CAT 2022 Slot 1 QA Question & Solution
AlgebraMedium
Question
The largest real value of a for which the equation $\mid x + a \mid + \mid x - 1 \mid = 2$ has an infinite number of solutions for x is
Options
-1
0
1
2
Solution
In the question, it is given that the equation \(\lvert x + a \rvert + \lvert x - 1 \rvert = 2\) has an infinite number of solutions for any value of x. This is possible when x in \(|x+a|\) and x in \(|x-1|\) cancels out.
Case 1:
\(x + a < 0,\quad x - 1 \ge 0\)
\(-a - x + x - 1 = 2\)
\(a = -3\)
Case 2:
\(x + a \ge 0,\quad x - 1 < 0\)
\(x + a - x + 1 = 2\)
\(a = 1\)
Largest value of \(a\) is \(1\).
The answer is option C.