CAT 2022 Slot 1 QA Q10 Inequalities & Modulus

Question 10 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

The largest real value of \(a\) for which the equation \(|x+a|+|x-1|=2\) has an infinite number of solutions for \(x\) is
  1. A
    -1
  2. B
    0
  3. C
    1
  4. D
    2

Answer: (C) 1 Inequalities & Modulus MCQ

Solution

  1. \(|x + a| + |x - 1|\) is the sum of distances from \(x\) to \(-a\) and to \(1\); its minimum value is \(|1 - (-a)| = |a + 1|\), attained for every \(x\) between the two points.
  2. The equation has infinitely many solutions exactly when this minimum equals 2: \(|a + 1| = 2\).
  3. So \(a = 1\) or \(a = -3\); the largest is \(a = 1\).

Same concept in other years

See every question on Sums of |x − a|: distance on the number line.

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