgmat wrote:

What is the value of a ?

a) \(a^2 = b^2\)

b) b - a = -12

To prove that each statement alone is insufficient, we present a BIFURCATION (algebraic, in this case):

\(? = a\)

\(\left( 1 \right)\,\,{a^2} = {b^2}\)

\(Take\,\,\left\{ \begin{gathered}

{a^2} = {b^2} = 0\,\,\,\,\, \Rightarrow \,\,\,\,? = 0 \hfill \\

{a^2} = {b^2} = 1\,\,\,\,\, \Rightarrow \,\,\,\,? \ne 0\,\,\,\,\,\,\,\,\left( {a = \pm \,1} \right) \hfill \\

\end{gathered} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{INSUF}}{\text{.}}\)

\(\left( 2 \right)\,\,a\, - b = 12\)

\(Take\,\,\left\{ \begin{gathered}

b = 0\,\,\,\,\, \Rightarrow \,\,\,\,? = 12 \hfill \\

b = 1\,\,\,\, \Rightarrow \,\,\,\,? \ne 12\,\,\,\,\,\,\,\,\left( {a = \,13} \right) \hfill \\

\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{INSUF}}{\text{.}}\)

\(\left( {1 + 2} \right)\,\,\,\,\,\left\{ \begin{gathered}

{a^2} - {b^2} = 0\,\,\,\,\, \hfill \\

a - b = 12 \hfill \\

\end{gathered} \right. \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered}

\left( {a + b} \right)\left( {a - b} \right) = 0 \hfill \\

a - b = 12 \hfill \\

\end{gathered} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \begin{gathered}

a + b = 0 \hfill \\

a - b = 12 \hfill \\

\end{gathered} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,2a = 12\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a\,\,{\text{unique}} \Rightarrow \,\,\,\,\,\,{\text{SUF}}{\text{.}}\,\)

The above follows the notations and rationale taught in the GMATH method.

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Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)

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