isi-entrance None Q9

isi-entrance · India · subjective_collection Simultaneous equations
Consider the system of equations $x + y = 2$, $ax + y = b$. Find conditions on $a$ and $b$ under which
  1. [(i)] the system has exactly one solution;
  2. [(ii)] the system has no solution;
  3. [(iii)] the system has more than one solution.
$\Delta = \begin{vmatrix} 1 & 1 \\ a & 1 \end{vmatrix} = 1-a$; $\Delta_1 = \begin{vmatrix} 2 & 1 \\ b & 1 \end{vmatrix} = 2-b$; $\Delta_2 = \begin{vmatrix} 1 & 2 \\ a & b \end{vmatrix} = b-2a$.
(i) For exactly one solution: $\Delta \neq 0$, i.e. $a \neq 1$.
(ii) For no solution: $\Delta = 0$ (i.e. $a = 1$) and $\Delta_1 \neq 0$ or $\Delta_2 \neq 0$, i.e. $a = 1$ and $b \neq 2$.
(iii) For more than one solution: $\Delta = \Delta_1 = \Delta_2 = 0$, i.e. $a = 1$ and $b = 2$. 
Consider the system of equations $x + y = 2$, $ax + y = b$. Find conditions on $a$ and $b$ under which
\begin{enumerate}
\item[(i)] the system has exactly one solution;
\item[(ii)] the system has no solution;
\item[(iii)] the system has more than one solution.
\end{enumerate}
Paper Questions
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